# Linked Questions

164answers
49k views

### What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) [closed]

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament,...
26answers
25k views

### Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
34answers
17k views

### Easy math proofs or visual examples to make high school students enthusiastic about math [closed]

I'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or ...
15answers
12k views

### Intuitive understanding of the derivatives of $\sin x$ and $\cos x$

One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) ...
17answers
3k views

### Irrational numbers in reality

I have a square tile which measures 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another will be $\sqrt 2$ metres. However $\sqrt 2$ is an irrational number, could ...
4answers
5k views

### What is the explanation for this visual proof of the sum of squares?

Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
5answers
5k views

### What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of ...
20answers
475 views

### Proving $x^2+x+1\gt0$

I was doing a question recently, and it came down to proving that $x^2+x+1\gt0$. There are of course many different methods for proving it, and I want to ask the people here for as many ways as you ...
5answers
5k views

### Pythagorean Theorem Proof Without Words 6

Your punishment for awarding me a "Nice Question" badge for my last question is that I'm going to post another one from Proofs without Words. How does the attached figure prove the Pythagorean ...
7answers
50k views

### Why $\cos(-\theta)$ gives positive values while in case of sine it is negative?

Why $\cos(-\theta)$ gives positive values while in case of sine it is negative? I mean $\cos(-\theta) = +\cos(\theta)$ $\sin(-\theta) = -\sin(\theta)$ $\tan(-\theta) = -\tan(\theta)$ and please ...
4answers
598 views

### What areas of math can be tackled by artificial intelligence?

Artificial intelligence is nearing, with image/speech recognition, chess/go engines etc. My question is, what areas of math that are interesting to mathematicians, is likely to be the first to be able ...
5answers
3k views

### Why do some series converge and others diverge?

Why do some series converge and others diverge; what is the intuition behind this? For example, why does the harmonic series diverge, but the series concerning the Basel Problem converges? To ...
1answer
2k views

### Textbooks for visual learners

Perhaps this question has already been asked (if so, please let me know) but I am looking for books that appeal to visual learners. I discovered that I am able to understand concepts much quicker ...
2answers
748 views

### If square waves are square integrable, why doesn't fourier expanding work?

If square waves are square integrable, then why does expanding on a fourier basis not recover the equation?
5answers
3k views

### Why is cosine a sine function with offset pi/2?

I was looking at this question and stumbled across this answer stating that the picture proves "Why cosine is simply sine but offset by pi/2 radians" I realized I don't know why this is true. So! ...

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