# what is the definition of numbers? [duplicate]

Well the question may seem obvious but I can't really find a proper answer to this. Mathematics all seem kind of difficult to understand so please help. I believe it is a quantity. Thanks in advance.i am mainly confused about using numbers and using numbers with units.the second one makes sense but i cant understand the first one

## marked as duplicate by Thomas Andrews, mweiss, Simon S, JMoravitz, user147263 Jun 29 '15 at 19:44

• The question in the following link is a little different, but I think the answer is spectacular, and also answers your question. math.stackexchange.com/questions/199676/… – Alex S Jun 29 '15 at 16:53
• In general, the term "number" is used fairly loosely in mathematics, unfortunately, so there isn't a general rule. – Thomas Andrews Jun 29 '15 at 16:57
• In my opinion, "numbers" can be added, subtracted, multiplied and divided; roughly speaking, numbers is elements of a some field. – Michael Galuza Jun 29 '15 at 17:12

Perhaps rather strangely, there is not a simple standard definition of "number".

In the third century BC, Euclid took "number" to mean one of $2,3,4,5,\ldots$, a sequence starting with $2$ and closed under the operation of adding a unit, i.e. if $n$ is a number, then so is $n+1$, so the sequence continues infinitely. Euclid did not consider the "unit" to be a number. That is consistent at least with the meaning of the expression "A number of students asked this question." One wouldn't say that if just one student had done so.

Euclid proved a number of propositions about numbers, including things about divisibility. He introduced what is now the oldest algorithm still in standard use, his algorithm for finding the largest divisor that two numbers share in common. He proved that the list of prime numbers keeps going no matter how far it is extended. (His proof was that if $S$ is any set of prime numbers then $1+\prod S$ has at least one prime factor other than the prime numbers that are in $S$. In fact every prime factor of that number fails to be in $S$.)

Today "numbers" are taken to include $0$ and $1$ and negative integers $-1,-2,-3,\ldots$ and rational numbers, e.g. $10/7$, that result from dividing one integer by another, and "real" numbers that are not rational. "Real" numbers are identified with points on what you may have heard called the "number line". (In a sense Euclid dealt with irrational numbers and in another sense he didn't. He showed that no pair of what he called "numbers" had the same ratio as the ratio of the length of the diagonal of a square to the length of the side. But he didn't consider such ratios to be "numbers".
("$\alpha\rho\iota\theta\mu\omicron\sigma$" is the Greek word used by Euclid, usually (or always?) translated as "number".))

Conventionally "numbers" are taken to include "imaginary" numbers and "complex" numbers. A complex number is a sum of a real number and an imaginary number.

The concept of "number" is sometimes taken to include cardinal numbers of infinite sets, and sometimes taken to include "nonstandard" real numbers, some of which are infinitely large or infinitely small. Occasionally, someone considers quaternions to be "numbers", and that's really only a question of how words should be conventionally used.

Formally, we would describe numbers as "mathematical objects used to count, measure or label". However, this definition seems to apply more to real numbers, that is numbers we use in everyday life (2, 5, $\pi$ ...). There exist certain quantities, which we call numbers, that cannot be assigned to a measure or label in real life. Therefore, depending on how exact you want to be, different definitions will be applicable. Here are some I have found, and the link corresponding to them.

"Mathematical objects used to count, measure or label" - wikipedia

"an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification." - New Oxford American Dictionary