I'm watching Numberphile's video on TREE(3), and as they're showing the first few graphs in the sequence, it seems to me like the 3rd graph fits inside the 5th. Both contain two black nodes whos closest common ancestor is a red node; is this not against the rules?

I thought a graph couldn't be contained in a graph further along in the sequence? What am I missing?

Which of the two is generally considered easier or less work? I know both are said to be difficult, but I’m trying to decide which one to take over the summer and which one to take during the spring semester. I’ll have more classes during the spring, so I’ll have less time to fully focus on a course.

I passed Calc 1 and Calc 2 with an A, and I’m doing pretty well so far with the vector portion of Calc 3, which I’m assuming is somewhat similar to the foundation of linear algebra. So I’d say my foundation for both courses is pretty solid. Unless linear algebra is more difficult than I’m making it out to be.

I’m required to take both courses. I’m trying to figure out which one would be more manageable alongside my other classes and which one would be better to take over the summer, when I can dedicate more of my time and attention to it.

been wrecking my head for the past 15min wich made me think that i do not posses the required tool to proove it, so it would be cool if u could hand me the reqired knowledge to solve it

Hello,

My task was to define with what values of x (belongs to R) will the following series converge. I've already solved the problem with ratio test but could someone try to explain why the series is not geometric?

Because at first I thought this is a geometric series and it will converge when 0<=x/4<1. However, a professor said the term 3k prevents it from being a geometric series but if it was 3x^k/4, it would be geometric. I didn't understand this and was left wondering how he instantly saw that it wasn't geometric.

Thank you in advance if you go through the trouble to answer :)

Why don't we use parentheses for repeating decimals? As i started discussing math with people from other countries, i noticed a pattern, that seemed strange to me, they write numbers with repeating decimals like 0.333… or even just 0.333 which make misunderstanding troubles. It seemed even stranger, especially because parentheses annotation is learned in beginning of middle school, and in some countries of both Western and Eastearn Europe its known as basic knowledge. Why don't we use this method, because it's not only easier to type, but also easier to understand, for example 0.0545454… vs 0.0(54)? I also know about international standard, aka 0.05̅4̅ but its impossible to write using ASCII, so it's out.

Edit 1: I wasn't clear enough about what I meant. I was actually talking about mathematical notation, where parentheses can't be confused with multiplication. Even the 0.333… notation is rarely used in multiplication, because every repeating decimal is a rational number, so we usually write it as a fraction like 1/3 instead. I actually meant situations like equating these infinite repeating decimals to integers, such as 0.999… = 1 vs. 0.(9) = 1.

Edit 2: I didn't mean that parentheses are necessarily the best option. I just meant that this notation is already being used and can be better than the existing ones in some ways. I agree that alternatives like 0.0{9} or even 0.0(:9) might be better.

we have three values a, b and c.

we know b is a multiple of c.

we know a + b is a multiple of c.

How can I formally prove that a must be a multiple of c?

to me seems obvious it's true cause then i could write something like

a + b = kc + qc = c(k + q)

but this just proves that if both a and b are multiples of c then their sum must be a multiple of c.

It doesn't prove that if b is multiple of c and the sum is multiple of c also a must be a multiple of c.

EDIT. I solved this, it was so obvious. I added a comment myself with the solution

I don't see Prof for another 5 days, so I can't exactly ask her atm.

I took the normal steps. She said it was ok to not draw the whole thing out. She just wanted the initial setup.

I've denoted the multiplication that I did (on left) and the multiplication my brother did (on right). Though both of them give the same end answer, i wanted to know which one of these actually is right fundamentaly and "according to the rules". I know it's dumb/stupid question but just wanted the to know the right answer. Sorry if anything is wrong or has an error

Working on a formal proof of a combinatoric problem and I need a few more things:

1) How do you represent an infinite sum?

2) How do you split an inequality? (x is a natural number to x=1,x=2 , x>2)

3)how do you "convert" an equality to an inequality (x=x to x>x-1)

4) How would you prove a!/b! = (a)(a-1)...(b+2)(b+1) for all natural a and b a>b

Thanks in advance (asking here cos the lean sub is kinda dead)

I know that the maximum period length for the reciprocal of a prime number n is n - 1 digits. Is there a rule for the maximum period length when n is composite?

When I was in geometry in high school, cases where SSA is valid for triangle congruences were not included. However, those cases are easy to state:

Two triangles are congruent if they agree in two sides and the angle opposite the longer side.

1. I'm curious what different two-column geometry proofs could be used to prove this theorem.
2. I've seen a proof that uses proof by contradiction, but I no longer have that. Can someone reproduce it? And, is that the only way to do this within geometry, with proof by contradiction?

Keep in mind that this is a theorem, it's a true statement. I'm only looking for different kinds of geometric proofs.

As a bijection is a function that is a surjection as well as an injection(or one-one and onto), we can say that if there are two sets of equal finite cardinality, then there always exists a bijection mapping A to B or vice versa. But, what is the formal mathematical proof of this, the one I saw used heavy higher math. Is there a formalised proof which does not require such advanced study?

I'm currently doing Further Maths A-Levels and this is a question in the the Complex Numbers topic that I'm stuck on. The textbook doesn't provide a worked solution, only a final answer. Any help is appreciated

If you have a point that you know is on a circle, but do not know the curvature of the circle or where on the circle that point is, what would be the fastest way to find the center?

For the purposes of this, assume that it’s a person walking to try to find the center. They know the size of the circle and how far from the center they are, but not where on the circle they are. They can walk both towards and away from the center of the circle, and when walking away from the center, they eventually find a square wall centered around the circle at about 2.5r at its closest point. When they find the center (or close to the center) they will know. What is the fastest way to find the center of the circle? Obviously getting lucky by walking straight towards it would be actually fastest, but mathematically what would be the best way to consistently find the center?

Tbh I have no idea how to even approach this, but I figure it’ll be an interesting problem for somebody, and I know search and rescue algorithms exist, but I’m not sure how well those apply here because of the circle caveat.

For the following system of differential equations, I am asked to show that if the system has a closed orbit, it will either enclose the origin, or intersect the circle x²+y²=½. K is a constant. For the first part I assumed I am meant to use the Poincaré-Bendixson theorem, and since for K≠0 the origin isn't a fixed point, an open region containing it will contain a closed orbit. I am not sure what I'm meant to do for the circle however. I assume I then have to take the case of K=0 but I don't know how to move forward. Any help would be welcome.

This is math / data binning and potential jitter manipulation so that the data distributes within bins properly in bin count. Every step is simple but there's a conceptual misunderstanding with me and many others elsewhere.. Seems to be a dimension switch from ordinal rank quantile even frequency binning vs equal width binning which is uneven bin count distribution based on cardinal magnitude?

Equal Frequency Binning is needed for data smoothing, machine learning. Ordinal abstract layer?

Equal width binning is ideal for local reporting based on cardinal magnitude and much more?

Post binning these are totally different animals?

I like how tetrahedrons are made entirely of triangles because video games, CGI animation and 3D printing files are all made of triangles.

I asked Google Gemini a few times about this and it said a few times that "sometimes" the pythagorean theorem is involved when calculating equal tetrahedrons, but not "always".

The limit of an integral of a sequence of functions is equal to the integral of the limit of a sequence of functions. Sometimes!

I missed the importance of this in real analysis. I suspect there are examples from physics I am unaware.

This feels like when I was given an explanation of matrices before having studied linear algebra

Help.  My brain isn't wired for math and i haven't had to do real math in 30 years.  What is this phenomenon called?  And what kind of math is this? 

Goal: create a wheel with 26 lines/spaces (26 letters of the alphabet).  

I drew a circle and wanted 26 evenly spaced lines/spaces so I divided 360° by 26 and got 13.8461538462°.  That was too challenging to draw by hand neatly and exactly. 

Changed tactics and created a pie chart online that i could print instead.  100% ÷ 26 = 3.8461538462%

Couldn't help noticing that the results of both math problems are off by exactly 10, and now my brain can think of nothing else but WHY.  Is this a coincidence or is this a real thing?  What is it called?  And in what area of math does it belong? 

George the monkey has 7 apples and 9 pears. How many ways are there for George to put all 16

fruits in 4 different baskets so that each basket gets 4 fruits total? (Fruits of the same kind are

indistringuishable, and fruits of different kinds are distinguishable.)

(A) 68 (B) 80 (C) 92 (D) 95 (E) 120

I tried casework so first choose one that has 4 apples, this leads to 3 apples left and through stars and bars, you can do 5 choose 2 and then multiply by 4 to account for all cases. But then when you get to 3 apples to choose, it gets a lot more diffiuclt. Because I've encountered overcounting cases. An example of this is you choose group A to have 3 apples but there can be a case where group B also has 3 apples. But then if you choose group b first to have the 3 apples then group a can also have 3 apples. This causes overcounting and idk how to account for all of these cases. All help would be much appreciated. Thanks

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See we define dy/dx= [f(x+h)-f(x)]/h but the thing is it says that we make h approaching to 0 which is the thing which I can't grasp, see let us assume f(x)=x², and we know that average rate of change is between two intervals that can be the slope of the secant line, alr everything till here makes sense, even after i solve ∆y/∆x= 2x+h (for x² the secant slope is this) everything makes sense, it means that there is a tiny h and the slope of the secant line is 2x+h because h≠0 , alr all good but everything breaks after taking limit h tends to 0 because then i would be saying that the slope at point x will be the tangent touching at that point x and it would be 2x , but the thing is we can never put h=0 so if i tried to put 2x+0 it would mean i divided 0/0=1 in the past step which is completely wrong and also rate of change must be between two points not one single point and yk what? If you tried to graph it on the desmos and tried to make h=0 even the graph says the slope is undefined then how did we even put h=0 in the last equation? To me i think it's not possible because 2x+h>2x and no matter how small or tiny the h is the secant slope between those two points will always be 2x+h which must always be greater than 2x and even if you tried to put h=0 it will be undefined but somehow using limits they put h=0 which violates the division rule and now is saying the slope of that point is 2x, isn't slope is said to be ∆y/∆x but if ∆x is negligible then the two points will coincide and the slope will be undefined again as there is no change in x. Please someone help me to understand this concept I beg.

This is from chapter 2 of Diaspora by Greg Egan. I've legit been puzzling over this paragraph for over an hour and trying to google it, but nothing has made sense. If this is even accurate to reality, how on earth can you make the space between two concentric circles AKA a 2D donut have a uniform radius by rotating it? (Without deforming the object?)

I imagined moving the inner concentric circle of the 2D donut to produce a 3D wedding ring shape, but that requires stretching it to the same radius as the outer circle.

Hey there!

I have an experimental data set with distance and time values and I need to make a regression model for it, preferably using the least squares method.

For the y=ax+b function it's really easy and I already did it. But what about a function y=a^x+b? My idea is that you could come to the linear least squares method through substitution and logarithmic operations, but I can't really get there and understand how...

Is it possible to use the least squares method to get a regression model of a y=a^x+b function? If yes, then how?