Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [intuition]

Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

To have intuition about a mathematical truth is to have some insight into why it is true, and to understand the motivation for talking about that truth in the first place. This is usually stated in contrast with merely having a superficial knowledge of a mathematical truth as a fact, or only having skills at applying a mathematical truth to solve a problem without having the conceptual understanding of solution.

For a nice explanation of mathematical intuition with examples, and links to other articles on developing mathematical understanding, see Developing Your Intuition For Math on BetterExplained.com.

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What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a…
Jamie Banks
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Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ , coming from the Gamma function. I have a mathematical…
Qiaochu Yuan
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Please explain the intuition behind the dual problem in optimization.

I've studied convex optimization pretty carefully, but don't feel that I have yet "grokked" the dual problem. Here are some questions I would like to understand more deeply/clearly/simply: How would somebody think of the dual problem? What…
littleO
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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of the guys in my office, and despite a very shady…
Asaf Karagila
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Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra. I was wondering if anyone knew a…
WWright
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What are some counter-intuitive results in mathematics that involve only finite objects?

There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem counter-intuitive is our intuition not working…
Burak
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Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from…
FireGarden
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Counterintuitive examples in probability

I want to teach a short course in probability and I am looking for some counter-intuitive examples in probability. I am mainly interested in the problems whose results seem to be obviously false while they are not. I already found some things. For…
MR_BD
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How to intuitively understand eigenvalue and eigenvector?

I’m learning multivariate analysis and I have learnt linear algebra for two semesters when I was a freshman. Eigenvalues and eigenvectors are easy to calculate and the concept is not difficult to understand. I found that there are many applications…
Jill Clover
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What Does it Really Mean to Have Different Kinds of Infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help…
Allain Lalonde
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What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 \text{ cm}^2$ $C^{2} = 74 \text{ cm}^2$ ${x} =…
user123399
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Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it in terms of repetitive addition. What is the…
Happy Mittal
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Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof on Wikipedia and I understand the proof, but I don't "get it". Can someone help me out with this ? I find it hard to wrap my head around…
hari_sree
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Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and…
Emi Matro
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Intuitive interpretation of the Laplacian Operator

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian Operator (a.k.a. divergence of gradient)?
koletenbert
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