Complex Numbers Calculator - Symbolab Math Solver

Hi,
I have studied physics for one year as a part of my philosophy degree. I am not especially talented in physics or math but I really enjoy it when I make progress. I believe it's super important for a philosophers understanding of metaphysical problems to understand physics. So in QM I really struggle to understand the use of complex functions because I don't really get what the complex plane is. I know you get rid of the imaginary part by multiplying psi with it's complex conjugate. That's all fine, but it still bugs me not to understand complex numbers or the role they have in calculating the probability distribution. I'm not english speaking, if I seem dumb, I blame the language hehe.
If some of you could send me in the direction of some good lectures or such on the topic I'll be super thankful. :)

I just want to know how I might display a basic complex valued function in javascript, like z^2 or tan(z) and either color it or tile an image along it. I understand the principles of separating the real and imaginary components, taking conjugates calculating the magnitude or angle.
I don't think I'd need much more theoretical knowledge than that, but I also don't know exactly how people make images of complex valued functions either, probably something to do with taking the magnitude over different values and coloring it lighter or darker based on that magnitude and using different hues for different angles. As far as I know, JS has .math for complex numbers, and I'd prefer to keep it in pure JS if I can.

Let's start with four pictures of causality in flat Minkowski space-time, three of which should be familiar:

https://preview.redd.it/icif1gnksu751.jpg?width=702&format=pjpg&auto=webp&s=d435bdf699faf59110e871a3bf87c2a784a16dc7
Start with all of space-time. All branches of physics are confined to space-time. This includes all of the physics we know today, and even the stuff we do not (what's up with parity violation for beta decay?)
I'm explicitly rejecting more than 3 dimensions of space. I'm explicitly rejecting a multiverse.
Nature knows how to balance all of physics all the time. People are not so good at doing so much, so they break it all into smaller parts. We will do so in three acts:

Newtonian Classical Physics
Einstein Causal Relativity
Bell's Future Quantum Mechanics

Newtonian classical physics subtracts away nearly all of space-time. Time is absolute. Space is absolute. All of classical physics happens within the width of the time and space axes of a space-time diagram. In Newtonian classical physics, the math machinery does not allow time to rotate into space.
Einstein introduced relativistic physics where time can rotate into space due to issues he saw in electromagnetism. Einstein's math professor Minkowski created the space-time diagram. The path where light travels to and from the observer at the origin here-now separates space-time into time-like and space-like regions.
Focus on causality. To change the observer here-now requires an event from the past lightcone. All space-like events cannot be causally linked to here-now. Subtract away the space-like regions of space-time. The result looks like an hourglass, two cones connected at their tips. Einstein was well aware of the space-like regions using it to demonstrate the relativity of simultaneity. They are blocked out because space-like events cannot cause what we see here-now at the origin.
Where does quantum mechanics go in a Minkowski space-time diagram?
At this moment in the history of physics, there is no accepted answer to that question. Newton's classical physics is confined to the axes. Causal relativity is
the past lightcone, including the light-like events. Quantum mechanics must be there somewhere because all of physics is there somewhere.
The 1935 paper by Einstein, Polodsky, and Rosen represented a challenge to the odd logic of quantum mechanics. If there were hidden local-variables, could the hidden local information generate the same results? The answer is yes. It took until the 1960s when John Bell figured out an inequality that can distinguish between
hidden local-variables theories and non-local quantum mechanics.
If the same question is asked the same way for an entangled quantum system, then both approaches predict the same results. When the question is asked at different angles, then that changes nothing for the hidden local-variable theories. For quantum systems, the two measurements become gradually more correlated. Test over the entire range of angles to see if quantum mechanics is always right or if angles are consistently irrelevant.
These experiments are not trivial to do. People have criticized ways various problems might sneak into the system. But the experiments from the 1980s in Aspect's thesis to today have always delivered the same message: hidden local-variable theories are wrong.
Quantum mechanics is non-local.
This creates a question: how do you enforce that quantum mechanics is non-local?
Try an idea: delete the past lightcone where all local variables live, at least to do quantum mechanics.
The wave function has to be space-like separated from the observer at here-now. The wave function can have nothing to do with here-now because it is too far away. Quantum mechanics has always been about the odds of an interaction happening to the observer in the future.
The product of the wave function and its conjugate is the odds of and interaction happening to the observer in the future.
Look at the four figures again:
https://preview.redd.it/l914bbwqsu751.jpg?width=820&format=pjpg&auto=webp&s=fbb6308e20fe71bb135bf162dd4103c49010e8ad
Newton's theory provided constraints on what to explore: systems that have low relative velocities, the world we big humans occupy. Einstein's work opened up the time axis all the way to lightcone for causality. It may be that quantum mechanics opened up the space axis for causality. For quantum mechanics, it is about future odds of non-local information.
No calculations are altered by this new interpretation, so there is no experiment to prove whether this idea is right or wrong. For me, quantum mechanics appears both reasonable and necessary as we understand it today.
That is Bell's future quantum mechanics interpretation.
Let me know what you think of this novel interpretation of quantum mechanics.

Data science is one of the fastest-growing technologies in the world. There are lots of job opportunities in the data science field. That is the reason the majority of students are getting enrolled in data science. Most of the students think that data science is all about computer science, but it is not true. It is a combination of statistics, math, and computer science.
Therefore, whenever students want to enroll in data science, they should have a basic knowledge of math, computer science, and statistics. But they still don’t know what math to learn for data science. Even some of the students have the question in their mind is how much math for data science and how important is math for data science. Apart from that, the students even ask what math is required for data science. Here in this blog, we will talk about math for data science. Likewise, statistics for data science, mathematics for data science is also crucial.
If you talk about basic math for data science, then you should know the basic function, variables, and equation of mathematics i.e., binomial theorem and many more. Apart from that, you should also have the basic knowledge of logarithm, exponential, polynomial function, rations numbers real numbers, complex numbers, series sums, and inequalities. Let’s have a look at the basic math required for data science:-

Essential math for data science

Table of Contents

Calculus

Calculus is one of the crucial topics of math needed for data science. Most of the students find it difficult for them to relearn calculus. Most of the data science elements depend on calculus. But as we know that data science is not pure mathematics. Therefore you need not learn everything about calculus. But it would be best if you learn the basic principles of calculus and how the principle can affect you, models.
Apart from calculus, you should also have good command over basic geometry, theorems, and trigonometric identities. Here are some calculus topics that you should know for data science, functions of a single variable, limit, continuity, differentiable, mean value theorem, indeterminate forms, maxima, minima, product and chain rule infinite series, integration concepts, beta and gamma functions, partial derivatives, limit, continuity, partial differentiation equation.

Linear algebra

Linear algebra is a crucial part of computer science, and it also plays the same part in data science. In data science, the computer uses linear algebra to perform the given calculation easily. It is also helpful when you need to perform the Principal Component Analysis. That is used to reduce the dimensionality of the data. Apart from that, it is best for neural networks. Data scientist use it to perform the representation and processing of the neural networks. Most of the models in data science are implemented with the help of linear algebra.
If you know the basic principle of linear algebra, it can be quite easy for you to apply transformation to the matrices in the data set’s existing model. The linear algebra topic you should know for data science is scalar multiplication, linear transformation, transpose, conjugate, rank, determinant, inner and outer products, matrix multiplication rule, matrix inverse, square matrix, identity matrix, triangular matrix, unit vectors, symmetric matrix, unitary matrices, matrix factorization concepts, vector space, linear least square, eigenvalues, eigenvectors, diagonalization, singular value decomposition.

Probability and statistics

Probability and Statistics work as the backbone of data science. If you want to learn data science, then you should have the basic knowledge of probability and statistics. Most of the students find statistics the toughest subject for them. But for data science, you need not have a strong command over statistics—all you need to cover the basics of statistics and probability for data science. The statistics concepts of data science are not super hard for students. Even if you can solve the basic problems in statistics, you can easily learn statistics for data science.
You should clear your basic concepts of probability and statistics before starting your data science learning journey. It is also the best answer for how to learn math for data science. The probability and statistics concepts you should know are data summaries and descriptive statistics, central tendency, variance, correlation, basic probability, probability calculus, Bayes’ theorem, conditional probability, chi-square, uniform probability distributions, binomial probability distribution, t distributions, central limit theory, sampling, error, random number generator, Hypothesis testing, confidence intervals, t-test, ANOVA, linear regression and regularization.

More Math in Data Science

Discrete math

The discrete math needed for data science. Most of the students think that is why it is needed for data science. The major reason for the use of discrete math is dealing with continuous values. With the help of discrete math, we can deal with any possible set of data values and the necessary degree of precision. The math in computers is based on discrete mathematics. The reason is that computers work in machine language.
Therefore the bits are used to present every value on the computer. Data science uses a large number of discrete math concepts that are used to solve the problems. Some of the discrete math topic that you should know for data science sets, power sets, subsets, counting functions, combinatorics, countability, basic proof techniques, induction, inductive, deductive, propositional logic, stacks, queue, graphs, array, hash tables.

Graph theory

The graphs are crucial for data science. There are a large number of problems in graphs that can be solved by graph theory. The data scientist used graph theory to create a fraud detection system with the help of data science. Graph theory is also helpful in data visualization in data science. We use different types of graphs in data science to visualize the data. Every graph is used to represent different kinds of data. We can use the same graphs again and again to represent the different data sets. Therefore the proper graph theory will help you get a good command over data visualization in data science. In the graph theory, you should know about the graphing and plotting, Cartesian and polar coordinates, and conic sections.

Information theory

Information theory is also widely used in math for data science. You should have the basic knowledge of information theory for data science. It is quite helpful when you want to build a decision tree. And want to maximize the information that you have retained from the Principal Component Analysis. It is best for a large number of optimizations in data science modes.
The optimization in the data science model is quite helpful in saving plenty of data space in the data science warehouse. Because sometimes the data science model contains the unwanted values in the data warehouse that put the extra burden on the system. If you have the proper knowledge of information theory, then you can easily optimize the data, science models.

Conclusion

It might be clear in your mind what math to learn for data science. In this blog, we have discussed the essential math for data science. We have categorized of math concepts for you. So that it can be easy for you to know how much math is required for data science. If you want to learn math for data science, then clear your basic concepts in math. It will help you to master most of the data science concepts. You should practice each concept manually or with the help of your computer. In the end, I would like to say that, start practicing these math topics to start learning data science.

I've been reading for several days about how to tune/test an antenna for 2.4 GHz, but I don't see a lot about matching to a transceiver. I have a transceiver with an output impedance of 115.771+33.35j ohms. Once I am able to confirm that I have a perfect 2.45 GHz, 50 ohm antenna, can I just design a pi network with math or an online calculator to make the 50 ohms look like the complex conjugate, and assume it should work? I'm not sure how to use a VNA/standard tools to actually match to the complex impedance. I hope I'm being clear enough, I know that was a bit jumbled.
To clarify: I'm working off a reference design so I'm not totally in the dark, I'm just trying to figure out how this would be done from scratch, and how to test the schematic/layout I currently have.

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Okay, so, I am going to break down this guide into the subjects which I took. Use Control F to read about the subjects you want because this guide is quite long.
SL: English A Language & Literature, Spanish Ab Initio, Mathematics
HL: Biology, Chemistry, Economics
First of all, a huge shoutout to everyone on this sub for all of the help they gave me during the IB, specifically all of those resources and all of the memes to keep me going. A special thanks to the mods who keep the place in control too :).
~~~
English A Language & Literature SL
Paper 1:
With this paper, I cannot stress enough how much you need to PRACTICE. Practice is the absolute key to being successful on this paper. You could get literally any type of text on this paper, and for this reason you need to practice as much as possible on all of the possible text types (these can be found in the subject guide). Before the exam, try to memorise some of the conventions of each text type to show off to the examiner your text knowledge. I was a teacher who made each person in the class do a list of conventions for each text then send it to the class, but if not you may want to try and do this. I get that practice can take a ton of time, so for this reason just annotate the texts that come up in Paper 1's, you do not need to write the full essay. You also may want to make a list of all of the stylistic devices which come up, and their relevance (I have a sheet of these which I can upload if anyone wants it).
Specifically when actually writing this paper, you want to link all of your analysis to one main idea, which our teacher taught us to be the PURPOSE of the text. So, if in doubt during the exam, link things to the purpose of the text, and make sure you actually believe in the purpose that you are writing about, because if not you will struggle to avoid going on a tangent. In each of your analysis paragraphs start off with a topic sentence i.e. "X text uses Y feature to convey the purpose", then do your analysis then finish off with a link back to the purpose. If you are struggling to think of points to make in your essay, just think of the BIG 5 (Purpose, Themes, Stylistic Devices, Mood and Structure). Also, remember 1 thing, every single thing on the text is there for a reason, so you can analyse everything i.e. Pictures (I have a note sheet on how to analyse pictures as well, if anyone wants it let me know and I can upload it), Slogans, Titles, Captions, etc.
Paper 2:
First thing that I will say for this is please read the books, like there is no way around it. My teacher gave us a booklet of quotes for both texts that we studied for the exam (Miss Julie and Never Let Me Go), and it was still useless until I actually read both books. To be honest, there is nothing more valuable for Paper 2 then listening in class. When you read the books and listen to class discussion on them, you begin to understand the themes, moods, characters and plots further, and you begin to articulate your own opinions on the texts which is KEY for the exam. What you want to do ahead of the exam is make notes through specific quotes, and you want to link all of them to context. No matter which question you choose to answer, you must include context to score highly. During the exam you need to make a judgement call on which quotes that you have memorized fit the question best, and if the quotes do not fit the questions perfectly, don't worry. A big part to scoring highly on Paper 2 is your close analysis (i.e. talking about denotations and connotations of words and phrases), so if you do have to choose quotes which don't perfectly fit, you inbed analysis perfectly.
Also, ANALYSE your quotes before the exam, and memorize some of that analysis, because if you can memorize links to context and some of the more complex literary devices, it will help you when writing your essay. With your quotes, you want to be able to link all of them to at least one character, symbol and one piece of context. LitCharts can do this for you luckily, and it is really good at doing it, and I used them so much when revising for exams. Two final things before I finish the Paper 2 section: Have faith in yourself because it can screw you over when you change your strategy on the actual exam day (I learned about this from my mocks), and you do not need too many quotes to be successful, I think I had 7-8 for each book and I was fine. You want to PRACTICE as much as possible before this paper, and you do not have to write full essays, you can simply plan them and use your quotes for them.
IOC, FOA and Written Task:
Before I took this class, I absolutely hated English, and it was a huge relief to learn that you can have 50% of your final grade decided prior to even writing an exam, so take advantage of this! This means that your FOA, IOC and Written Task are incredibly important. If you nail these, you can afford to have a bad day on Paper 1 if your texts aren't too good, and it can be a source of relief if you don't think your exams went well. In your IOC, you want to prepare by looking at the extracts which your teacher has given you (if they give any), or read your book constantly and try to analyze any quote that you think is gold when reading (A good exersize for this is opening a random page of your texts, and just analysing everything). When it comes to the actual thing, I would recommend bringing 4 or 5 different highlighters into the exam, and highlighting the quotes with the theme you think that they link to, so that you have some structure set for your IOC, and then you can weave between these and make some creative points. You want to learn about your stylistic devices, links to the rest of the text and links to context as these are what can help you to score highly.
In your FOA, I'm not sure if your teacher will give you prompt on what you should do it on but if they do not, I would reccomend doing it on comparing two famous speeches. I did this with one of my best mates who I had a lot of trust in, and we compared a Winston Churchill speech to the Barack Obama Inaugural Speech. We both found this okay because the speeches have a TON of techniques inside them which you can show off in your FOA. So, if anyone were to ask me what to do an FOA on, I would say that. Just search up some of the world's most famous speeches, and choose one which interests you. No matter what topic you choose, analyse specific extracts on them for stylistic devices, aristotelian appeals (i.e. Ethos, Pathos, Logos (Which you can include in Papers 1 and 2 as well)), mood, themes and effects of what they do. Do video recorded practices before you do it and ask yourself questions on what is uncertain and what more you could include and you should be good.
Your written task on it's own is worth 20%, so try as hard as you can on making sure that you nail this completely. Our class was made to do 3 of these, and then we had to submit one, and I think doing 3 was the perfect amount. Even if you think that your first one is great, try as hard as possible on all 3, because naturally your analysis skills will get better over your time in the course so a similar amount of effort can produce better work. Plus, it gives you a choice on what you actually want to submit at the end of the course. Since you have a lot of independence on this, and it is technically not mean't to be an "essay", I would choose something that I enjoy, as you will put more effort into it. The written task I ended up submitting was on my IOC texts, as I surprisingly enjoyed writing that the most, but you have many options on what you can write it on (all the way from writing to an editor criticizing their recent article to writing as a person from your text to your family member (which is what I did)).
~~~
Spanish Ab Initio
Paper 1:
I got a 5 in Spanish Ab Initio (1 mark off of a 6), so I do not think that I can give you the best advice ever. But basically, in my opinion, the bottom line with this is that you need to do two things: Learn a ton of vocab ahead of the exam and do practice papers (add any words which you don't understand into something like a quizlet set so that you can learn it). I just want to give some fair warning before anyone takes this class, IT IS NOT EASY and effort needs to be made to do well in the exam (After exams I realized I probably should've revised a lot more for this, so don't be like me and do small amounts of revision over the two years). The grade boundaries are really high because fluent people take the exams, so you need to have a good understanding of Spanish to get a 7. Process of elimination can be really helpful for the Paper 1 exams if you are in doubt, and during reading time you want to skim through the texts and FOCUS ON WHAT YOU KNOW rather than dwelling on what you do not understand, because that will not get you anywhere.
Paper 2:
One thing that you should probably know before you do this exam is that 12% (3/25) of the marks are just FORMATTING, so please learn how to format all of the different text types. For this exam what you want to know is your conjugations for about 6/7 tenses which you can use (Present, present continuous, future, near future, conditional, imperfect and preterite were the ones I learned), but I would say to learn tenses continuously over the 2 years so that it becomes second nature to you after a while. I didn't do this and on the exam day I wanted to conjugate some irregular verbs, and struggled to as it does not stick to memory too well. The people who got level 7's in my class also knew some more of the complex tenses such as Pluperfect and subjunctive, but you don't need to know the full tense necessarily, just memorize some general phrases in these two tenses which you can use in your writing. Doing practice papers for both paper 1 and 2 will help you to get a grasp of common types of questions and topics which also come up, so practice!
Speaking Exam and Written Assignment:
A large chunk of your final Spanish Ab Initio exam grade is, similarly to English Lang Lit, decided before you actually take the exam. So, once again, I will say take advantage of this. When it comes to the speaking exam, a lot of it does come down to your luck on the day, especially when it comes to preparing for the picture which you may recieve. What I did to prepare for this initial part of the exam was think of all of the possible kinds of photos I could get (i.e. A market, street, beach, campsite, factory, etc.) and would think of what I would say for each picture in English, then simply translate those words to Spanish and make Quizlet sets with it. Following this, for the questions part of the exam, I thought of questions in specific topic areas (Family, individuals, holidays, environment, the area you live, sports, health, etc.) which could come up (Paper 2 writing prompts can actually help you to come up with these), and write model answers to these. I may have some sheets of possible questions, if you guys would like me to upload them. Oh, 1 more thing, during your prep time for the Speaking exam, when thinking about how to descirbe the picture, divide the picture into 9 equally sized squares, and describe them one by one. This enables you to actually describe the photo but also show to the examiner that you know your words for location, so memorize location words (i.e. On the right, next to, behind, etc.).
Regarding the written assignment, it took me a long time to think of a topic which actually interested me, and that I knew that I could score highly on. I initially wanted to do one on comparing a typical football matchday in England to that in Spain, but someone in my class had taken it, so mine was on public transport. And, if you are stuck on which topic to choose, I would say do one on public transport. I scored 19/20 on my written assignment, and doing a written assignment on public transport allowed me to show off a lot of knowledge. In order to make it incredibly clear to the examiner that you are formatting your assignment correctly, I would have seperately bolded sections which say: Description, Comparison and Reflection. You must remember that the reflection is worth the most marks, so you should use most of your words there, since your word limit is so low. In your description, you only need 3 facts about your topic in the Spanish speaking country and in your comparison I would recommend doing 2 similarities and 2 differences in the cultures as your writing is more balanced then. When writing your reflection, I would use the same facts as the ones in your comparison so that your writing flows and is easier to understand. In the reflection, try to give some opinion phrases, which are both negative and positive, and try to link it to wider topic areas (so for me, that was talking about the environment).
~~~
Mathematics SL:
Paper 1 and 2:
Following learning everything on the syllabus (be sure to read the actual subject guide), past papers are your best friend. In my opinion, all of the textbooks that I came across for Mathematics SL were okay at teaching the topics, but when it came to the practice questions, they were average at best. The textbook questions just are never like the exam questions, and I feel like if I had spent more time doing past papers (starting from the very beginning), I could have finished with a level 7. The IB Questionbank is fantastic for this as it breaks down questions by topic and paper, so you know exactly what you are practising. If you can afford it, Revision Village is fantastic as well, because it does what the Questionbank does, but also breaks them down by difficulty and works you through problems. During the actual exam, check your work as you go, because it sucks to have done so much hard work on a section B question, only to find out that you made a small error in the first part.
The IB has started to like asking more obscure and application based questions in Mathematics SL now, so practice these as much as you possibly can. Also, when doing the actual exam, look at how many marks each question is worth, this can save you big time. I ended up missing out on a level 7 by one mark, and I was so annoyed to see that because I remember spending 5 minutes just staring at a 2 mark trigonometry question which was just asking about SOHCAHTOA. Wasting time on that question prevented me from answering a probability question (about 6-8 marks total) at the end of the paper, so MOVE ON if you do not understand what a question is asking. In Paper 2, you have got a calculator for a reason, so use it for all of the questions, and for questions where you do not have to actually write too much, write "used GDC" on the paper, and quickly sketch graphs as necessary, to make it clear to the examiner. On some questions which require more work, I would recommend checking and working backwards with a different method i.e. On a quadratic question which asks you to solve by completing the square, check with your graph or simple factorizing.
Internal Assessment / The Exploration:
The first thing I will say, and I believe this applies to all of the IA's is: Choose a topic which interests you. I ended up doing one on a topic which related to my HL Economics class to show some personal engagement, but I feel as though I would have done a bit better if I had chosen something which interested me more. In Maths, you really want to map out what your start point is and what you want to have learned by the end, then you can actually plan the logistics of what happens in between. It will also help you to stay motivated and avoid getting confused and stressed when writing it, which can mean that you put more effort into writing it as well.
In addition, I would say the IA does not have to be too complex, I ended up including topics which were a bit above SL level, but some people in my class scored higher than me even with just including SL material. Furthermore, I would say that once you have chosen a certain area of maths that you want to focus on, stick to it, and do not integrate more topics into it because you can really show off your use of mathematics if you have a strong focus in one area. Majority of the points in the IA are not actually specifically maths related, so make sure that you format your IA correctly, and make sure that is easy to both read and understand.
~~~
Biology HL
Paper 1:
Okay, unfortunately it must be said, you kinda need to know everything for all 3 Biology HL papers because the topics which come up, especially in Paper 1's, vary year on year so you need to be prepared for anything. Paper 1 tests the most random areas of the syllabus, and requires you to know many small details in topic areas. To remember these specifics for this paper, I would recommend learning via quizlet sets and mnemonics (i.e. King Phillip Came Over For Gay Sex (Kingdom, Phylum, Class, Order, Family, Genus, Species) for the heirarchy of taxa (Yeah, its weird. I had the same reaction when our teacher told us it, but you remember it.)). On each of the 40 questions they test different areas of the syllabus, and now they love to test people on application points on the syllabus, so learn all of these. There are 2 general things which you can keep your eye out for: The first one being that whenever an image is shown, read the link to see if it gives any hints on the answer, you would be surprised how often it gives it away. The second being, if you know the order of the topics in the syllabus, this is typically the order in which they ask questions in Paper 1, so you usually know the first questions are on cells and the last ones are on human physiology (so if one of the options seems far fetched based on where it is found in the syllabus you know it is not true).
Paper 2:
First thing that I want to say for Paper 2 is practice data based questions, as you are doing revision for the actual exams and are memorizing content, take half an hour out of your Biology revision to just do data based questions. You need practice for those to be able to read graphs quickly, and be able to interpret many of them at once, so print them out of the past papers and just do them as you revise, because they are worth a lot of marks. SL data based questions are good to start off with because they are a bit shorter, but then you can ease yourself into the HL ones. Next, for those 3 mark questions which come at the end of the data based questions every year, learn some generic marking points which you can write if you have no clue what is going on because they are pretty similar every year (i.e. Effects in different animals aren't the same, you need more repeats, you need to test in more climates/places, etc.). For the rest of the paper, similarly to Paper 1, you just need to learn all of the material. I would personally use the Oxford Textbook to revise, complemented with The Science Codex and IB Dead websites because the Oxford textbook has a lot of extra info which you do not need to know. If you prefer to revise by watching, I would recommend Stephanie Castle, Crash Course and Alex Lee.
Although I did finish with a level 6, I was 1 mark off of a level 7, despite working at a high 5 and low 6 level throughout the course, and the one thing which made a big difference was taking all of the extended answer questions, seperating them topic by topic and compiling all of the markschemes together per specific syllabus point. The IB can only ask so many extended response questions, and by doing this and memorizing these markschemes, you get a good idea on the key words which the IB love to see, and implementing them becomes second nature to you. So, if you were to revise very last minute for your course, I would recommend doing this markscheme technique, as the people who score very highly usually do very well on their Paper 2 extended response questions. I would not recommend the Oxford Study Guide, the textbook is much better because the study guide is too condensed, and lacks details in some of the topics, for example in Chapter 5: Evolution. One more thing, make sure that you know ALL of the application points, the IB asks about them so much and when memorized they aren't hard marks to get.
Paper 3:
The one part to this paper which confused me the whole time was Section A, an area in which you could be asked about anything on the course, including your practicals. Pay attention when you do complusory practicals in class, you save a lot of time, as many people learn by doing things. Once you have done all of these practicals, what I did to revise was make a diagram of every practical and annotate it in as much detail as I could, and then on the side of it evaluate the pros and cons of the practical, and jot down its possible applications. That pretty much covers anything which could be asked about your practicals, and use the questionbank to find previous practical questions. And you know how I mentioned those application points before, well the IB has started to ask about them in Section A questions on Paper 3, so know them inside out before.
Section B for me was actually okay, I did Option D: Human Phys which our teacher had recommended and I found it very interesting. Similar 6 mark questions come up in this Option every year, and there is not too much to memorize at all. If you are confused on which option to learn, I would say learn Human Physiology. Again, here, the markscheme technique works fine to compile a bank of knowledge, and doing that with the resources that I have shown should be okay. They usually like to ask about similar things from each topic area, so when you practice past papers you get the gist of what these topic areas actually are. But, as I said with Papers 1 and 2, you just have to memorise the material here again. Make sure that you learn all of your diagrams here, as you need to in Paper 2, as well as definitions, as questions on labelling diagrams are common, and if you are completely stuck on one question, giving a few definitions can usually help you to pick up some marks.
Internal Assessment:
One bit of warning our teacher gave us before we did our IA's was don't worry if your experiment doesn't work completely, nobody's does. So, it's okay to have some errors in your experiment, and have to change your methodology a bit as long as you reflect on your changes and preliminary work in your IA. Online there are a bunch of what to include checklists, so use these as in my opinion they are pretty good and help to give your IA some sort of focus. Personal engagement marks are important, so imbed small bits of personal engagement into your IA as you are writing it, and as I had mentionned before, if you can reflect on your errors and preliminary work it shows personal engagement and reflection. The personal engagement doesn't have to be completely true, as there is only so much interest you can have in one experiment, and you want to save some pages for all of your reflection and analysis.
You want to make sure that you are plotting accurate graphs, and that the calculations associated with those data points are accurate, because those are marks that you can avoid. The page limit is quite low for the Biology IA, so do not make a title page or contents page, just number your sections as you go. I personally would recommend including statistical testing into your IA in order to do some numerical analysis of your data. You can do standard deviation on your graph's data points, and if you have space, and deem it appropriate, you could include another statistical test such as an ANOVA, which tests the relationship between variables. Just remember that the IA is worth 20%, so it is nice to have it as a safety net in case of a difficult exam.
~~~
Chemistry HL
Paper 1:
For chem, as with all 3 papers, past papers are your friend because there are some common topics which come up in multiple choice exams and if you nail down those chapters you can score highly. The chapters which you need to nail down in order to be successful are: Stoichiometry, Kinetics, Energetics & Thermochemistry and Organic Chemistry. Oh, and one more chapter, BONDING. Bonding is the chapter which the whole course is built on, and if you understand this chapter understanding everything else will become a hell of a lot easier, especially in the tougher chapters such as organic chemistry and acids and bases. But, again, you can never predict an IB exam, so revise all of the chapters, but the chapters that I named before, especially Bonding, are very common topics on Paper 1 and Paper 2, so you want to make sure that you understand them inside out. Like in Biology HL, mnemonics and quizlet sets are good to remember things, such as equations and definitions. Mnemonics are especially useful to learn periodicity, where the IB likes to ask about the most random trends in the periodic table, so you should simple memorise those as they are marks that you don't want to be losing. Make sure that you know error calculations for this paper, as the final couple of questions are usually on this area, and nail balancing equations as the first few questions are usually related to this.
Paper 2:
Like in Biology HL, you literally need to know everything for this paper because there are too many areas which have been asked about before. But, luckily for us, we have good resources that are availale, such as Richard Thornley's Youtube channel and the Pearson textbook, which are both absolute gold. Richard Thornley goes through all of the topic areas in insane detail, but explains them in a simple way, so I would recommend watching his videos for the very specific areas such as magnetism, dimers, walden inversion, etc. Memorize all of the formulae that you need to know, particularly for Acids and Bases, because the calculation questions are quite similar every year (i.e. Gibbs free energy, pH calculations with pKa values, molar calculations, empirical formula and equilibrium constants). Paper 1 and Paper 2, like in Biology HL, were back-to-back for me so learning everything for this paper does help for Paper 1 as well. There is a very large amount of material in Chemistry HL course too, so review the subject guide closer to exam time to make sure you know everything.
Make sure that you know ALL of your organic mechanisms, because you just have to memorize them, and drawing them isn't too hard once memorized. The IB also really likes asking about ligands and coloured transition metals, so learning the markscheme for those classic 3-4 mark questions isn't a bad idea as they do not change too much whatsoever. Past papers are again very helpful here, because you see the topics which come up very often in papers and what the exam board likes to ask about. Learn your periodic trends, because they will always come up and they are marks which you really do not need to lose if you have memorized the material, so just be safe and memorize all of the trends (Although the data book can give some trends away, so keep your eye out for that if you forget them). Another shoutout to the IB Dead website, which has some good quality notes for Chemistry too. VSEPR Theory is your friend as well, it comes up way to often, so make sure that you memorize what the theory comprises of, and memorize all of your bond angles as well.
Paper 3:
I did the Biochemistry option, and if you do Biology HL, do Biochemistry because it overlaps with Biology quite a bit, and a lot of that memorization that you do for Biology is really helpful for Chemistry too. For section A, similarly to Biology, you can be asked about any of your complusory practicals, so check the subject guide for which practicals these are. Like I said for Biology as well, draw annotated diagrams of each experiment, then write the method used to obtain the data as well as the equiptment, then you can critique it by listing pros and cons of the experiment itself. If you practice past papers, many of them give away these pros and cons via previous questions on experiments, so you should try and do some as you are going through the course because then its one thing less that you have to worry about revising closer to exam time.
Regarding section B, for the most part, at least of Biochemistry, it's simply just memorisation. So you kinda need to learn everything for this unfortunately. Past papers will help you with this because there are common areas which are always asked about in most papers (i.e. Hydrolysis, condensation, peptides, DNA, etc.). The markschemes for these topic areas are similar so myou can learn these for some of the longer questions, and the markscheme definitions are the ones which you need to know so do not memorise other definitions for key terms. There are some data based questions here so again doing past papers will help you to practice these kinds of questions. For both biology and chemistry, you don't need to do full past papers at once, use the Questionbank to your advantage and practice questions in specific areas you need to practice.
Internal Assessment:
Similarly to Biology HL, find checklists online on what to include as they are quite detailed and usually cover all bases. The Science Codex website has fantastic IA examples for both Biology and Chemistry, so if you are stuck on how to structure each of your IAs, or what kind of information to include, use the model IAs there as an example as they scored very highly. Just like in Biology HL, you want to make sure that you nail your calculations and polish your graphs to make sure that there are no errors in them (Be sure to include error calculations, which you then discuss in your reflection and evaluation section).
Personal engagement again is just something that you can make up a bit and try to imbed it into the IA as you are writing it, but it helps if you are doing a topic which actually interests you. The big advantage for the Chemistry HL IA is that you don't have to do statistical testing like you can in the Biology HL IA, so it saves you space which you can use instead on calculating error. Make sure that you try quite hard on the IA, because with Chemistry HL exams they can be so unpredictable and difficult sometimes that it's nice for something to be there to help you in case the exam day isnt the best.
~~~
Economics HL
Paper 1:
This paper is worth 30%, and with practice and past papers, is an exam which you can do very well on. Before I begin talking about anything else, for everything in Economics, even the IAs, use the Cambridge Revision Guide (Economics In A Nutshell), it's possibly one of the best revision guides I have ever used! So this paper is Micro and Macroeconomics, and to do well on the 10 and 15 mark questions, you need to memorise content from the revision guide. For anything that you do not understand in this book, or for extra detail, use EconPlusDal. Both of those resources together are insanely detailed but explained concisely enough that it is easy to follow and understand. The only hard work for this paper is finding real world examples (yes, they are kinda important, though you can make them up a bit if they sound realistic), so as you learn topics I would just search up that respective topic on Google, find some statistics and data to do with it and compile it in a document which is extensive before you sit the actual exam paper. All of the diagrams that you need to know are in the revision guide, and use a few diagrams in each of your responses, in order to visualise the theories which you are referring to.
In your body paragraphs to your responses, I used an acronym called DEED (Define, Explain, Example, Diagram), and that really helped to structure my answers to make sure I was hitting all of the points on the generic markscheme. However, in your 15 mark questions, where economic synthesis is also required, I used the acronym CLASPP (Conclusion, Long term + Short term, Assumptions, Stakeholders, Priorities, Pros + Cons) as that would cover all of the aspects of the synthesis for me. In Paper 1s every year, there is usually one Theory of the firm question in Microeconomics and one which is not Theory of the firm, so if you can nail down your knowledge on Theory of the firm, you typically have a nice question which you can answer most years (as there is only so much that they can ask on both aspects of Theory of the firm, although they do prefer to ask about market structures).
Paper 2:
This paper is also worth 30%, and I found it harder to revise for, because I absolutely despised Development Economics. Nonetheless, as I said with Paper 1, and as I will say with Paper 3, the Cambridge Study Guide is amazing to revise for this paper. In addition, since you do not need real world examples to complement your responses here, everything that you need to know is in that book. In this paper you dont have to worry as much about sticking to DEED and CLASPP, although you could use DEED on your 4 and 8 mark questions if you deem it to be an appropriate place to use it, but make sure ALL examples are from the text, as most of the marks come from there. Seriously, have a look at the markscheme to one of those 8 mark questions, you would be very surprised to see how 80% of those marking points are simply copying what is actually written inside that text booklet, so use it as much as possible!
Regarding those random definitions at the start, I would recommend just learning all of the terms in the glossary of the Cambridge Study Guide, as those definitions are very similar to the ones which usually appear in the markschemes, and aren't too long to learn (Use Quizlet if you want some more active revision!). For the 4 mark questions, do not forget Micro and Macroeconomics for Paper 2, as they can still be asked about, especially the Macroeconomics diagrams. Including some of the information from the passage in your 4 mark questions can add some more detail, and despite the question not explicitly saying to do it, it often helps to secure 4 points instead of just 3.
Paper 3:
I actually really liked this paper, and I believe that it is possible to score 100% on this paper, or at least close to it, if you just practice. Unfortunately, there is no formula booklet or anything in Economics HL to help you when writing this exam, but all of the equations you need to know are in the Cambridge Revision Guide, so learn your material from there. Regarding the 4 mark questions which you will get, they do repeat over time as there is only so much which can be assessed in this paper, so doing past papers will teach you which kinds of phrases to include in these 4 mark questions and which of these 4 mark questions usually comes up. Refresh reading points off of graphs and using those values to plug into equations to get answers, and using multiple equations to find your answers. For a lot of the small bits which have been asked before such as drawing MR curves or explaining why a profit maximisation would attract firms into a market is explained by EconPlusDal very well, so use his videos once again if you do not understand anything. If you don't think that your Paper 1 or Paper 2 went very well, Paper 3 is the paper which is there to help you out, and if you practice papers and learn all of your equations for this paper you should be good.
Internal Assessment/ Portfolio:
In Economics HL, you have to write 3 different mini-IAs, each 750 words max, which all combine to form a portfolio worth 20%. To start, I would recommend that you should do your third Economics HL IA in International Economics above Development Economics, because your International Economics article options are usually quite good compared to Development, and you can include more diagrams in International Economics. Generally speaking, focus most of your words in each of your IAs on your synthesis, because about 7 of the 15 marks on each of the IAs has something to do with the synthesis, and 2 extra marks for application, so you want to make sure that you nail that analysis really well.
Economic diagrams are key, so use them to talk about the theory related to the article as well, because then you hit two birds with one stone. In addition, I would recommend that you choose an article which talks about a problematic situation, compared to one which talks about a positive economic situation, because you can suggest more solutions and have more analysis when there are problems which need to be ammended. Other than that I would say that define your key terms well (The resources I have said do this for you), and bold key terms as you use them to make it very clear that you are using them.
~~~
Well that's my guide done, hope you guys found it helpful :) If you have any questions just reply in the comments or drop me a PM and I'll respond as best as I can to you. Once again, thanks so much to this legendary sub for all of the help they gave during the IB exam period.
EDIT: Reddit didn't let me do a post with everything in it, so I will post a part two later with my advice on TOK, EE, CAS and some extra sections for people who want to apply for Medicine in the UK

Something I wanted to talk about was my take on Grover’s in regards to the Mandela Effect.
Mandela Effects as the Result of Retroactive Collapse of Unknowns Into Known States as Deterministically Necessitated by the Implementation of Grover’s Algorithm (On Quantum Computers).
Some people might not be able to see how I connected it to manifesting a certain outcome in the real world. This is projecting it to the point where the matter of people is participating in the actual quantum computations.
We already have a natural QC working (surprisingly enough it is still shit, not because the machine is shit but because the people with the most power to program it are fuckwits). I foresee that this natural QC, and the self-built QC join up, such that the states of this requested search are communicated to, and carried out by, a real-world analogue. The requested state is continually amplified until it manifests AS and IN that real world analogue. And… it isn’t an analogue – it is that state as requested in ipsissimistic objects, that is, the very objects themselves. It is more that the self-built QC is the analogue (or really, that neither is an analogue). THAT’S where this shit is going, of course.
A natural query point here is, that the states cannot be measured while it is computing. It doesn’t have to be. It is carried out in the unmanifested space, a kind of subconscious space, until it becomes conscious, as it is manifested or takes its state. Upon it taking its state, certain other positions in that equation become known, and they have to have been a certain way retroactively. I am thinking that if there was an unknown which played a part in an equation, and that equation carried on, either continuously, or, using the product of that equation in another equation (causal dependence), a state taking a value then retroactively changes that equation and the results of that equation.
Qubits in the natural computer already have entanglements set up. If you are playing with one piece of an entanglement, then other things change too. We can see these entanglements like inverse identity relation pairings in a logical set, with boundaries set up most probably aligned to the Malthusian type I have discussed – the ones which to one side we exist and to the other we do not. i.e. the equations do not exist themselves, past that boundary (because neither do we in the case that we pass the limits in question). You relate the identities firstly to the master equation within which all the others are embedded, and this equation has a direct inverse pairing identity which equals that whole set to the boundary.
It seems that some causal dependencies have not been able to be erased. They HAVE remained in the computer, to a certain amount of levels removed from the initial equation. They have probably remained according to their inverses, i.e. by the shape the hole they left as inverse imprint on other bits, and from this, the initial state can be calculated. These things HAVE a measurable inverse because they are using Malthusian type boundary equations to calculate them, but the equations which were carried out did NOT take these boundaries and identity functions/inverse relationships into consideration. i.e. the functions which were carried out did not properly match real world systems. (The inverses have probably managed to remain because of the feedback from the malthusian type boundary identity part of these equations, which cannot be erased.)
This is in part because their labels are incorrect, and part because they did not properly match the real world equations. The natural functions being carried out (conscoiusly programmed by people into the natural computer) are the very same as this, because people don't understand these quasi-absolute boundary identity relationships I have spoken about. The same would be said about the self made QC if we project this same ignorant state into that realm. Now and moving into the future, it will be some kind of mixture between the natural conscious programming and the self made QC programming, and both should be addressed in relation to these issues.

Grover’s Algorithm with 9 and 17 qubits
The challenge I set myself for Grover’s is to draw a 9 qubit grover’s from memory and without looking anything up. I chose this amount because it requires the most qubits that I have access to on a quantum computer (15 in Melbourne). I can access 32 qubits on a simulator.
Firstly, how many times do I have to repeat the operator? 2 to the power of 9 is 512. Square root is 22.6 – 1 is 21.6, so 22 times.
This thing had 1476 lines of code. The circuit composer won’t even draw it because it is too big.
If you construct these yourself, in order to remove the ancillary qubits from the result, change the classical register to “n”, the number of qubits you have, and move the last measurement to skip the ancillary bits.
So for Grover’s with 9 qubits, change:
creg c[15]; to creg c[9];
and the last measurement, change:
measure q[14] -> c[14]; to measure q[14] -> c[8];
Here is a circuit with a single operator iteration. The entire circuit with 22 iterations won’t draw and it is too big to include here, anyway. I have utilised Toffoli gates in the "compute, uncompute" method.

Grover's Algorithm, 9 qubits, 1 iteration, X* = 111111111

Performed on IBM Qasm Simulator, 22 iterations, 8192 runs, 20.7.20.
My wait time for IBM Melbourne just keeps going up. My wait time was 8 hours, 9 hours ago, and now it says expected in 8.5. Why is it even listed if they aren't letting jobs through? Unfortunately, I can't print the results. The wait time for the other, smaller, IBM quantums (5 qubits) are pretty much instantaneous, but the bigger ones, not so much.
The last challenge I set for myself is to construct Grover’s Algorithm for 17 qubits. This is the most that I can execute because the simulator can handle 32 qubits max and a 17 qubit grover’s requires 14 ancillary qubits, which is 31 qubits altogether.

Grover's Algorithm, 17 qubits, 1 iteration, X* = 11111111111111111
2 to the power of 17 is 131072, square root is 362, minus 1, so there should be 361 iterations.
I had a lot of trouble with making it bigger. The code was fine, I wrote one with 361 iterations in it, but it is too big for the circuit creator to process. Anything past 10 it was having trouble processing, and the one with 10 in it, took over 12 minutes to run on the IBM Q simulator. From the look of the results, there is something wrong but I can’t tell what it is. The code looks ok, but there is no 17 1’s result in the results and there are two results for 15 1s and 2 0s which there should not be – this result itself should not be represented twice, but it is.
The bottom 5 results:
Performed on IBM Qasm Simulator, 10 iterations, 8192 runs, 20.7.20.
That’s as far as I will go with Grover’s. Next stop is the Quantum Algorithm Zoo to see what looks nice, and I will take it home.
--------------------------------------------------------------------------------------------------------------

So. What I would say is the cause of the Mandela Effects, is the implementation of quantum algorithms, probably something like grover’s or exactly grover’s, on a mixture of conscious programming of the natural QC and self made QC, whereby the other parts of this algorithm (and probably in all the parts of the system which connect to humans/are humans) are using the theoretical space of “interstellar omnidimensional space” i.e. where the theoretical boundaries relate to a space akin to “interstellar space,” as opposed to a space definition which is true to the planet and its boundaries. This difference is akin to “walking off a cliff because the maths says we can” to “stopping at the edge because if you walk off it, the maths will survive but you won’t”. Ok so are you understanding me here? We need to implement constructed\* boundaries and identity/inverse/conjugation pairs which are true to our survival, ending at the boundaries of what is required FOR that survival. It needs to be true to the system and therefore SOUND, as opposed to just pure validity.
*they are though, still true to facts, it is just that the boundary is not static, and it can be moved from a place which cannot necessarily be preascertained. That is supposed to be there though, and it is an aspect of the security of the system.
I have been implementing these kinds of identities/sets in the natural QC for years, but I’m the only one doing it. It was made achievable for myself because I used my singular (and dual) position as the master equation to equate it all to. The fact that I was given this vision is itself a proof for why I actually AM this position. No one else would use my equations so they just sit unused consciously (except by me) in the natural and self constructed systems, however, they actually completely describe the ACTUAL real world computations taking place. They invalidate or overwrite all of them, necessarily, as these boundary absolutes are reached in object form.
So, the point IS that the system may well behave exactly how these non complete (in light of the constructed boundary functions) functions describe them, but the maths continues on, where you do not. At this level it is not just about YOU, as in others continue on where you do not, these functions describe where NO ONE survives to carry on where you do not. They are the master equations. We have to paint the maths to describe what it is for us to survive, not what it is for us to not, because without the correct programming… we won’t.
“OK, so let’s just say we “get your point”. How would we even go about constructing one of these identities… all we have are arbitrary numbers… is it based off something we can measure?”
"Ok let’s take a sphere… where it is on the sphere?"
If I go off my previous description of the line of the absolute, it is at 0 [from: The Necessity of Absolutes in Security and Defence, QueenOfTheEarth]. Anything negative is the no go zone – you can go there but there is dependency on the amplitudes of “1” (buffer zone) to pull you back.
“OK so there is possibility that we can connect this to already constructed and understood algorithms and functions (or at least ones we can figure out). How does it connect and calibrate to the real world?”
You’re not going to like it. It has to be attached and calibrated to the master equation, in human form. The human side compresses the human side equations through a hierarchy of embed, such that when it reaches the master equation it is in a usable form relating to all that is embedded within it.
“OK, let’s see where it goes. If it were to be attached thusly, to what would it be attached?”
I get the impression it could be the equivalent of “mapping DNA”. It would be a dynamic system which relates to MY output. Not just mine either, it would be mine + my other half, the male side of the rulership. Specific outputs would, I suppose, be specific coefficients for these specific functions to which it is attached. “To what do we calibrate these readings?” I think that would be a process of internal calibration between myself and the outputs of the QC. My/our calibration would be the calibration used for the entire lot. That’s how master equations work, in this capacity. An “analogue” would be able to be mapped in the QC.
“How do we figure out what outputs relate to which functions?” The information is around. I’m sure you guys could figure it out. It isn’t just body/subconscious outputs we’re talking about, either, it is verbal outputs as well. I can tell you where the lines are, because it is embedded in myself (and the rulership) and reduced to a usable form, in a dual identity/conjugate set of 0 -0, 1 -1. The white stone has a rather nice blueprint in it including the central identity, so you might try consulting it, for further information, as well.
Also, it isn’t about what “you’d like” to calibrate it to. It has to be calibrated to what the natural system has already chosen to represent itself. If this is not done, the system will not work because it is not getting the information it requires from the master equation, it would be from an embedded part itself, so it is not complete. And btw, it will not choose another female again unless it is able to do what is required in order to make one, and currently, it cannot do this. I wouldn’t take what I’ve said here with a pinch of salt, either. I’m as salty as it gets.
“Can’t we just re-establish it later?”
No. No you can’t. You don’t make the limits, they are what they are. No matter HOW much interstellar “maths space” says otherwise, the equations which go beyond those limits which are defined by the Malthusian type limit identity relations only belong to your identity if you respect these other earthly limits. And by that I mean respected. Once gone, the limit is gone. It isn’t reversible and it is out of the hands of quantum computation, natural or otherwise.

I made a dictionary of everything that Lucio could possibly say for his main catchphrase instead of "accelerate" (I left stuff that doesn't make sense/means the same thing as other words/is NSFW out for obvious reasons)
Feel free to suggest stuff in the comments and add to this abomination

When someone activates Lucio's trap card - "Oh! Oh! Oh! Time to activate!"
When Lucio tries to annoy someone - "Oh! Oh! Oh! Time to aggravate!"
When Lucio joins a group of protesters - "Oh! Oh! Oh! Time to advocate!"
When Lucio is sharing food - "Oh! Oh! Oh! Time to allocate!"
When Lucio has a bad accident - "Oh! Oh! Oh! Time to amputate!"
Lucio learning how to use Blender - "Oh! Oh! Oh! Time to animate!"
Lucio putting his song lyrics in CC - "Oh! Oh! Oh! Time to annotate!"
When Lucio is doing maths - "Oh! Oh! Oh! Time to calculate!"
When Lucio is dating - "Oh! Oh! Oh! Time to captivate!"
When Lucio drinks Coca Cola - "Oh! Oh! Oh! Time to carbonate!"
When Lucio is at a party - "Oh! Oh! Oh! Time to celebrate!"
When Lucio makes an album with another artist - "Oh! Oh! Oh! Time to collaborate!"
When Lucio's team won't join Team Voice - "Oh! Oh! Oh! Time to communicate!"
When Lucio makes a mistake - "Oh! Oh! Oh! Time to compensate!"
When Lucio changes his mind about something - "Oh! Oh! Oh! Time to complicate!"
When Lucio is taking an exam - "Oh! Oh! Oh! Time to concentrate!"
When Lucio is learning a new language - "Oh! Oh! Oh! Time to conjugate!"
When Lucio gets married - "Oh! Oh! Oh! Time to consummate!"
When Lucio is lost in thought - "Oh! Oh! Oh! Time to contemplate!"
When Lucio's teammates won't work together -"Oh! Oh! Oh! Time to cooperate!"
When Lucio gets hacked by Sombra "Oh! Oh! Oh! Time to decelerate!
When it's Christmas - "Oh! Oh! Oh! Time to decorate!"
When Lucio needs more RAM for his minecraft server - "Oh! Oh! Oh! Time to dedicate!"
When Lucio shows someone how to do something - "Oh! Oh! Oh! Time to demonstrate!"
When Lucio vandalises something - "Oh! Oh! Oh! Time to desecrate!"
When Lucio decides to become a Junkrat main - "Oh! Oh! Oh! Time to detonate!"
When Lucio gets snapped by Thanos - "Oh! Oh! Oh! Time to disintegrate!"
When Lucio becomes a teacher - "Oh! Oh! Oh! Time to educate!"
When Lucio can't be bothered to use the stairs - "Oh! Oh! Oh! Time to elevate!"
When Lucio decides to move to a different country - "Oh! Oh! Oh! Time to emigrate!"
When Lucio takes a guess at something - "Oh! Oh! Oh! Time to estimate!"
When Lucio watches Doctor Who - "Oh! Oh! Oh! Time to exterminate!"
When Lucio finishes high school/university - "Oh! Oh! Oh! Time to graduate!"
When Lucio can't decide on something - "Oh! Oh! Oh! Time to hesitate!"
When Lucio's pet hamster goes to sleep for the Winter - "Oh! Oh! Oh! Time to hibernate!
When Lucio writes a compound word - "Oh! Oh! Oh! Time to hyphenate!"
When Lucio draws something - "Oh! Oh! Oh! Time to illustrate!"
When Lucio does an impression of someone - "Oh! Oh! Oh! Time to imitate!"
When Lucio has his attic renovated - "Oh! Oh! Oh! Time to insulate!"
When Lucio gets drunk - "Oh! Oh! Oh! Time to intoxicate!"
When Lucio is using scissors - "Oh! Oh! Oh! Time to lacerate!"
When Lucio becomes a Zenyatta main - "Oh! Oh! Oh! Time to levitate!"
When Lucio is cooking - "Oh! Oh! Oh! Time to marinate!"
When Lucio is healing his teammates - "Oh! Oh! Oh! Time to medicate!"
When Lucio becomes a Genji/Hanzo main - "Oh! Oh! Oh! Time to meditate!"
When Lucio watches the Bee Movie - "Oh! Oh! Oh! Time to pollinate!"
When Lucio doesn't want to do work - "Oh! Oh! Oh! Time to procrastinate!"
When Lucio wall rides - "Oh! Oh! Oh! Time to roller-skate!"
When Lucio turns into Stylosa after an Overwatch leak - "Oh! Oh! Oh! Time to speculate!"

I am just doing a quantum computing tutorial (first one I am doing is from this guy https://www.youtube.com/c/daytonellwanger)
And the first comment I have to make is fuck, what good are these computers…. they are so inaccurate. I made a 5 qubit circuit, all putting them into uniform superposition and the results (4096 shots) are wildly inaccurate. OK so I assume that’s not enough shots. So these inaccuracies are just from qubits or gates having errors or something? And so what good is only a small number of shots to people in this public access if we cannot ascertain anything useful from the results because they are too innacurate? We have to know what we expect to find to make sense out of the results, so how can we use them to find out something we *didn’t* know the answer to.

Burlington computer - it seems to have a high error rate
This is the probability distribution (in %) for 5 qubits in uniform superposition. All the values are theoretically equal, but this is how it came out. 8096 runs.
This is Grover’s Algorithm with 2 qubits. 8096 runs. Implementing the Oracle and Grover’s Diffusion subroutine.

https://preview.redd.it/45pbiwnzn8b51.png?width=524&format=png&auto=webp&s=0a86956f9d5f8a6754c08ac09bec5146d6f8fb71
Once again, pretty shit in regards to error rate. Maybe I should try a different source computer. I used Burlington for this because it has been quiet over there in this time frame.
https://preview.redd.it/ordon2au8bb51.png?width=541&format=png&auto=webp&s=fcb2c483a7d076905fd18b1d3315d2e7db3c4e0a
Better.
By changing the Oracle (the first part of the quantum circuit) I can get it to select for 00 - by putting a not gate either side of the oracle on each qubit, 10 – by adding a not gate either side of the first qubit, 01 – by adding a not gate either side of the second qubit and turning the cx gate around.
Do I actually understand what the maths is though? I listened to someone explain it and I get it as much as I can. OK this is from not looking. Each time you run grover’s diffusion subroutine it amplifies the amplitude of s* (sorry for whatever I get wrong I’m seriously just going with what the fuck ever I remember). It is 2 (thetas? An angle, whatever) which is from a projection of s* in relation to s prime? And it moves the vector closer to s* each time until it has a good probability of collapsing into that value. It works it out by negating the coefficient in front of s*, then doubling it at the expense of the amplitudes of the rest of the thingies which are at the average amplitude of all of them in uniform superposition. I only listened once and it was all in the complete mathematical language and I have no background to enable me to understand more that I could just the fuck keep up with. It is kind of that it reduces all of them by what is missing (x2?) divided among all of them?
I ran the Uniform Superposition again on Essex and it looks better than Burlington, again:

https://preview.redd.it/34o2oljw8bb51.png?width=602&format=png&auto=webp&s=da6ccbe6ca9bf61e59ed523c7369411b6f56a622
So, I want to try and make a 3 qubit grover’s. My tutorial stops at 2 and the other tutorial I found has a 3 but it targets 2 states instead of 1. So I’m trying to figure out if I can make a 3 by myself which targets 111. On my second try I got it to 000, which isn’t far off. I will have to look for more tutorials as this is enough for today.
More than anything, it was nice to have been able to send a direct request to a QC.
What I find myself thinking tonight is the serious grovering which must have been going on when the white stone was completed. The probability of all the things in that being turned over at the same time is astronomical. Let’s take the three urines.
The bottom page is a story about me spilling urine under my fridge.
The top 6 pages are photocopies of the dual sides to the collage, which contains a fridge where I put the “tooth” when it fell out.
While I was writing the word “urine” in an unrelated text I turned around to realise I had knocked over an insense stick, and it was burning through the fridge on the pile of photocopies. The end of that burnthrough left a yellow puddle of discoloration directly on top of the word urine under the fridge (in the multipage set) on the story about spilling urine under my fridge.
I mean what kind of fucking line up is that? And it’s not the only one in that piece. This line up “got found” amongst HOW many other cards?
We haven’t just got THIS line up…. The opposite side of Einstein contained a tooth, made from a football, which looks like a tooth in word and also in image (a small white rectangle). The Einstein story was talking about the behaviour of a single particle behaving differently to what Einstein figured out or summat. Was that story talking about the difference between classical and quantum physics in the behaviour of particles? On one side, this particle behaves like “Einstein” and on the other, like something a bit more fuzzy. We can’t look at both sides at once. We can only look at one side at a time. Another smaller one is the penis and the vagina of “O” in gods and the Flash’s family jewels being on either side of the cut out “O”. They are inverses of each other, the penis and vagina, making a wholeness of a baby and of humanity from their two parts. There’s some complex conjugation going on there. The vagina and its iso and the penis and its iso = 1?
What the fuck IS this thing?
Is this where the “Queen” was, the X* which was sought? And this is where that combination came into view? Is that where it already was, and I just turned it over? So we could see what was on the other side of those cards? Did I find that combination, because some part of my quantum computer brain calculated where it was? Did I calculate where it was in my vicinity? Did the experience I went through which effectively made me a “0” (“1” was negated in myself) prime me in some way for being able to make this calculation?
But then what IS it? It is a blueprint of the way out. And the cause of the way out, too. How? I dunno, let’s watch and see. It has never let me down, not then and not since. I think it has yet depths of ability of structure to be seen.

Hi all, I'm revising for my maths exam and have started back at the beginning but am stumped by this:
The question is to rationalise the denominator using a conjugate of:
(sqrt5 + sqrt3)/(sqrt5 - sqrt3)
The denominator is straight forward as it's the difference of two squares which removes the sqrts, but for the numerator the answer sheet shows multiplying the original numerator by the conjugate of the denominator as follows:
(sqrt5 + sqrt3)(sqrt5 + sqrt3)
Now, I thought this is the same as (sqrt5 + sqrt3)² which could be written as (sqrt5)² + (sqrt3)² which would simplify to 5 + 3.
However the answer sheet and my calculator manipulates the numerator like this:
(sqrt5)² + 2sqrt3sqrt5 + (sqrt3)²
Simplified to 5 + 2sqrt15 + 3.
In other words, they multiply out the two brackets in the numerator the long way instead of squaring both terms straight off.
Why can't you do this? What am I doing wrong here?
If you have any advice that'd be great, otherwise I think I'll just end up taking hits on simplification in a lot of my answers, doh.
Thanks guys

When I say ICE table, I know that it doesn't mean I have to create a table, just when I need to isolate a variable (X) in order to find the unknown concentration- ICE table I used just for the specificity.
I am in my 2nd year of chemistry, so this acid/base chemistry stuff should come easy to me. It does not. I know what a buffer solution is and how to identify the acid/base and their conjugates. So, I would say I understand most of the conceptual ideas behind buffers, pH and acids/bases.

When it comes to some of these buffer and volumetric word problems (instrumental class), I struggle. I feel like I've always sucked at the math-problem solving in chemistry. I remember equations, know my math well, but can never *solve* or use logic correctly.

Quick example of problems I deal with: "Calculate the volume of concentrated ammonia and the weight of ammonium chloride you would have to take to prepare 100 mL of a buffer at pH 10.00 if the final concentration of salt is to be 0.200 M. Ammonia Kb= 1.75 x 10^-5"

So I'd imagine not only ICE tables, but use of problem solving where I manipulate equations to find certain qualities in the correct subsequent order. For example, some problems I found that I needed to use the ionization constant (Kw) to get a certain value. I usually used the 1.0 x 10 -^14 Kw to convert from Kb to Ka, but sometimes the solutions show it's use when not even converting those values.

Any tips of how to become better at these types of problems and approach them is greatly appreciated. The problem I listed, I had trouble matching what the solution sheet had. I feel it may be a mistake- also, the Kb value for ammonia is different than what I see online.

WARNING: LONG POST. USE COMMAND/CONTROL + F TO YOUR ADVANTAGE.
SL: English A L&L, Spanish Ab
HL: Psych, Math, Phys, Chem
EE in Psychology
~~~
Hi everyone. First of all I'd like to thank this community for keeping me sane through my two years of IB. I wouldn't have been able to recover from my 5 mental breakdowns during exams without you guys and your memes. So as a thank you, I'll write some tips for some subjects that I did for the future generation of IB sacrifices...
As a young and innocent IB Y1 student, I was looking everywhere for posts that would help me understand what the heck was necessary for me to do and... I couldn't find any. So here goes.
~~~
English A Lang Lit SL
To be completely honest, I hated this class with a passion. The pros in this class are such that 50\% of your grade is decided before you walk into that exam. I'll talk about this in a moment. The cons? Well. The cons in this class is literally the same as the pros.
You probably know what I'm trying to get at. Your Oral work and WT are very very very important before you walk into that exam. They make up that 50% that I was talking about. You can either walk into that exam knowing that you have strong orals and WT, which will give you extra confidence during exams (and who doesn't want extra confidence?). On the flip side, you can walk into the exam knowing that you don't have strong orals or WT. Take your pick.
For my IOC (half of the oral), I panicked halfway through. There's literally 10 seconds of silence in my recording. So I guess what I'd tell you all who are scared of your IOC or that you panicked during your IOC is to not worry. As long as you manage to tear the text apart with some high level vocab and overthinking, you're good. There's no real "right answer" in english. So express your opinion. Similar to the P2, make sure you READ THE BOOKS (I did Othello and The Things They Carried). For the IOC, know the specific excerpts (your teacher may give them to you) by annotating and make sure you connect them to the rest of the book (bonus points!!).
For my WT, I nearly gave up on that. Don't. I swear. After walking out of my English P1 in the end, I was going to burst into tears because all I analyzed was the first half of the passage. However, knowing that I put out a strong WT, I settled a little. Your teacher will make you do multiple WT (I think I did 3 in total) despite only submitting one in the end, although I'm some schools your teacher may choose not to (I don't think they are supposed to but welp). Do them all as best as you can. It'll give you the freedom of choice in the end on which to brush up and make better. You have a lot of freedom for the WT (like the EE) so do something that you are passionate about. It'll be easier to work with.
During exams, I flopped the P1 (or at least I felt I did). So I went home and worked my ass off to study for English P2. Unlike P1, where you can get literally anything, you can prepare for P2. For my P1, my teacher already gave me a lot of practice on how to write essays so I kinda knew how to analyze and identify textual features and stuff, but dumb old me wanted to organize my essay differently for the first time during my exam. DO NOT TRY SOMETHING NEW DURING EXAMS. I flopped. But for P2, I did pretty well on it. I'm not the kind of student who needs to make notes for every single thing. So I didn't make notes for plot and all that. However, I did pay attention in class and READ THE BOOKS. Reading the books really helped me to understand character and plot development, which I wrote about in my final P2 essay for sure. Instead, I made notes for specific quotes and examples from the text that I wanted to use in my essay. Aha! Here's the part where you need to use that brain to decide which quotes to pick. The quotes that you pick will be different from others just because you may have your own way of writing or opinion of the texts in comparison to others. Make sure you pick your own quotes that you analyze the shit out of PRIOR to the exam. For me, I had quotes connected to characters and symbols and themes and plot. I also added stylistic feature analysis. So if there was a simile in a quote that had characters/symbols/etc. BING to the O. This made it so. much. easier. during the exams when I had to analyze the quotes. Also, litcharts/sparknotes/shmoop are literally your best friends. I abused them so much when I studied my two books (A Doll's House and A Streetcar Named Desire). Similar to the IOC preparations, make sure you can connect these quotes to other events in the book. You don't need quotes for the other events but just know what one event leads to, how it leads to the other event and why. I think I had about 30 quotes altogether for both books. You don't need that many. But seeing as my P1 went badly, I wanted to be very prepared for my P2.
Exam Prep: Annotate past papers for both P1 and P2 (maybe write a timed essay or two) and make outlines. For P1 annotate the text and for P2 annotate the question.
~~~
Spanish Ab Initio SL
Review the vocab to prepare for P1. That's about all there is to it. Also make sure you know some basic conjugations for your P2. Like English A Lang Lit SL, 50\% of your grade will be in before your exam. What you want that to mean for you would depend on your own effort.
So you may or may not know. The Group 2 language grade boundaries are screwed up because of native or near-native speakers taking the same exams. Don't believe me? Take the English Ab Initio paper and have a look. Then compare that to the English B papers.
Anyway, you basically need to achieve a near-native interpretation level yourself if you want to get good grades in this class. It's pure hard work. I don't want to say too much about this subject as none of my peers and I did stellar in this class. It was the only class of mine where my actual was lower than my predicted.
Exam Prep: Do past papers for P1 and try to begin to get a hang of the kinds of questions IB asks (like T/F questions etc). Make sure you have outlines for P2 text types in order to prevent the loss of easy marks.
~~~
Psychology HL
Notes. Notes. Notes. Personally, the reason why I feel like I survived this class was because I made the most in depth notes ever (a feat I wish I did in my other classes tbh...). I took 4 textbooks and made notes using all of them, condensed into one document. This really helped me for my mocks, exams, and overall understanding of the subject.
What I did, contrary to popular opinion, was to learn the LAQ for all the levels of analysis in the core. This was because I like the freedom of choice and, plus, LAQ topics can be tested in SAQ anyway. But that's what worked for me.
Make sure you know the command term. IB Prepared for Psychology was my book of God. It really helped me learn what each command term wants in the essay. So maybe take a look for that. It'll help.
Fundamentally, for the new syllabus even, these should still stay the same. You want to be able to understand the subject by making notes. The only difference between the new syllabus and my syllabus (final exams 2018) is the way you'll organize your notes and the way the exam format may be minutely.
A lot of my friends made complete outlines for learning outcomes that were given (which I don't think should be a problem due to IB removing testing according to the learning outcomes). This was probably one of the reasons why the examiner reports always involved insufficient completion of following the demands of the command term just because the learning outcomes that are given do not necessarily have the same essay structure needed to get those 6/7 grades in your exam due to the different command term. I'd suggest to understand the command term, the content, and evaluation in order to BS your way into structuring an appropriate well-scoring essay for your P1 and P2. I never made one essay outline ever in my notes to prepare for IB Psychology examinations.
Most teachers probably won't give a shit about P3. But make sure you know the definitions for P3. It'll get you some easy points. Don't forget to know the command term for this paper too. Describing inductive content analysis and explaining it are two completely different things.
Command term understanding takes a lot of practice, so make sure you take your teacher's advice to heart and try to make your writing better each time. A 7 essay from one student won't look the same as another 7 essay, but both will check the boxes for a 7. So practice, practice, and practice.
The IA is simple to secure high marks. I don't know if they will change it in the new syllabus yet, but as of my syllabus, you need to make sure you follow the rubric very closely. If you tick all the boxes, you'll be able to get a 7.
Exam Prep: Notes! For the new syllabus P1 and P2, maybe take a look at the specimen/past papers for the new syllabus and ensure that you understand how your exam works. For P3, annotate past paper excerpts and get a feel for what IB usually asks (e.g. inductive content analysis).
~~~
Mathematics HL
Practice is so key. Your teacher may or may not assign practice questions to you. Do them all the time. It'll be tempting to look at questionbank and do all those questions, but the issue with that is that when you see those exact same questions on your exam in school, you will already know how to do the question and get it right. So leave questionbank and past papers to your second year so you can at least get a feel for where you may truly stand in your first year. However, the reality is that since you are faced with questions that you have never encountered before in the final IB examinations. This could lead you to getting a predicted grade higher than your actual.
So maybe you're wondering what you can practice with. Make sure your fundamentals are sound and you understand what you need to know. The syllabus can suffice for this. But eventually you'll do past papers and you'll begin to get a feel for what you need to know and what topics commonly go together.
I did the calculus option for P3. It's quite difficult but that's where the fundamentals come into play. Once you're able to identify what fundamentals are needed in a question, you're ready to solve. There's a lot of youtube videos related to the calculus option. You just have to find them. They're generally the equivalent of college calculus year 1 and year 2 courses I think. Can't really remember.
For your maths IA... how do I put it... It's better to do something that you understand and lose a mark or two in criterion E (use of mathematics) rather than do something you don't understand and lose a mark all across the board in communication, reflection, etc. Personally, I did something very simple that required barely any complicated thinking but I only lost a mark in criterion E and ended up with a 7 for my math IA.
Also, knowing your formula booklet will help you a TON. Losing 3 seconds every time you look for a formula during an exam is a nono. Don't be that kid. Know where the equations are. Trust me. It'll give you the chance for an extra few marks.
Exam Prep: Past papers. Past papers. Past papers. Know your calculator's capabilities for P2 and P3. It can make or break a grade boundary. Know your formula booklet in and out.
~~~
Physics HL
Your data booklet is your best friend. This is in a similar way to what I wrote about the formula booklet for math but different. The IB Physics data booklet is an asshole. Literally. They try to trip you up by using different notations here and there. I fell into that trap in my P3 exam. So make sure you know the tricks of the data booklet because it can save your grades during an exam.
Understanding the science behind obtaining certain values in calculations is very important. For this class, the IB Oxford study guide is the book of God. It goes over so many things that are necessary and it's hassle-free compared to the textbook. Before my exams, I would take this book and read through the entire thing. I think that's what saved my ass during my exams.
Personally, my IA went through some dumb problems and I had to rewrite it so I didn't do very well on it in the end. I won't comment much about the IA but just work hard on it.
I think that for me personally in order to prepare for my exams I did like one or two past papers until I realized that the questions didn't help me much personally as there were so many different scenarios that the IB could test you on. So my advice would be to do some recent past papers to get a feel of the progression of the exam. On P2 for instance, the first question is generally about mechanics followed by a big question on electricity or whatever and there's usually a big question on modern physics etc. The goal of the two years in IB physics is to get a feel for what topics go together well. The goal of exam study and doing past papers near the end of the course in preparation is to get a feel for the big topics.
As for P1, just know that the IB will be tricky so try your best to identify some tricks that they'll attempt and common mistakes that students to.
For P3, I did astrophysics. Please do not forget Section A. Review your practicals. Astrophysics wasn't too bad. You just need to memorize some stuff so... yeah. There's not a lot of difficult math in it either so just do some practice problems and understand what's going on. You're basically gonna have to memorize the IB Oxford study guide to regurgitate onto the exam page for this option.
Note: I've heard good things about the Cambridge Physics textbook, so give that a try. I didn't use it but apparently it's pretty good.
Exam Prep: Past papers and read through a source of notes (either your own or someone else's or the study guide, which I recommend) to make sure you refresh your brain on the fundamentals as almost everything that is tested on the exam is related to fundamental physics. (Literally almost everything can be related to F=ma lol)
~~~
Chemistry HL
Know the periodic trends. This is the trippiest topic (for most of my friends and I know I get tripped up a lot by this as well). When you get your P1, there will be a periodic table behind the first page. DON'T RIP IT OUT I ALMOST GOT INTO DEEP SHIT, but grab your pen and draw in the trends. It'll make the P1 questions faster. Similarly to physics, make sure you know what common mistakes you have on your own (Richard Thornley has videos about this I think... not too sure cuz I didn't watch them but I remember seeing them).
Just like physics, everything that you learned in IB sciences is built off pre-IB sciences, which are very simple and fundamental. So if you ever get stuck on a concept, think back to pre-IB and try to build from that. Believe me, it makes some concepts a lot easier in chemistry HL as some of these concepts can be so triggering and difficult to understand.
My IA has a funny story behind it... I nearly made my whole classroom evacuate. But that's not what I'm here to talk about. What I learned after my physics IA (which I didn't work too hard on tbh, which may explain why I made the dumb mistake and had to rewrite it and got a crappy grade) was that you really should work hard on your IA cuz they can literally save your ass when you walk out of that exam room thinking that you didn't know half the exam. I worked so hard on my chemistry IA and restarted collecting data 3 times (it took me 3 days and a lot of money) until I got the perfect data. I then took a long time to research some concepts that were quite difficult but tried to understand them nonetheless. A few weeks after I submitted my IA, I was told that my IA would be an exemplar for future years. It's one of my proudest pieces of work. Work hard on your IA and make it something you can be proud of.
There's nothing much else to say for chemistry except for the way to prepare for papers is relatively similar to physics. Do a few past papers in order to get a feel for what the IB likes and mixes.
I did medicinal chemistry for P3 and it's quite similar to what I said for astrophysics. There's a lot of memorization so get your IB oxford study guide and give it a hard read through. Understand what's going on and it should suffice. (Also don't forget Section A practicals as well!)
Note: If you want an in-depth textbook for chemistry, Pearson does the trick.
Exam Prep: Past papers and read through a source of notes (either your own or someone else's or the study guide, which I recommend) to ensure you know the concepts well.
~~~
ToK
First thing. If you are writing your essay and have literally no idea what is going on. You're probably doing it right.
But how do I know that this is right? What is right? What is wrong? How do we separate right from wrong? Is it moral to do so?
Yup. Glad I'm done with that shit. Okay in all seriousness though, make sure to plan your essay out and BRAINSTORM. You'll be able to think of some ToK level idea after brainstorming for a while to weave into your essay.
Second thing. Pick a knowledge question that you are passionate about for your presentation. Just like what I've said before about being able to pick your topics in other subjects for your work. If you like the topic, you'll work harder. So do that.
The essay and the presentation are quite similar in the way you are supposed to develop your ideas and points. Keep that in mind.
~~~
EE
Personally, my EE was on psychology just because I couldn't be bothered to stay in school over the summer and do experiments for science EEs. Overall, the entire process for me (research and outlining and writing) took 2 weeks over the summer. Of course, I didn't procrastinate much during the day (except for the occasional gaming break). What I'm trying to say is that your EE doesn't actually take long and you can enjoy your summer. I'm not shitting you. If you need a break, take it. But try to capitalize on those moments when you are productive because I had a crappy writers block at 3000 words where I didn't know what else to write about, so I'm quite glad that I completed it until that point quite early because I had the spare time to take a break.
Make sure that you outline your entire EE before you write. Yes, you can change your outline as you write. But it's much better to be able to visualize and plan your EE before you get started.
Also, finish your EE before the end of the summer. Maybe leave a week or so for you to relax before Y2 starts. It helped me a lot. Make sure that the EE that you have before you give yourself the "done with EE" status over the summer is the third or fourth draft of your EE with your own proofreading and editing. Take a day off in between and look at it with fresh eyes. I'd suggest doing the second draft after a day of rest and the third draft after a week or so. It'll help you see where you make mistakes and stuff. By doing this and making your own EE the best it can be before Y2 starts can really prevent you from burning out early in Y2. Imagine having to do all your IAs and EE in the first semester of Y2. Prevent that from happening cuz it sucks.
~~~
I hope that this post will help others on this reddit as posts from others have helped me during my two years. If there are any questions that you have, feel free to PM or comment down below. Good luck to all future IB takers!!

I am trying to understand optics for Macro lenses ( e.g. short working distance imaging), can anyone recommend a good place to start. I am not a professional optics person, but have ok math and decent image sensor knowledge

The context is this, I'm using fairly standard machine vision CMOS sensors ( 1.4 - 4+ micron pixel size) and imaging monochrome features that are say 30 microns ish and fairly close ( 5-30mm distance) all on a flat plane ( which I can control to say 0.1mm and can illuminate). I would like the optical magnification to be roughly in the 0.25 - 0.5 range . I have two setups which work, one is a cheap USB 2.0 microscope (VGA) from amazon, the other is a C-Mount 2x mag finite conjugate objective like this https://www.edmundoptics.co.uk/p/2x-0-10na-ultra-compact-objective/38498/ with a working distance of 11mm on a 2 micron imager which also works (but has a very very narrow depth of field and is optical mag 2x).

I'm now trying to translate this to the world of s-mount optics and having a hard time, because 'standard' lenses don't seem to be designed for very short working distances with narrow depths of field.

I've got my head around depth of field calculations from here http://www.dofmaster.com/equations.html, and "working F number ( aperture)" for near focussed targets, and just about understand Rayleigh criterion, all of which suggest that the criteria are all really tight ( trade off of Circle of Confusion and Depth of Field).

What I can't work out is, is this an area where there are cheap ' off the shelf' solutions ( e.g. you can buy macro lenses for mobile phones for $20 which look pretty good - how do they work crisply without having an infinitesimally small DOF?) , or is it an area where I need to assemble a mini-microscope, or do I need to pay for a custom optics person (fairly cash strapped currently but could pay for a day or two's work) ?

Any pointers to books, webpages, products, online courses most helpful.
Thanks

I am trying to compute the Homography without using the Opencv library.
I am doing as mentioned in this link:
https://math.stackexchange.com/questions/494238/how-to-compute-homography-matrix-h-from-corresponding-points-2d-2d-planar-homog
Also, went through the link below :
https://alyssaq.github.io/2015/singular-value-decomposition-visualisation/

After getting a matrix from the above link, say matrix A
I am calculating the SVD of A using
S,U,V = np.linalg.svd(A, full_matrices=True)

and the homography matrix H then is the last column of conjugate of V transpose.

But the answer given by cv2.findHomography and my above method dont seem to match.
Can someone help me with this?

View original blog post for pretty formulas and code highlighting.
(highly recommended)
This post is part of my Game Math Series.
Source files are on GitHub.
Shortcut to sterp implementation.
Shortcut to code used to generate animations in this post.

An Alternative to Slerp

Slerp, spherical linear interpolation, is an operation that interpolates from one orientation to another, using a rotational axis paired with the smallest angle possible.
Quick note: Jonathan Blow explains here how you should avoid using slerp, if normalized quaternion linear interpolation (nlerp) suffices. Long store short, nlerp is faster but does not maintain constant angular velocity, while slerp is slower but maintains constant angular velocity; use nlerp if you’re interpolating across small angles or you don’t care about constant angular velocity; use slerp if you’re interpolating across large angles and you care about constant angular velocity. But for the sake of using a more commonly known and used building block, the remaining post will only mention slerp. Replacing all following occurrences of slerp with nlerp would not change the validity of this post.
In general, slerp is considered superior over interpolating individual components of Euler angles, as the latter method usually yields orientational sways.
But, sometimes slerp might not be ideal. Look at the image below showing two different orientations of a rod. On the left is one orientation, and on the right is the resulting orientation of rotating around the axis shown as a cyan arrow, where the pivot is at one end of the rod.

If we slerp between the two orientations, this is what we get:
https://i.redd.it/kfemvdlz4rx01.gif
Mathematically, slerp takes the “shortest rotational path”. The quaternion representing the rod’s orientation travels along the shortest arc on a 4D hyper sphere. But, given the rod’s elongated appearance, the rod’s moving end seems to be deviating from the shortest arc on a 3D sphere.
My intended effect here is for the rod’s moving end to travel along the shortest arc in 3D, like this:
https://i.redd.it/f8x7jyi15rx01.gif
This is where swing-twist decomposition comes in.

Swing-Twist Decomposition

Swing-Twist decomposition is an operation that splits a rotation into two concatenated rotations, swing and twist. Given a twist axis, we would like to separate out the portion of a rotation that contributes to the twist around this axis, and what’s left behind is the remaining swing portion.
There are multiple ways to derive the formulas, but this particular one by Michaele Norel seems to be the most elegant and efficient, and it’s the only one I’ve come across that does not involve any use of trigonometry functions. I will first show the formulas now and then paraphrase his proof later:
Given a rotation represented by a quaternion R = [W_R, vec{V_R}] and a twist axis vec{V_T}, combine the scalar part from R the projection of vec{V_R} onto vec{V_T} to form a new quaternion:
T = [W_R, proj_{vec{V_T}}(vec{V_R})].
We want to decompose R into a swing component and a twist component. Let the S denote the swing component, so we can write R = ST. The swing component is then calculated by multiplying R with the inverse (conjugate) of T:
S= R T^{-1}
Beware that S and T are not yet normalized at this point. It's a good idea to normalize them before use, as unit quaternions are just cuter.
Below is my code implementation of swing-twist decomposition. Note that it also takes care of the singularity that occurs when the rotation to be decomposed represents a 180-degree rotation.

public static void DecomposeSwingTwist ( Quaternion q, Vector3 twistAxis, out Quaternion swing, out Quaternion twist ) { Vector3 r = new Vector3(q.x, q.y, q.z); // singularity: rotation by 180 degree if (r.sqrMagnitude < MathUtil.Epsilon) { Vector3 rotatedTwistAxis = q * twistAxis; Vector3 swingAxis = Vector3.Cross(twistAxis, rotatedTwistAxis); if (swingAxis.sqrMagnitude > MathUtil.Epsilon) { float swingAngle = Vector3.Angle(twistAxis, rotatedTwistAxis); swing = Quaternion.AngleAxis(swingAngle, swingAxis); } else { // more singularity: // rotation axis parallel to twist axis swing = Quaternion.identity; // no swing } // always twist 180 degree on singularity twist = Quaternion.AngleAxis(180.0f, twistAxis); return; } // meat of swing-twist decomposition Vector3 p = Vector3.Project(r, twistAxis); twist = new Quaternion(p.x, p.y, p.z, q.w); twist = Normalize(twist); swing = q * Quaternion.Inverse(twist); }

Now that we have the means to decompose a rotation into swing and twist components, we need a way to use them to interpolate the rod’s orientation, replacing slerp.

Swing-Twist Interpolation

Replacing slerp with the swing and twist components is actually pretty straightforward. Let the Q_0 and Q_1 denote the quaternions representing the rod's two orientations we are interpolating between. Given the interpolation parameter t, we use it to find "fractions" of swing and twist components and combine them together. Such fractiona can be obtained by performing slerp from the identity quaternion, Q_I, to the individual components.
So we replace:
Slerp(Q_0, Q_1, t)
with:
Slerp(Q_I, S, t) Slerp(Q_I, T, t)
From the rod example, we choose the twist axis to align with the rod's longest side. Let's look at the effect of the individual components Slerp(Q_I, S, t) and Slerp(Q_I, T, t) as t varies over time below, swing on left and twist on right:
https://i.redd.it/o2dkpq7e5rx01.gif
And as we concatenate these two components together, we get a swing-twist interpolation that rotates the rod such that its moving end travels in the shortest arc in 3D. Again, here is a side-by-side comparison of slerp (left) and swing-twist interpolation (right):
https://i.redd.it/co1m19nf5rx01.gif
I decided to name my swing-twist interpolation function sterp. I think it’s cool because it sounds like it belongs to the function family of lerp and slerp. Here’s to hoping that this name catches on.
And here’s my code implementation:

public static Quaternion Sterp ( Quaternion a, Quaternion b, Vector3 twistAxis, float t ) { Quaternion deltaRotation = b * Quaternion.Inverse(a); Quaternion swingFull; Quaternion twistFull; QuaternionUtil.DecomposeSwingTwist ( deltaRotation, twistAxis, out swingFull, out twistFull ); Quaternion swing = Quaternion.Slerp(Quaternion.identity, swingFull, t); Quaternion twist = Quaternion.Slerp(Quaternion.identity, twistFull, t); return twist * swing; }

Proof

Lastly, let’s look at the proof for the swing-twist decomposition formulas. All that needs to be proven is that the swing component S does not contribute to any rotation around the twist axis, i.e. the rotational axis of S is orthogonal to the twist axis.
Let vec{V_{R_para}} denote the parallel component of vec{V_R} to vec{V_T}, which can be obtained by projecting vec{V_R} onto vec{V_T}:
vec{V_{R_para}} = proj_{vec{V_T}}(vec{V_R})
Let vec{V_{R_perp}} denote the orthogonal component of vec{V_R} to vec{V_T}:
vec{V_{R_perp}} = vec{V_R} - vec{V_{R_para}}
So the scalar-vector form of T becomes:
T = [W_R, proj_{vec{V_T}}(vec{V_R})] = [W_R, vec{V_{R_para}}]
Using the quaternion multiplication formula, here is the scalar-vector form of the swing quaternion:
S = R T^{-1}
= [W_R, vec{V_R}] [W_R, -vec{V_{R_para}}]
= [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_R} + W_R (-vec{V_{R_para}})]
= [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R (vec{V_R} -vec{V_{R_para}})]
= [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}}]
Take notice of the vector part of the result:
vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}}
This is a vector parallel to the rotational axis of S. Both vec{V_R} X(-vec{V_{R_para}}) and vec{V_{R_perp}} are orthogonal to the twist axis vec{V_T}, so we have shown that the rotational axis of S is orthogonal to the twist axis. Hence, we have proven that the formulas for S and T are valid for swing-twist decomposition.

Conclusion

That’s all.
Given a twist axis, I have shown how to decompose a rotation into a swing component and a twist component.
Such decomposition can be used for swing-twist interpolation, an alternative to slerp that interpolates between two orientations, which can be useful if you’d like some point on a rotating object to travel along the shortest arc.
I like to call such interpolation sterp.
[Edit]
An application of swing-twist decomposition in 2D just came to mind.
If the twist axis is chosen to be orthogonal to the screen, then we can utilize swing-twist decomposition to use the orientation of objects in 3D to drive the rotation of 2D elements in screen space or some other data. The twist component represents exactly the portion of 3D rotation projected onto screen space.
However, in terms of performance, we might be better off just projecting a 3D object's local axis onto screen space and find the angle between it and a screen space axis. But then again, the swing-twist decomposition approach doesn't have the singularity the projection approach has when the chosen local axis becomes orthogonal to the screen.

I'm doing the optimization of a function that takes some time to evaluate (around 40 seconds). Analytical derivatives are not available (Dynamical parameter estimation). I'm currently using Nielder-Meads. When the function is close to the optimum the space between the vertices is quite small, but not "alligned" to the coordinates. Is there anyway to use these calculated vertices to estimate the jacobian and improve convergence?
My colleagues suggest Nielder-Meads until close to the optimum and then use some derivative based method (like conjugate gradient descent) to "fine tune". I'm imagine somebody invented a method that somehow use boths.
EDIT: I'm thinking about doing a linear regression and take the jacobian of the linear function, Does it makes sense? I'm not really formally trained in math.

First:I am a math student who is also interested in computer science. Through the last year I learned many new algorithms in numerics as well as in basic (not BASIC) programming techniques, and was surprised about certain algorithms, or about its 'beauty' or simplicity for a seemingly 'non-trivial' task and i began to like some of then particlularly (while others did not catch my interest very much) here some of them:

Conjugate Gradients-Algorithm: It is an numerical algorithm for solving certain (SPD) systems of linear equations that solves the system exactly after a number of iterations that equals the size of the system. And it uses just very little memory.
Fast Fourier Transformation: A very fast algorithm for decomposing e.g. a recorded sound wave into it's frequencies. (O(n log n))
Marching Squares: An algorithm that is used for tracing edges (or drawing countour lines) on a discrete grid.
Not really an algorithm: In 2D or 3D computer graphics u use '3D' or '4D' matrices for calculating transformations, that means turns but also translations which I consider as a very cool idea. (In math you get used to represent a turn + translation as y = Ax+b while with that idea you can just do 'y = Ax' which is pretty neat for chaining transformations.)

Did you also come across algorithms that you liked very much or you found that used just brilliant ideas? Merry Christmas!

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