# Tagged Questions

For requesting, clarifying, and comparing definitions of mathematical terms.

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### What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
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### Is infinity a number?

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school &#...
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### Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
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### Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
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### What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
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### What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
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### Why do we not have to prove definitions?

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
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### Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
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### What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition. My ...
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### Given real numbers: define integers?

I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following: Integer numbers are just special cases (a subset) of real numbers. ...
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### difference between class, set , family and collection

In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
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### Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
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### Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
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### Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
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### What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
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### Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
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### Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) limit....
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### Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
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### Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?

I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is ...
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### Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: Commutativity ...
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### Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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### Definition of “simplify”

In mathematics the word "simplify" is used a lot. In a lot of cases it is obvious what actually makes an expression simpler, but not always. Is there a measurable definition of simplicity or is it ...
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### How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than x"...
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### GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also:  \gcd\left(\frac{13}{...
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### How can an ordered pair be expressed as a set?

My book says $$(a,b)=\{\{a\},\{a,b\}\}$$ I have been staring at this for a bit and it is not making since to me. I have read several others posts on this, but none made ...
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### Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
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### Is some thing wrong with the epsilon-delta definition of limit??

In the epsilon-delta definition of limit which is: For all $\epsilon>0$ there exists a $\delta>0$ such that, whenever $|x-a|<\delta$ then $|f(x)-L|<\epsilon$ . Now since $\epsilon$ ...
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### What is a negative number?

I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...
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### Proof that the empty set is a relation

In the book Naive Set Theory, Halmos mentions that the "The least exciting relation is the empty one." and proves that the empty set is a set of ordered pairs because there is no element of the empty ...
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### Is a line parallel with itself?

Simple Question, but I'm finding a lot of dispute on the "lesser" internet. Basically, given a line, is it parallel with itself?
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### Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have $\mathbf{F}(\... 5answers 1k views ###$\epsilon, \delta$…So what? Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using$\epsilon$,$\delta$definitions" or "by this we can omit those ... 2answers 294 views ### To whom do we owe this construction of angles and trigonometry? I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ... 7answers 3k views ### What is combinatorics? I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is. Could anyone give me a definition/explanation of combinatorics, ... 3answers 693 views ### Do you know of any Calculus text defining the exponential and logarithm functions in an alternative way? The story in nearly every introductory Calculus book is well known by everybody: you don't have the "right" to raise a number to an irrational power, so forget exponents for now and let's take a look ... 4answers 1k views ### Is this alternative notion of continuity in metric spaces weaker than, or equivalent to the usual one? I will try to be as clear as possible. For simplicity I will assume that the function$f$for which we define continuity at some point is real function of a real variable$f: \mathbb R \to \mathbb R$... 6answers 2k views ### How exactly can't$\delta$depend on$x$in the definition of uniform continuity? I'm told that a function defined on an interval$[a,b]$or$(a,b)$is uniformly continuous if for each$\epsilon\gt 0$there exists a$\delta\gt 0$such that$|x-t|\lt \delta$implies that$|f(x)-f(t)|...
Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty set,...