Linked Questions

1
vote
1answer
115 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?
131
votes
25answers
27k 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 ...
89
votes
34answers
22k 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 ...
77
votes
3answers
15k views

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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
54
votes
4answers
6k 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.
7
votes
21answers
586 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 ...
15
votes
6answers
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 ...
35
votes
3answers
3k 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 ...
11
votes
5answers
779 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 ...
2
votes
7answers
68k 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 ...
19
votes
4answers
960 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 ...
6
votes
5answers
5k 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 ...
7
votes
3answers
3k views

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 ...
51
votes
1answer
2k views

Pattern “inside” prime numbers

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 ...
8
votes
5answers
3k views

How similar are circular and parabolic paths?

I'd for a long time thought of parabolas as semi-circles. However, if you take half of a circle the ends will - look - parallel, where as parabolas continue to extend horizontally and infinitely. ...

15 30 50 per page