Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...
233
votes
16answers
41k views
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 ...
38
votes
8answers
4k views
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 ...
35
votes
9answers
2k views
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. ...
33
votes
8answers
1k views
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, ...
29
votes
14answers
2k views
What exactly is infinity?
On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
19
votes
9answers
2k views
How do you define functions for non-mathematicians?
I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
19
votes
3answers
495 views
Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals
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 ...
18
votes
3answers
1k views
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 ...
17
votes
11answers
2k views
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 ...
17
votes
6answers
1k views
Why does the Dedekind Cut work well enough to define the Reals?
I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals.
I just can't get the ...
17
votes
3answers
961 views
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) ...
17
votes
3answers
511 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 ...
16
votes
6answers
1k views
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 ...
16
votes
3answers
1k views
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?
15
votes
6answers
820 views
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.
15
votes
5answers
962 views
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 ...
15
votes
7answers
948 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, ...
15
votes
3answers
605 views
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:
$$ ...
15
votes
3answers
508 views
Generalization of a ring?
I've just started learning about rings. Rings are one additive abelian group strung together, through the associative law, with another structured operation.
Couldn't we continue stringing together ...
14
votes
3answers
948 views
Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...
14
votes
2answers
145 views
Is this an equivalent definition of a normal subgroup?
Let $G$ be a group and $N$ a subgroup. Consider the condition $$(\forall g\in G)(\exists x,y\in G)\ NgN=xNy.\tag1$$
If $N\lhd G$, then for each $g\in G$ we have $NgN=gNN=gN=gN\cdot1$, so the ...
13
votes
5answers
480 views
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. ...
13
votes
2answers
465 views
What is it to be normal?
I'm interested to find out why the word 'normal' crops up so many different areas of mathematics, especially but not exclusively in abstract algebra, and how the definitions are related, if at all.
...
13
votes
5answers
1k views
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 ...
12
votes
2answers
618 views
What's the name for the property of a function $f$ that means $f(f(x))=x$?
I can think of several examples of functions such that twice application of the function is equivalent to no application of it.
Additive inverse
Multiplicative inverse
Fourier transform
Complex ...
12
votes
6answers
360 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 ...
12
votes
2answers
2k views
What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, ...
11
votes
4answers
706 views
$\sqrt 2$ is even? [closed]
Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ ...
11
votes
3answers
261 views
What's wrong with this “backwards” definition of limit?
Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:
$\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if ...
11
votes
4answers
776 views
Which is the “proper” definition of a geodesic curve?
I'm taking a course on differential geometry, and up until now I'd always thought that the definition of a geodesic is (loosely speaking) a curve on a surface with the minimal length between its ...
10
votes
3answers
359 views
What is the purpose of the $\mp$ symbol in mathematical usage?
Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please ...
10
votes
5answers
318 views
what is the definition of $=$?
what is the definition of $=$?
Above is the question that I would like to be answered, below are some of my thoughts.
I've been thinking about what it means to say
$A = B$
I came to this from ...
10
votes
3answers
255 views
It is possible to define our intuitive notion for probability in subsets of $[0,1]$
I've always heard and read the sentence:
If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$.
What is the meaning for that? Is this the "real" ...
10
votes
3answers
922 views
What is a special function?
When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged ...
9
votes
8answers
693 views
9
votes
4answers
188 views
Is the free group on an empty set defined?
I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.
9
votes
3answers
202 views
Why does the definition of limits of a function have strict inequality?
Definition (As written in Michael Spivak's Calculus)
The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon >0$ there is some $\delta > 0$ such that, for all $x$, if ...
9
votes
3answers
828 views
Question about basis and finite dimensional vector space
I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5)
I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
9
votes
2answers
346 views
Does *finitely many* include the option *none*?
Does finitely many include the option none?
Say I have a sequence $(x_n)$ and I want to say that there can only be $0$ or $n\in \mathbb N$ non-zero terms. Can I say that the sequence has finitely ...
9
votes
3answers
480 views
Differentiable at a point
My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
9
votes
2answers
98 views
Why do we use open intervals in most proofs and definitions?
In my class we usually use intervals and balls in many proofs and definitions, but we almost never use closed intervals (for example, in Stokes Theorem, etc). On the other hand, many books use closed ...
9
votes
3answers
5k views
What is the difference between Singular Value and Eigenvalue?
I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is singular value just another name for ...
9
votes
2answers
374 views
Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials
[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below]
in my attempt to find a precise version of the 'definitions' usually given when ...
9
votes
2answers
233 views
Check my workings: Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.
Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.
Precalculations
My goal is to show that for all $\epsilon >0$, there exist a $\delta > 0$, ...
8
votes
4answers
410 views
If a function can only be defined implicitly does it have to be multivalued?
What is the general reason for functions which can only be defined implicitly?
Is this because they are multivalued (in which case they aren't strictly functions at all)?
Is there a proof?
...
8
votes
6answers
404 views
What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?
I thought I'd bring this question to math.SE, as it could spark some interesting discussion.
When I first learned vectors - a long time ago and in high school - the textbook and teachers would always ...
8
votes
5answers
683 views
Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?
Context: Rings.
Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?
Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
8
votes
5answers
397 views
Proving that $a + b = b + a$ for all $a,b \in\mathbb{R}$
Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting ...
8
votes
3answers
326 views
Why is closure omitted in some group definitions?
In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?
8
votes
3answers
1k views
How can an ordered pair be expressed as a set?
My book says
\begin{equation}
(a,b)=\{\{a\},\{a,b\}\}
\end{equation}
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 ...
