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1500 questions
62
votes
19 answers
How do I convince my students that the choice of variable of integration is irrelevant?
I will be TA this semester for the second course on Calculus, which contains the definite integral.
I have thought this since the time I took this course, so how do I convince my students that for a definite integral
$$\int_a^b f(x)\ dx=\int_a^b…
ireallydonknow
- 1,415
- 2
- 14
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62
votes
9 answers
Intuition behind "ideal"
To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these points:
Why is the name "ideal" coined?. In English…
IDOK
- 5,300
62
votes
24 answers
How would you explain to a 9th grader the negative exponent rule?
Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = \frac{1}{a^x}$ from the pattern method: $2^2 = 4, 2^1 =…
alo
- 321
62
votes
3 answers
What does E mean in 9.0122222900391E-5?
I often find this at the bottom of pages.
Page generated in 0.00013899803161621
Sometimes, I come across
Page generated in 9.0122222900391E-5
What does that time mean?
I tried searching Wikipedia for E and maths but found the e mathematical…
abel
- 739
62
votes
1 answer
Effect of elementary row operations on determinant?
1) Switching two rows or columns causes the determinant to switch sign
2) Adding a multiple of one row to another causes the determinant to remain the same
3) Multiplying a row as a constant results in the determinant scaling by that constant.
Using…
dfg
- 3,891
62
votes
7 answers
$A_4$ has no subgroup of order $6$?
Can a kind algebraist offer an improvement to this sketch of a proof?
Show that $A_4$ has no subgroup of order 6.
Note, $|A_4|= 4!/2 =12$.
Suppose $A_4>H, |H|=6$.
Then $|A_4/H| = [A_4:H]=2$.
So $H \vartriangleleft A_4$ so consider the…
Stephen Cox
- 759
62
votes
14 answers
How do I motivate myself to do math again?
I have been thinking of asking for help for a few months now but posting in a public forum like this is intimidating.
Still, I am currently in a university studying mathematics as an undergrad. I took quite a few knocks a few months back when I…
Dust
- 235
62
votes
6 answers
Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we…
Srivatsan
- 26,311
62
votes
8 answers
Is there a way to get trig functions without a calculator?
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?
Sometimes I don't feel right when I can't do…
Jonathan Lam
- 803
62
votes
12 answers
I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2
I didn't know how to phrase the question properly so I am going to explain how this came about.
I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I am sure that if there was anything that we…
Anonymous
- 701
62
votes
27 answers
What are some conjectures of your own?
Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow), which gather a lot of attention to the site are about informal lists. So, in the theme of, but in contrast…
Graviton
- 4,462
62
votes
16 answers
Rigour in mathematics
Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious.
Can you give me examples of conjectures/theories that seemed true but through rigorous mathematical proving it was shown otherwise?
please delete me
- 1,201
62
votes
4 answers
Are there open problems in Linear Algebra?
I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory.
There are a lot of open problems and conjectures in…
Mebat
- 1,281
62
votes
15 answers
Why does an exponential function eventually get bigger than a quadratic
I have seen the answer to this question and this one.
My $7$th grade son has this question on his homework:
How do you know an exponential expression will eventually be larger than any quadratic expression?
I can explain to him for any particular…
John L
- 1,493
62
votes
5 answers
How do you rotate a vector by a unit quaternion?
Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to rotate the vector from local space to world…
Narf the Mouse
- 793