# Questions tagged [motivation]

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

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55 views

### Are there any real & decent mathematics video games?

Personally, I like to play video games from time to time, especially arcade games (e.g. Tetris, pinball) or fast-paced games like Super Hexagon, which is known to be quite challenging. However, for ...
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### Is there a von Neumann type theorem for the $\sigma$-algebra generated by the set of all the eigenfunctions?

Defintions Let $(X, \mathcal X, \mu, T)$ be a measure preserving system. Let $U_T:L^2_\mu\to L^2_\mu$ be the associated Koopman operator. We will write $\mathcal X_0$ to denote the $\sigma$-algebra ...
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### Jacobian Matrix at a non-singular point

I am currently studying singularities in Algebraic Geometry and wanted to understand why the rank of the Jacobian matrix would characterise a point of singularity/non-singularity (assuming we start ...
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### Series equal to e

I'm having trouble convincing myself why $$\sum_{k = 0}^{\infty} \frac{k}{k!} = e.$$ As I was under the impression that only $$\sum_{k = 0}^\infty \frac{1}{k!} = e$$ by definition. By writing out ...
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### A question on nowhere dense sets.

Consider the $2$ definitions: A set $A$ in a topological space $(X,\tau)$ is said to be a nowhere dense set if it is not dense in any nonempty open set. A Set $A$ in a topological space $(X,\tau)$ ...
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### On the beginnings and motivations of certain branches of set theory

I am planning to give a talk in my university on descriptive set theory, large cardinals and inner model theory. And the target audience are undergraduate students. I am trying to roughly explain what ...
108 views

### What lies behind the definitions of split monics and epics?

Is there an easy way to memorize the definitions of split monics and split epics, and not to confuse the domains/codomains of the arrows from those definitions? For example, is there a mnemonic rule?...
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### How to visualize $(A\cap B)^\mathrm{o}=A^\mathrm{o} \cap B^\mathrm{o}$?

We all know that $(A\cap B)^\mathrm{o}=A^\mathrm{o} \cap B^\mathrm{o}$, where $A,B \subset X$ which is a metric space. The proof is not also difficult, but actually I cannot visualize or feel ...
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### Visualization of a “Not so intuitive” problem of linear algebra.

I recently encountered a problem in Hoffmann-Kunze linear algebra: If $(.,.)$ is the standard inner product on $\mathbb C^2$ then show that $(Tv,v)=0 \forall v\in \mathbb C^2 \implies T=0$, I think ...
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### Why do functional analysts want their spaces to be complete?

Why is one in functional analysis only investigating complete spaces (like Banach or Hilbert spaces)? I heard someone saying that analysts in general like to work with limits, which makes sense. But ...
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### Why Do We Need Initial Conditions to Solve PDEs?

I am looking for further clarity on why solving PDEs without any specified initial values was not "good enough." For example: say we had the ODE \begin{equation} y' = y \end{equation} without ...
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### why commutative integral with limit is important in real analysis? [closed]

why commutative integral with limit is important in real analysis ? Why $\lim_{n\to\infty }\int f_n=\int \lim_{n\to\infty } f_n$ is important ?
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### What motivates the arithmetic-geometric mean?

What motivates the arithmetic-geometric mean? What inspires it? I understand how to calculate this mean but do not understand what might prompt a mathematician to pursue such a mean in the first place....
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### What is a good way to introduce Euler's totient function?

I was thinking of this question and when I googled I couldn't find any MSE results, but I found one from Reddit. I just wanted to ask the question here and post the answer as community wiki just so ...
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### Turán’s Graph Theorem: Motivation behind the Weight Function

If a graph $G = (V, E)$ on $n$ vertices has no $p$-clique, $p \geq 2$, then $$|E| \leq (1- \frac{1}{p-1}) \frac{n^2}{2} \;\;\; \;\;\; \;\;\;(1)$$ We get a proof from the book "Proofs from THE BOOK" as ...
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### What are the applications of the Mean Value Theorem?

I'm going through my first year of teaching AP Calculus. One of the things I like to do is to impress upon my students why the topics I introduce are interesting and relevant to the big picture of ...
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### An idea to teach equation except balance model

I looking for an idea to start teaching the equation of algebra, but not use the balance model. I am looking for a new motivational idea or a pedagogical method to start teaching equations. The link ...
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### Topic for a math talk [closed]

I have to make a talk about mathematics for first and second year undergraduate students of maths. If someone could help me with a topic or an idea, it would be helpful. Preferably it is something ...
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### Why and how algebraic structures emerged in mathematics? [closed]

For example, why we study Group Theory to prove general results for all instead of specifically studying $\mathbb Z$ (or any other set) closed under some operation? What makes algebra and those ...
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### What are the application of universal property of subspace topology?

I come across following theorem: Universal property of subspace topology: $X$ is any topological space $Y\subset X$ $Z$ is any another topological space if there is continuous map $g:Z\to X$ such ...
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### The idea of a bundle chart and bundle atlas.

The definition of a bundle charts and bundle atlas is rather obscure in my opinion. Is it fair to say that: 1) the purpose of a bundle chart is to give coordinates to each tangent space? 2) the ...
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### Motivation behind point set topology [closed]

Why should I study point-set topology? What initially interested me in topology was the pop-sci rubber sheet stuff or coffee cup-donut stuff or proving fundamental theorem of algebra using curves but ...
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### Kernel and image of matrix: What are they? Why do they exist?

I've been trying to get an understanding of the Kernel of image of matrices. I'm studying them in college right now, but the problem is, while I can find a ton of resources on how to find them given a ...
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### Motivation of the Logarithm

Suppose that someone wants to calculate approximately the product of 101,123,958,959,055 and 342,234,234,234,236 without using a computer. Since these numbers are so long, completely carrying out ...
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### Interpretation of Fourier Transformation - what is it?

What exactly is Fourier Transformation? For functions on the Schwartz Space $S(\Bbb R^n)$, we may define, $$\hat{u}(\xi) := \int e^{-ix\xi} u(x) \, dx$$ This formula seems to come out of no where ...