Hello again,

Yesterday was our third Lecture where I introduced Fractions. If you never had a good understanding of what a fraction is I highly recommend watching this lecture. After it explains what a fraction is, it then goes to what is Reducing Fractions and Improper Fractions. The math itself is easy, but understanding Fractions fully is different. It also gives you a huge advantage in your math and sciences courses because most of the problems, lab work, or any class where they require you to measure things, the values are never whole numbers. It is usually a fraction or a decimal.

Fraction Lecture: https://www.youtube.com/live/fuAZ8zeTGlk

Playlist (includes Lectures 1-3, future lectures are uploaded weekly): https://www.youtube.com/watch?v=MppI9i3RGH0&list=PLFrt4v_t-kU4

If you want to watch the other lectures visit our channel. If you want to join our class send me a DM or watch them on our YouTube channel. Classes are once a week and it all started here in this reddit community. This is the original post that started this: https://www.reddit.com/r/learnmath/s/GEsyzHJ6Xp

I have a real and complex analysis course and its endsem is in 3 days. Syllabus include Chapter 1 - Chapter 5 from baby Rudin. I have quite frankly no interest in this course, and just want to pass the exam with decent marks, but I'm finding it really difficult to even understand the proofs. Can someone please guide me on what simpler materials can I refer to?

Edit : I have studied Chapter 1 and Chapter 2 somewhat for midsems, kindly be kind and help🙏🏽😭

Hi everyone, I have a cyclic/Latin-square-style counterbalanced repeated-measures design (as shown in image above), and I’m trying to work out the right way to use a Generalized Linear Mixed Effects model to analyse the data for it as I’m new to GLMMs. My aim is to compare how different cell-culture compositions affect the firing rates (expected to be of lognormal or tweedie distribution) of neural cultures while they’re being “trained” to perform tasks.

The study uses 100 neural cultures split evenly into two experiments of 50. Experiment 1 uses Culture Type A and Experiment 2 uses Culture Type B (so culture type is nested within experiment, not crossed). Each experiment’s 50 cultures are prepared in two independent batches of 25, and within every batch the cultures are made up with one of three cell-culture compositions (A, B, or C) - giving four batches and all three compositions in both experiments.

Each culture is then trained under 9 environmental conditions in a counterbalanced order, performing 2 tasks per environmental condition at 600 trials per task.

The GLMM model I currently planning to use is:

Firing rate ~ Culture Type + Batch + Cell Culture Composition + Environmental Condition + Task + (1 | Culture ID)

However, I feel that culture ID is insufficient as a random effect and it should include the other variables too, although I am unsure of what combination to use, especially since there’s so many of them. I am also worried about having to add dispersion models in the case of over/under-dispersion as well, since my response variable is not normally distributed. Hence, I would like to seek comments and advice on my current model from those who are experienced in working with GLMMs. Thank you very much in advance for the help.

Link to an image of my experimental design: https://imgur.com/a/v4pU3tN

Hi there,

Real Analysis by Jay Cummings really helped me to understand Analysis I. I am wondering if there are books equivalent to this for Analysis II :)

Any recommendations are appreciated(!)

Disclaimer: The purpose of this post is to explain why people get two different answers for these type of notations, not to agree with one answer or the other, and to explain why this is not even a math problem and should not be posed as one.

If you posted 12 ÷ 3(4) online, people’s answers to it would either be 16 or 1. The math notation 6 ÷ 2(1 + 2) has the same issue. People argue whether the answer to it is either 1 or 9. Both sides have their own reasons for their answers being correct. The question is: Why are people’s answers split? Why is this even a debate?

People think this has something to do with BODMAS vs PEMDAS or the division symbol. To clarify, it has nothing to do with BODMAS vs PEMDAS, and it has nothing to do with the division symbol. It has something to do with one thing and one thing only - Juxtaposition. What is juxtaposition? Juxtaposition (as in implied multiplication) means multiplying two different numbers close together without using the multiplication symbol (x). Some people were taught by their math teachers in school to treat juxtaposition as having higher priority over regular multiplication/division, while some were taught to treat all multiplication operations equally. Even calculators disagree with eachother for this reason if you use ambiguous notations. However, if you rewrite 12 ÷ 3(4) as 12 ÷ 3 x 4, it completely removes the ambiguity and everyone would agree the answer to this is 16.

Why is juxtaposition being a priority over regular multiplication/division even a thing in math for some people? My personal theory is that juxtaposition is a consequence/implication of translating math notations into spoken language and vice versa. What do I mean by that? For example, imagine a third-grade math teacher instructing a student to write ‘twelve over three x’ in math notation. If you were the student, would you write it as:

A.

12

___

3x 

or B.

12

___ x ?

3

Notice how it’s the same syntatic ambiguity as the poorly written math notation. If you chose either notation A or B, how would you reconcile with the other option? If you chose A as the correct answer, you might just change B into 12x/3 to remove the ambiguity instead of trying to explain it in spoken language. If you chose B, you might instead describe notation A as ‘twelve over in brackets three x’. Now, I am sure that most of you chose notation A in this instance, **juxtaposing the 3 with the x**. Why? Because your math teacher described it that way in spoken language. 

Let’s take another example. Write ‘twelve over three times four’ in math notation. Do you write it as:

A.

12

_____

3 x 4

or B.

12

___ x 4 ?

3

Here it gets a bit trickier. There is a good argument for notation B being the right answer, and I agree *mostly* with that. But what is the implication of taking notation B as the right answer? How then would you describe notation A in spoken language? You could say it as ‘twelve over in brackets three times four’, but as far as I know, no math teacher described notation A this way in spoken language (because as you can see, there are no brackets in notation A).

Now let’s take a look at a similar syntatic ambiguity outside the field of math. Here is an equivalent syntatic ambiguity in the English language:  

“I watched the man with the telescope.” 

Meaning A: You used a telescope to watch the man.  

Meaning B: You watched a man who happened to be holding a telescope.

Like the ambiguous math notations, both meanings *can* be correct, and have different reasons as to why it’s correct. Saying that one meaning is completely wrong without understanding the thought process of the other side is a shallow way of thinking. I’m not saying your answers to these math notations are incorrect. They are, but brushing off this issue and saying there is no ambiguity in the aforementioned math notations is not the correct way to deal with this issue.

On a side note, notice how translating between math notation and spoken language causes a lot of confusion. If you keep translating between the two, you’ll end up with completely different notations. It’s like one of those videos where a movie scene is google translated 100 times and ends up being a complete word salad compared to the original scene.

What does all of this prove? It shows that 12 ÷ 3(4) is not a math issue but a semantic/syntatic issue. One side understood it as ‘twelve divided by three fours’ while another side understood it as ‘twelve divided by three times four’. In conclusion, don’t use ambiguous notations.

Hello, i know this topic or question pops up a lot. I have even read a few posts here about it.

But I still dont really know how to start or where to start. Because i feel like I need to start from beginner math. Yes, division and subtraction, those beginner stuff.

When I was young I was pretty good at math, but late high school and college, I had a hard time keeping up. I could do some of the math problems that my class threw at me, but i felt that i lacked the common sense or foundation to truly understand math. I was just doing math. I hope this makes sense.

I know that 9 when divided is just three 3s. But i cant exactly apply that to bigger numbers and always rely on calculators. The same goes for more advanced math topics. I could do the simple problems but when it gets more complex, my brain doesnt know where or how to start solving it. I want to fix that.

Sometimes in my physics class, there would be some word problems and we have to solve for stuff with certain formulas and I remember myself getting so excited and happy whenever I got correct answers. But strangely enough, ive never felt that way with math.

I WANT to feel that way with math, because some people think its fun when you start to understand it, and I want to understand it. And i think it would really help me in the long run.

Basically, i want to get good at math. Generally. From basic to more complex stuff. Maybe math that ill have to learn in college.

Can i ask for some pointers on what to do or what to read or watch. I dont want to be lazy chud anymore bro😭

So I think I'm understanding partial fraction decompositon which in a way is like asking "Given this rational function, what partial fractions could've been combined to produce what I have now?"

For distinct linear/quadratic factors, it makes sense. Since when combining fractions their denominator becomes the product of each fraction's denominators', if you have multiple factors in the denominator of a rational function, then there needs to be a term for each of those factors.

What I don't get however, is if we have repeated factors, why we have to include a term for each power of each repeated factor. Here's an example, Suppose my fraction is (x^2+1)/ (x-2)^2. The partial fraction decomposition is A/(x-2) + B/(x-2)^2. Why is the B/(x-2)^2 there?

57) f(x) = 4<x

A) The graph lies above the x-axis, rises from left to right, with the negative x-axis as a horizontal asymptote; (0, 4) and (1, 16).

B) The graph lies above the x-axis, rises from right to left, with the negative x-axis as a horizontal asymptote; (0, 4) and (1, 16).

C) The graph lies below the x-axis, falls from right to left, with the positive x-axis as a horizontal asymptote; (0, 4) and (1, 16).

D) The graph lies below the x-axis, falls from left to right, with the positive x-axis as a horizontal

asymptote; (0, 1) and (1, - 16).

So this is from my exponential function exam and im kind of confused as to which one is the correct answer.
The points i got was (0,1) and (1,16) but none of the answer choices seem to match, am i missing something?

Hi! I'm a high school student from South Korea, and I'm hoping to apply to colleges in the US in a couple of years.

One thing I want to work on before then is learning math in English. I've been thinking about using Khan Academy instead of my Korean math textbooks, following the courses that match what I'm currently learning in school. But I'm not sure if this is a good idea, so I wanted to ask people who have actually used Khan Academy.

A few things I'm curious about:

- Is Khan Academy's high school math curriculum easier than what students usually learn in the US? I've heard that Korean high school math can be pretty challenging, so I'm wondering if the courses might be too easy for me.

- Does it cover everything you'd normally learn from high school math textbooks, or are there some important topics or areas that aren't covered in enough depth?

- If I complete all the lessons and exercises, would that be enough to build a solid understanding of high school math? Or would I still need other textbooks or problem sets? I'm also curious about how American students actually use it.

Do students usually use Khan Academy alongside their regular school classes, especially when preparing for college? Or is it mostly something they use for extra practice and review?

And here's probably my biggest question:

Has anyone here used Khan Academy as their ONLY math learning resource throughout high school?

I'm not just talking about using it alongside school lessons or other textbooks. I mean relying entirely on Khan Academy to learn high school math. If you have, how did it go? Did you feel prepared for college-level math? Did you notice any gaps in your knowledge when you started college? I'd especially like to hear from people who have actually tried this, rather than just people who are familiar with the platform.

Thanks!

I learn topic and when I forgot it , I revise but it takes lot of time to find the required topic in the book. Any easy way to revise which is less time consuming? By the way I study for fun not for any exam.

How much math would you say you knew by the time you started in HS?

Hello all, I got homeschooled via Penn foster when COVID hit, I was in the 10th grade.
I always struggled in math because it was uncommon someone was able to tell me the “whys” like, why we move a number, why this step in the equation makes sense, so it was very hard to get it to stick.
Anyways, geometry fckn sucked and, I have no education in math past that point. I did however take an accuplacer and scored the second highest in math on it recorded from that teacher. So I know I can learn it. I want to take my SAT and ensure I get a good enough score to go to college. What are some of the best videos, books, etc. that can be recommended for me?

Olá! Gostaria de apresentar uma hipótese que surgiu de uma reflexão pessoal. Não sou matemático e não tenho conhecimento suficiente para formalizá-la completamente, mas gostaria que pessoas com experiência na área tentassem investigá-la.

1. A estrutura inicial

Minha hipótese parte de três elementos distintos: X, Y e Z.

A relação entre eles seria:

X = {Y, Z}

Y = {X, Z}

Z = {X, Y}

O princípio fundamental é que nenhum dos três elementos contém a si próprio diretamente. Cada elemento é constituído pela relação com os outros dois.

2. Representação da relação

A estrutura também pode ser representada pela seguinte matriz:

    X  Y  Z
X   0  1  1
Y   1  0  1
Z   1  1  0

Onde:

• 0 representa a ausência de uma relação direta do elemento consigo mesmo

• 1 representa a relação entre dois elementos distintos.

Assim, todos possuem relação com os outros dois, mas nenhum possui relação direta consigo mesmo.

3. O princípio da manifestação

Existe também outra parte da hipótese que considero importante.

Se um dos três elementos se manifestar individualmente, ele não deixa necessariamente de representar a estrutura dos três.

Em outras palavras, mesmo que apenas X se manifeste, X continua relacionado a Y e Z. Quando os demais voltam a se manifestar, não seriam elementos completamente diferentes, mas os mesmos componentes da estrutura inicial.

A ideia é investigar se a manifestação individual de um elemento pode coexistir com a interdependência dos três.

4. O que gostaria que fosse investigado?

1. Essa estrutura pode ser representada por alguma teoria matemática existente?

2. Existe alguma contradição nas três relações propostas?

3. É possível desenvolver um modelo matemático que represente essa interdependência?

4. O princípio da manifestação individual pode ser formalizado?

5. Existe alguma aplicação dessa estrutura em matemática, lógica, computação ou outras áreas?

Não afirmo que essa hipótese seja uma descoberta ou que esteja matematicamente correta.

Ela surgiu de uma reflexão pessoal, e minha intenção é permitir que pessoas com conhecimento na área possam investigá-la, criticá-la, desenvolvê-la ou demonstrar suas limitações.

(A concepção original é minha. Estou aberto a contribuições e gostaria que a autoria da ideia fosse preservada caso ela venha a ser desenvolvida por outras pessoas.)

I’m 15 years old and looking for some guidance. I have a pretty big goal of becoming an anesthesiologist, however, I have some deficits in math which cause me to be at a 6th grade level of math, and in order to become an anesthesiologist, I need to be at a trigonometry level in math.

I have aphantasia (inability to visualize things in my head, I can think in words only) which makes math very difficult for me, in addition to the fact that I didn’t even learn to read until I was seven years old which also caused me to be behind.

My main reason for this post is to get some insights on the way you guys may process mathematics.

How do you process or break down equations in your head without visualizing them?

Do you have any tricks or patterns that enable you to do math quickly in your head?

And how do you prevent your brain from freezinf up under pressure? (For example, I know what 5x2 is, but the second I was under pressure, I froze up and it took me 4 entire seconds to solve it until it clicked. )

Thank you!!! No pressure, I’m just looking for any and all suggestions/ ideas / advice that could help me achieve my goal.

Revising Mathematics

I am facing a problem, I read books but forgot old things. Also it is not a good idea to revise the topics by opening book because in this way lot of your time get wasted in finding where the topic is . Mathematics is infinite you can't revise all things by opening book.

So what is the best way to revise? Pls tell I will be very grateful

Hyperbolic numbers are of the form a + bj, where j² = 1, but j ≠ ±1, and a, b ∈ ℝ. Adding them works the same with complex numbers: you add their real and hyperbolic parts together. Multiplying them works a bit differently. For example, (a + bj)(c + dj) = (ac + bd) + (bc + ad)j whereas in the complex numbers, (a + bi)(c + di) = (ac - bd) + (bc + ad)i.

In complex numbers, solving quadratic equations of the form az² + bz + c = 0 is straightforward. If a ≠ 0, the solutions are z0 = (-b + √(b² - 4ac))/2a, and z1 = (-b - √(b² - 4ac))/2a. In hyperbolic numbers, a quadratic equation can have up to 4 roots, not just 2, when a, b, c, z ∈ 𝔻 (the hyperbolic numbers). Any hyperbolic number can be mapped into an isomorphic direct sum of two real numbers using the orthogonal idempotent basis:

u = (1+j)/2, v = (1-j)/2

these idempotent elements satisfy: u² = u, v² = v, uv = 0, and u+v = 1.

then, we express a, b, c, and z in terms of their idempotent components: a = a_1 u + a_2 v, b = b_1 u + b_2 v, c = c_1 u + c_2 v, and z = z_1 u + z_2 v. (where a_1 = a_real + a_hyper, a_2 = a_real - a_hyper, b_1 = b_real + b_hyper, b_2 = b_real - b_hyper, c_1 = c_real + c_hyper, c_2 = c_real - c_hyper, and z_1 = z_real + z_hyper, z_2 = z_real - z_hyper)

Next, we decouple it into two quadratic expressions with real components, for z = z_1 u + z_2 v:

1. a_1 z²_1 + b_1 z_1 + c_1 = 0

2. a_2 z²_2 + b_2 z_2 + c_2 = 0

These can be solved using the quadratic formula:

1. z_1 = (-b_1 ± √(b²_1 - 4 a_1 c_1))/2a_1

2. z_2 = (-b_2 ± √(b²_2 - 4 a_2 c_2))/2a_2

This gives 4 roots. Substituting them back to find z gives:

z = (z_1 + z_2)/2 + j(z_1 - z_2)/2

Likewise, a cubic equation in 𝔻 can have a maximum of 9 solutions, a quartic 16, and generally, an n degree polynomial can have up to n² solutions in 𝔻.

But what if the hyperbolic number z is in the exponent, not the base? Here's how such equations can be solved.

From the Taylor series, we know:

e^xj = 1 + xj + x²j²/2! + x³j³/3! + x⁴j⁴/4! + ...

Which equals:

1 + xj + x²/2! + x³j/3! + x⁴/4! + ... = (1 + x²/2! + x⁴/4! + ...) + (x + x³/3! + x⁵/5! + ...)j

The real and hyperbolic parts are the Taylor expansions for cosh(x) and sinh(x) respectively. This gives us a hyperbolic version of Euler's formula, where instead of sine and cosine, we have the hyperbolic versions of these functions:

e^xj = cosh(x) + sinh(x)j

We can use this to solve a simple exponential equation of the form a^z = b, with a ∈ ℝ⁺. We start by converting a^z into a^x+yj (x, y ∈ ℝ) and by exponent rules, this equals a^x * a^yj. a = e^ln(a), so a^yj = e^{(y ln(a})j) = cosh(y ln (a)) + j sinh(y ln (a)). Therefore, a^z = a^x cosh(y ln (a)) + j a^x sinh(y ln (a)) = b. This gives a system:

1. a^x cosh(y ln (a)) = b_real

2. a^x sinh(y ln (a)) = b_hyper

Solving this system isn't easy, but the problem is now reduced to ℂ.

If you have any questions, suggestions, or are interested to learn more about this system, comment below.

How do i prepare for a math exam? Any advice is welcome! i feel like im still not ready because i still solve problems on a pretty slow pace and im scared i might not be able to do good! Thanks!

For context, I'm a first year college student majoring in Physics and Math. I took Calc BC as a senior in high school, and I chose to take a proofs-based Calculus I class in college which all freshman math majors are required to take. This was instead of the Physical Sciences Math courses that started with MVC.

I really don't want to fall behind in my next quarter of physics, and my TA suggested that I self-study linear algebra, multivariable calculus, and some differential equations, since our next quarter (E&M) is apparently super reliant on that. I have no idea what to do to study though. I tried last summer to watch videos and stuff, but for most resources like Khan Academy or other online sites, it just takes a huge amount of time and/or goes too slowly. I previously learned calculus through the mathisfun website, but I don't think they go up higher in level that that. If anyone has any suggestions of how/where to start studying these fundamentals I would greatly appreciate it!

I have a math exam coming up that we need to get over a certain amount to continue in the program. I usually am decent at math, but i’m really worried about this one just because of the amount of formulas, and I tend to panic at word questions (especially if I have to work backwards, or it includes multiple equations)

we get a formula sheet, but our instructor won’t tell us what is on it.

i’m just not sure how to study effectively. do I just keep doing practice questions?

I’m a math major freshman. I love graph theory. I read many coloring theorems and extreme graph theory like forbidding a subgraph. But I feel like I’m just reading it and I couldn’t find an application. I’m also interested in something I think is called network theory. I love modifying algorithms like change the definition of a node or an edge and stuff and see what will happen.

Confused. What should I do. What should I learn. And any suggestions?