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1500 questions
445
votes
10 answers
My son's Sum of Some is beautiful! But what is the proof or explanation?
My youngest son is in $6$th grade. He likes to play with numbers. Today, he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, and the sum equals a later number in that same…
haugsire
- 3,431
400
votes
33 answers
Pedagogy: How to cure students of the "law of universal linearity"?
One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:
$$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$
$$ 2^{-3}…
Peter LeFanu Lumsdaine
- 9,434
391
votes
34 answers
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
Elementary row operations.
Linear dependence.
Row reduced forms and their…
Dilawar
- 6,125
379
votes
73 answers
'Obvious' theorems that are actually false
It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".
Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).
But…
beep-boop
- 11,595
379
votes
20 answers
Find five positive integers whose reciprocals sum to $1$
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was surprised that a teacher would assign this kind of…
Low Scores
- 4,565
378
votes
15 answers
Can every proof by contradiction also be shown without contradiction?
Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction?
As an aside, how is proving by…
sonicboom
- 9,921
- 13
- 50
- 84
372
votes
23 answers
Zero to the zero power – is $0^0=1$?
Could someone provide me with a good explanation of why $0^0=1$?
My train of thought:
$$x>0\\
0^x=0^{x-0}=\frac{0^x}{0^0}$$
so
$$0^0=\frac{0^x}{0^x}=\,?$$
Possible answers:
$0^0\cdot0^x=1\cdot0^0$, so $0^0=1$
$0^0=\frac{0^x}{0^x}=\frac00$, which is…
Stas
- 3,979
363
votes
118 answers
Collection of surprising identities and equations.
What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.
I'd request to avoid 'standard' / well-known…
Calvin Lin
- 68,864
361
votes
11 answers
What is the importance of eigenvalues/eigenvectors?
What is the importance of eigenvalues/eigenvectors?
Ryan
- 5,509
359
votes
23 answers
Why don't we define "imaginary" numbers for every "impossibility"?
Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such…
lily
- 3,727
357
votes
7 answers
How can you prove that a function has no closed form integral?
In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/subtraction
multiplication/division
raising to powers and…
Simon Nickerson
- 5,611
357
votes
31 answers
Is it true that $0.999999999\ldots=1$?
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
Michael Haren
- 318
355
votes
8 answers
Calculating the length of the paper on a toilet paper roll
Fun with Math time.
My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with small error) the total paper length of a toilet…
Enrico M.
- 26,114
351
votes
0 answers
Limit of sequence of growing matrices
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \\
1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 0 & 1/2\\
0 & 1/2 & 1/2 & 0
\end{array}\right),
$$
$K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right)$ and consider the sequence of matrices defined by
$$
K_L =…
Eckhard
- 7,705
345
votes
36 answers
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
The following is a quote from Surely you're joking, Mr. Feynman. The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally…
AgCl
- 6,292