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1500 questions
59
votes
5 answers
Expressing the determinant of a sum of two matrices?
Can $\det(A + B)$ expressed in terms of $\det(A), \det(B), n$
where $A,B$ are $n\times n$ matrices?
I made the edit to allow $n$ to be factored in.
Sidharth Ghoshal
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59
votes
5 answers
Every nonzero element in a finite ring is either a unit or a zero divisor
Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
rupa
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59
votes
2 answers
How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$
Nice Question:
let $x\in [0,2\pi]$, show that:
$$\sin{\sin{\sin{\sin{x}}}}\le\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}?$$
I know this follow famous problem(1995 Russia Mathematical…
math110
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59
votes
3 answers
What is the importance of Calculus in today's Mathematics?
For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a minimum, in most Universities worldwide?
Added…
Américo Tavares
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59
votes
3 answers
Why is it worth spending time on type theory?
Looking around there are three candidates for "foundations of mathematics":
set theory
category theory
type theory
There is a seminal paper relating these three topics:
From Sets to Types to Categories to Sets by Steve Awodey
But at this forum…
Hans-Peter Stricker
- 17,751
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- 128
59
votes
7 answers
How to tell if I'm good enough for graduate school?
I'm concerned about my level of preparation for graduate school. I have decent grades (3.64 GPA, 4.0 in math), great recommendations lined up, and some OK research experience (no publications), but I really don't feel like I'm ready for the next…
Sven
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59
votes
8 answers
Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?
The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot give rise to contradictions, and this bothers…
FireGarden
- 5,687
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59
votes
17 answers
Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
One of the first things ever taught in a differential calculus class:
The derivative of $\sin x$ is $\cos x$.
The derivative of $\cos x$ is $-\sin x$.
This leads to a rather neat (and convenient?) chain of…
Justin L.
- 14,054
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59
votes
12 answers
Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?
Are there mathematical concepts that exist in the fourth dimension, but not in the third dimension? Of course, mathematical concepts include geometrical concepts, but I don't mean to say geometrical concept exclusively. I am not a mathematician and…
jojafett
- 659
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- 4
59
votes
3 answers
Why is the localization at a prime ideal a local ring?
I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is the localization of the ring $ A $ with respect to…
Bryan
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59
votes
6 answers
What is the difference between homotopy and homeomorphism?
What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? Otherwise, is there any counterexample?…
liufu
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59
votes
15 answers
Unexpected use of topology in proofs
One day I was reading an article on the infinitude of prime numbers in the Proof Wiki. The article introduced a proof that used only topology to prove the infinitude of primes, and I found it very interesting and satisfying. I'm wondering, if there…
Miksu
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59
votes
8 answers
How can I find the points at which two circles intersect?
Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
Joe Elder
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59
votes
10 answers
Why does the discriminant in the Quadratic Formula reveal the number of real solutions?
Why does the discriminant in the quadratic formula reveal the number of real solutions to a quadratic equation?
That is, we have one real solution if
$$b^2 -4ac = 0,$$
we have two real solutions if
$$b^2 -4ac > 0,$$
and we have no real solutions…
user487950
59
votes
4 answers
Why can we use induction when studying metamathematics?
In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition…
183orbco3
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