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1500 questions
141
votes
46 answers
What are some examples of a mathematical result being counterintuitive?
As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition. My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite…
Steven-Owen
- 5,806
140
votes
24 answers
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 wrong. (e.g. missing square puzzle ) Do you know the…
puzzlet
- 785
140
votes
11 answers
Continuity of the roots of a polynomial in terms of its coefficients
It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed degree n (maybe monic? seems like that shouldn't…
Harry Altman
- 5,053
140
votes
26 answers
In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?
I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: $$\begin{array}{|c|c|c|} \hline p&q&p\Rightarrow q\\ \hline T&T&T\\ T&F&F\\…
Ethan
- 1,519
140
votes
11 answers
What is the difference between regression and classification?
What is the difference between regression and classification, when we try to generate output for a training data set $x$ ?
Bober02
- 2,604
139
votes
10 answers
Learning Lambda Calculus
What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus? Specifically, I am interested in the following areas: Untyped lambda calculus Simply-typed lambda calculus Other typed lambda calculi…
Noldorin
- 6,818
139
votes
5 answers
Getting better at proofs
So, I don't like proofs. To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert. Oftentimes the proof in the book is something that I get if I study, but hard to come up with on my own.…
bobobobo
- 9,822
139
votes
2 answers
How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$
How can one prove that $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
larry
- 1,549
139
votes
14 answers
How do I get the square root of a complex number?
If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
Macha
- 1,583
138
votes
8 answers
What is the difference between independent and mutually exclusive events?
Two events are mutually exclusive if they can't both happen. Independent events are events where knowledge of the probability of one doesn't change the probability of the other. Are these definitions correct? If possible, please give more than one…
Adnan Ali
- 1,553
137
votes
8 answers
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we define it to be $0$. But why characteristic zero? Why…
Srivatsan
- 26,871
137
votes
35 answers
Examples of mathematical results discovered "late"
What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple that it's surprising no one thought of it sooner.…
Hrothgar
- 639
137
votes
7 answers
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some…
AlexBrand
- 345
137
votes
3 answers
Expected time to roll all $1$ through $6$ on a die
What is the average number of times it would it take to roll a fair $6$ -sided die and get all numbers on the die? The order in which the numbers appear does not matter. I had this questions explained to me by a professor (not math professor), but…
eternalmatt
- 1,535
137
votes
9 answers
Is there an integral that proves $\pi > 333/106$?
The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves $\pi > 333/106$?
anon