Mathematics Stack Exchange
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Questions tagged [definition]

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

Definitions are at the core of mathematical precision; they answer the question "What is X?" in mathematics. Into this category fit questions regarding equivalence of definitions, clarification of complicated definitions, or proposed new definitions for mathematical notions, with requests for improvements or comments.

8139 questions
523
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22 answers

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 that has something to do with the square root of $-1$. When I tried to calculate the square root of $-1$ on my…
Sachin Kainth
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195
votes
7 answers

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 eigenvalue?
Ramon
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174
votes
19 answers

Mathematical ideas that took long to define rigorously

It often happens in mathematics that the answer to a problem is "known" long before anybody knows how to prove it. (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be…
Yly
  • 15,791
170
votes
19 answers

What actually is a polynomial?

I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial. I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more…
Travis
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164
votes
19 answers

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, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do cross product and dot product using two points instead…
6609081
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151
votes
15 answers

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 means "iff". For example I am reading "Essential…
fiftyeight
  • 2,757
139
votes
10 answers

What's the difference between theorem, lemma and corollary?

Can anybody explain me what is the basic difference between theorem, lemma and corollary? We have been using it for a long time but I never paid any attention. I am just curious to know.
HPS
  • 4,706
127
votes
7 answers

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 — but a difficult one to answer in an intelligent manner.…
Pops
  • 1,405
120
votes
2 answers

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby undermined the entire answer. However, it can be…
goblin GONE
  • 69,385
116
votes
3 answers

What are the differences between class, set, family, and collection?

In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about classes. What is their relation? Some background…
Asinomás
  • 107,565
110
votes
8 answers

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, logarithmic, exponential, and $n$th roots, and solving…
user23784
105
votes
8 answers

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 tedious details concerning duality. Suppose that $U:…
Hans-Peter Stricker
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101
votes
7 answers

Is zero positive or negative?

Follow up to this question. Is $0$ a positive number?
user8190
98
votes
7 answers

Where are the axioms?

It is said that our current basis for mathematics are the ZFC-axioms. Question: Where are these axioms in our mathematics? When do we use them? I have now studied math for a year, and have yet to run into a single one of these ZFC axioms. How can…
Imean
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95
votes
11 answers

Why not include as a requirement that all functions must be continuous to be differentiable?

Theorem: Suppose that $f : A \to \mathbb{R}$ where $A \subseteq \mathbb{R}$. If $f$ is differentiable at $x \in A$, then $f$ is continuous at $x$. This theorem is equivalent (by the contrapositive) to the result that if $f$ is not continuous at…
Perturbative
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