# Tagged Questions

For questions about the motivation behind mathematical concepts and results. These are often "why" questions.

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### What problems, ideas or questions first got you interested in algebraic geometry?

Obviously, a lot of people are very interested in algebraic geometry. I suppose this means it is a fascinating area. However the few times I have tried to read introductory books and/or articles in ...
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### What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?

The definition of the Lebesgue measure on $\mathbb R^n$ is fundamentally tied up with the following assumption: The measure of the cartesian product of $n$ intervals should be the product of the ...
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### What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
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### Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
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### Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
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### Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
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Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, I'...
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### Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
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### Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
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### Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0$ and $\mathfrak{c}$? It seems ...
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### Motivation of vector bundle of a manifold

I am studying about vector bundle from M.Lee but not getting the feel of it. Can someone explain me about the importance of vector bundle? Why do we need to study about it? Thanks! Also, i have to ...
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### Why do we require $X$ to be Hausdorff when defining $C(X)$?

In Efton Park's "Complex Topological K-theory", he begins a section by defining the Banach algebra $C(X)$ (continuous functions $f: X \rightarrow \Bbb C$), presupposing that $X$ is Hausdorff and ...
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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 do mathematicians sometimes assume famous conjectures in their research?

I will use a specific example, but I mean in general. I went to a number theory conference and I saw one thing that surprised me: Nearly half the talks began with "Assuming the generalized Riemann ...
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### Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
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### Motivation for construction of cross-product (Quaternions?)

I'm trying to present a narrative that brings the (3D) Cross Product into existence. "Given two vectors $\mathbf u$, $\mathbf v$, how to construct a vector perpendicular to both?" ... looks like ...
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### Why were limits introduced in calculus? [closed]

I have got the intuitive of the limits but why were they introduced in calculus, are they really helpful to us?
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### Practical motivation for analysis at school

I want to show pupils at an age of 14 to 17 and who do not like math what they can use analysis/analytic geometry in a job. I could not remember an impressive example for most topics. Let's assume ...
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### What does this theorem mean?

Let $(V,\|\cdot\|)$ be a finite-dimensional normed space. Define $\|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}$, for all linear operators on $V$ Define $\Omega$ to be the set of all invertible linear ...
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### Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
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### What is the importance of the spectral theorem?

I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that ...
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### For what reason, the surface measure represents the surface area?

Let $||\cdot||$ be a norm on $\mathbb{R}^n$ (It's an arbitrary norm, not 2-norm) Define $S^{n-1}=\{x\in\mathbb{R}^n : ||x||=1\}$ Let $\mu$ be the n-dimensional Lebesgue measure. Define \$\sigma(E)=n\...