# 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.

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### Tensor product over non-commutative ring.

I already knew the definition of the tensor product of two modules over a commutative ring : It takes two modules over the ring and spits out a module over a ring. I thought this was nice. I was ...
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### Is this a good justification for why integration by substitution works? [duplicate]

It is a well known fact that $$\int f'(g(x))g'(x)dx=f(g(x))+C$$ This follows directly from the chain rule. However, sometimes it is easier to perform the substitution $u=g(x)$: \begin{align} u&=...
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### Does everything in math has to be intuitively explain-able? [closed]

I always try to find an intuitive explanation for everything I study in mathematics. Sometimes, I come across an idea where I wonder if an intuitive explanation is needed for it. Maybe there are some ...
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### An (intuitive) explanation for the emergence of $\pm$ in the expansion of $\cos\Big(\frac{x}{2}\Big)$ in terms of $\cos x$?

For some angle $\alpha$, we have : $$\cos(2\alpha) = 2\cos^2\alpha - 1$$ $$\cos\Big(\dfrac{\alpha}{2}\Big) = \pm\sqrt{\dfrac{\cos\alpha + 1}{2}}$$ I want to gain a deeper and more intuitive ...
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### Intuition issue with independent events.

Take for example 100 students. Each student studies either French, German, French and German, or none. Let's say 40 students study French and 25 study German. If 10 study both then they are ...
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### Conceptually, what is the meaning / the idea of conjugacy? [duplicate]

My background is in geometry, and I have a basic level of understanding of topological conjugacy: It defines an equivalence relation in the space of all continuous surjections of a topological space ...
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### Intuition behind why rotating a point $180^\circ$ about the origin give the same result as reflecting it with respect to the origin

I understand that the outcomes of those two actions could be proven mathematically, and we can see that those results match. I also understand that I can visualize those actions and see how the ...
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### Intuition behind equation for volume of a cone without calculus [duplicate]

I have come back to study geometry a bit and I'm kind of stuck at deriving the volume formula for a cone. I have read the calculus-based derivation and it totally makes sense, but calculus has been ...
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### Given $n$ slots and $k$ objects to fill the slots, what is the probability of a given slot to be filled.

Problem Given: $n$ slots, numbered from $1\ldots n$ $k$ objects a slot can be filled by one object what is the probability that a slot at some position $i$ to be filled? Some Notation For a visual ...
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### Intuitive meaning of null-homotopic map between sphere an torus.

I am given an exercise where I have to show that every continuous map from $S^2$ to $T^2$ is null-homotopic. I don't really understand what is the implication of this result intuitively.