1 vote
172 views

### Visualising the sum of the first $n$ positive odd integers [duplicate]

Using the fact that $1+2+\cdots+n=\frac{n(n+1)}{2}$, we can deduce that sum of first $n$ positive odd integers is $n^2$. However, is there a way of finding the sum of $1+3+5+\cdots+(2n-1)$ visually?
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### 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 ...
26k 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 ...
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### Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
• 3,945
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### 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.
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### 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 ...
• 2,811
6k 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 ...
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### 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 ...
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1k views

### If sum of triangle angles is $180$ degrees, how $\sin(270)$ is possible?

I'm not new to trigonometry, but this question always bothers me. As it is in Wolfram MathWorld- $$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$ We know that the sum of the angles in a ...
81k 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 ...
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### 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 ...
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### 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 ...
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### Speechless mathematical proofs.

Do you have proofs without word? Your proofs are not necessary has zero word, you may add a bit explanations. As an example, I has a "Speechless proof" for \frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+....
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### Riddle: A special $6$-digit number

Here is a riddle: Riddle: I am thinking about a $6$-digit number $\underline{ }\, \underline{ }\, \underline{ }\, \underline{ }\, \underline{ }\, \underline{ }$ (no leading zeros). All digits ...
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Update $(2020)$ I've observed a possible characterization and a possible parametrization of the pattern, and I've additionally rewritten the entire post with more details and better definitions. It ...