ncmathsadist
  • Member for 11 years, 2 months
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Examples of patterns that eventually fail
103 votes

I am kind of partial to the old $n^2 + n + 41$ chestnut, namely that the expression is prime for all $n$. It fools an awful lot of people.

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Surprising identities / equations
81 votes

Here is a mathematical scherzo. $$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3.$$

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Why is $L^{\infty}$ not separable?
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78 votes

To be separable means to have a countable dense subset. Suppose that $(M, d)$ is a metric space and that $U\subseteq M$ be an uncountable subset and $r > 0$. Suppose that for all $x \neq y \in U$...

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Is there a mathematical symbol for "For every element"?
61 votes

This should work. $\forall x\in A\cdots$

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Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.
54 votes

The humps for $x\mapsto \sin(x^2)$ go up and down. Each has an area smaller than that of the last. The areas converge to 0 as you progress down the $x$-axis. By the alternating series test, this ...

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What is the purpose of the first test in an inductive proof?
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53 votes

Imagine a pond with an infinite linear progression of lily pads. You have a frog who, if he hops on one pad, he is guaranteed to hop on the next one. If he hops on the first pad, he'll visit them all....

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Your favourite application of the Baire Category Theorem
52 votes

The uniform boundedness principle of Functional Analysis is a very important application of the Baire Category Theorem. Added: (t.b.) See also Sokal's A really simple elementary proof of the uniform ...

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Prime factorization of 1
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43 votes

Let us remember that an empty product is always 1. Hence, 1 has the empty product as its prime factorization. This product is vacuously a unique product of primes.

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Interesting "real life" applications of serious theorems
41 votes

In a cup of coffee, one molecule of coffee is in its original location, even though the contents are undergoing convection. This is the Brouwer fixed point theorem.

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Why $0!$ is equal to $1$?
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41 votes

Here are two resons. We can define $n!$ to be the number of rearrangements of $n$ distinct objects in a list. The empty list has one rearrangement: itself. We can define $n!$ as the product of all ...

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Changing a number between arbitrary bases
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33 votes

Let $x$ be a number. Then if $b$ is any base, $x \% b$ ($x$ mod $b$) is the last digit of $x$'s base-$b$ representation. Now integer-divide $x$ by $b$ to amputate the last digit. Repeat and this ...

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When do we get extraneous roots?
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30 votes

Suppose you have two expressions $e_1$ and $e_2$ and you know $$e_1 = e_2.$$ Then, if you apply a function to both sides, you have $$f(e_1) = f(e_2).$$ However, this logic in general does not reverse,...

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Piecewise functions: Got an example of a real world piecewise function?
26 votes

Computing income taxes in a bracketed system.

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Summation Theorem how to get formula for exponent greater than 3
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23 votes

You can obtain these formulae recursively. Look at this $$n^5 = \sum_{k=1}^n \left( k^5 - (k-1)^5\right) $$ Expand the second term. Cancel the $k^5$ terms. Then apply the identities above. You ...

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Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$
21 votes

Look at the arithmetic operations and their actions. With + and *, these matrices form a field. And we have the isomorphism $$a + ib \mapsto \left[\matrix{a&-b\cr b &a}\right].$$

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$\sqrt{x}$ isn't Lipschitz function
20 votes

You have $${\sqrt{1/n} - \sqrt{0}\over{1/n - 0}} = {1/\sqrt{n}\over {1\over n}} = \sqrt{n}.$$ This ratio can be made as large as you like by choosing $n$ large. Therefore the square-root function ...

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Homomorphisms between fields are injective.
18 votes

A field homomorphism must in particular be a ring homomorphism, so its kernel is an ideal. The only ideals of a field are the zero ideal and the field itself.

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Inequality with (1-x) as denominator
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18 votes

Divide this into cases. The expression is not defined if $x = 1$. If $x > 1$, you can multiply both sides by $x - 1$ to get $1 >0$ So, if $x > 1$ the inequality is satisfied. If $x < ...

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Pseudo-Cauchy sequence
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18 votes

Take $$a_n = \sum_{k=1}^n {1\over k}.$$

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Does Bounded Covergence Theorem hold for Riemann integral?
Accepted answer
18 votes

No. Enumerate the rationals in [0,1] with the sequence $\{r_n\}_{n=1}^\infty$. Now define $f_n(x)$ by $f_n(x) = 1$ if $x = r_k$ for some $1\le k \le n$ and 0 otherwise. For all $n$, we have $$\...

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The square of an integer is congruent to 0 or 1 mod 4
18 votes

Suppose the integer $z$ is even. Write it as $z = 2n$, where $n\in\mathbb{Z}$. Then $z^2 = 4n^2$; $z$ is divisible by 4. Suppose the integer $z$ is odd. Write it as $z = 2n + 1$ where $n\in\mathbb{Z}$...

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Continuous bijection from $(0,1)$ to $[0,1]$
18 votes

Suppose that $f:(0,1) \rightarrow [0,1]$ is 1-1 and continuous. By the intermediate value theorem, the image of any interval under $f$ is an interval. Since $f$ is 1-1, it is either (strictly) ...

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Is $a^{\ln b} = b^{\ln a}$?
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17 votes

Do this: $$a^{\ln(b)} = e^{\ln(a)\ln(b)} = b^{\ln(a)}.$$

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Is a line parallel with itself?
17 votes

I editorialize here, but I think it is useful for parallelism to be an equivalence relation. Hence, a line should be parallel to itself.

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Negative 1 to the power of Infinity
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16 votes

This does not exist. If you have a sequence $\{x_n\}$, then if $x_n \rightarrow l$, for any open interval $I$ with $l\in I$, $x_n\in I$ for all but finitely many $n$. Intuitively, any open interval ...

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How to find a random axis or unit vector in 3D?
16 votes

You can also do this. Generate three random numbers $(a,b,c)$ in $[-1,1]$; if $a^2 + b^2 + c^2\le 1$, then normalize them. Otherwise try again and pick triplets until you have a usable triplet. The ...

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Show that $\mathbb{R}/\mathbb{Z}$ is isomorphic to $\{e^{i\theta} : 0 \le \theta \le 2\pi \}$
15 votes

The map $t\mapsto e^{2\pi it}$ is a homomorphism from $\mathbb{R}$ onto the circle group. Its kernel is $\mathbb{Z}$. Invoke the first isomorphism theorem.

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Distinct permutations of the word "toffee"
15 votes

Here is another way to think about it. You have the word TOFFEE and six blanks - - - - - - you want to place the letters in. Begin with the E. Choose two of the six blanks and pop the Es in. This ...

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Your favourite application of the Baire Category Theorem
15 votes

The open mapping theorem and closed graph theorem of functional analysis are two vital applications.

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Is it unlikely to get the same number of heads/tails?
Accepted answer
14 votes

It is fairly unlikely if $n$ is large. The probability of $n$ heads and $n$ tails is $${2n\choose n} {1\over 4^n}.$$ We have $${2n\choose n} = {(2n)!\over n!n!}.$$ Stirling's formula states that as ...

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