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1500 questions
138
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
- 5,556
137
votes
4 answers
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$?
This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I am unsure how they are relevant.
jaynp
- 2,151
136
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…
Hrothgar
- 639
136
votes
7 answers
Construction of a Borel set with positive but not full measure in each interval
I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere.
To be precise, if $\mu$ denotes Lebesgue measure, how would one construct a Borel set $A…
user1736
- 8,573
136
votes
16 answers
What are good books to learn graph theory?
What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory as part of a combinatorics course, and would like…
Dom
- 523
136
votes
16 answers
Why did mathematicians take Russell's paradox seriously?
Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most of them mark its formulation as an epochal moment…
Uticensis
- 3,311
136
votes
22 answers
Good Book On Combinatorics
What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a…
Anonymous
- 281
135
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
135
votes
6 answers
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$, but in $\mathbb H$, ${\bf i}\cos x + {\bf…
MJD
- 65,394
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- 580
134
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…
Noldorin
- 6,610
134
votes
28 answers
What are some examples of notation that really improved mathematics?
I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational conventions that simplify problem statements and ideas,…
Patrick
- 2,106
134
votes
24 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\\…
Ethan
- 1,459
134
votes
10 answers
what is expected from a PhD student?
As a PhD student in applied mathematics or mathematics in general, are you expected to be able to prove every problem, for example, in an elementary real analysis book? I know it sounds silly but I am wondering if I have high expectations of…
Tomas Jorovic
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- 3
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- 38
134
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
- 1,505
134
votes
16 answers
Division by zero
I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.
$\dfrac {x}{0}$ is Impossible ( If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are…
danielsyn
- 1,577