What is a number?
Solution 1:
It depends on the context. The word number could for example mean integer, rational number, real number or complex number, all of which have precise definitions. In some situations it could even mean something like "an element in some particular ring or field", again this is well-defined. There are lots of other "number systems" in use as well, and it's probably impossible to list all of them.
The problem with a dictionary definition is that they don't build the language using undefined terms, axioms and definitions.
Solution 2:
What is a number? That is a good question.
If you ask the somewhat educated layman, you might get answers such as "a representation of a physical quantity". You may get some other answers too.
But since mathematicians use words from natural language with a particular meaning, let's cut the foreplay, and skip right down to the mathematician part. However there is no agreed, or even common, definition for "number". The definition I have in mind, and I suspect that many mathematicians would agree with me, is the following one:
We say that $x$ is a number, if it is an element of a number system, which is a system representing and measuring a quantity of some form.
This definition allows for natural numbers, integers, rational numbers, real numbers, complex numbers, ordinals, cardinals, and so on. All these are number system. The only thing they have in common is that they measure some sort of quantity, and they represent it somehow.
Therefore, for me, the context "X numbers" means that "X" is some sort of form of measurement for mathematical objects.
Solution 3:
The word "number" means what I wish it to mean, nothing more, nothing less.
Depending upon what I'm doing, things I've wished "number" to mean have included:
- Natural number
- Integer
- Rational number
- Real number
- Projective real number
- Extended real number
- Complex number
- Projective complex number
- Hyperreal number (and the nonstandard versions of all of the above)
- Continuous real-valued function
- Ordinal number
- Cardinal number
- Polynomial over a finite field
- Rational function over a finite field
- Element of a particular ring I'm working with
- Abelian group
Solution 4:
In set theory, we begin by defining $0$ as the number of elements in the set having zero unique elements. Then, for each subsequent number that we wish to define, we say that $i+1$ is the number of elements in the union of the $i$th set and the set containing the $i$th set. If we label this series of sets as $u_i$, then we have $u_0=\varnothing$, $u_{i+1}=u_i\cup \{u_i\}$. Then our first few "numbers" are as follows:
$$u_0=\varnothing$$
$$u_1=\{\varnothing\}$$
$$u_2=\{\varnothing,\{\varnothing\}\}$$
$$u_3=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$$
...
This is rather like getting something from nothing, but it definitely gives uniqueness to each integer.