# Tagged Questions

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### Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
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### Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
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### What purpose does the use of functions serve in mathematics?

Ok, so I know the overview a bit. I would like to know why one should use one, what they're used for, and maybe even the history behind their purpose. X = X (Y) + 7 ... 5 = 5 (20) + 7 ... 32 is the ...
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### Mean Value Theorem Motivation

I am currently practicing presenting mathematics to various audiences and am considering the example of the mean value theorem. I was wondering how would I be able to motivate this theorem to a ...
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### Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
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### Intuition behind the definition of Adjoint functors

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ \eta : 1_D ...