What is a number in math?

Question

Before I begin, let me give you so background. I previously asked a question on "How to prove that −x is not equal to x just because they yield the same result when in $x^2$". This got me thinking. What is a number in math anyway?

For instance, aside from the fact that 5 looks the same as 5, how do we know that 5=5? How do we know that 2 numbers are equal, and 2 numbers are different?

A can't be because they give the same input when plugged into a function. If this definition is true, then x and -x would be the same.

So can some tell me what is the definition of a number in a math? And please, can you also use that definition to prove/show that −x is not equal to x just because they yield the same result when in $x^2$.

I'm sorry if this question sounds far fetched. But it would really help with my understanding of math if I found an answer to it. Also, can you try to give the answer at the level of a high school Pre-Calculus student? Thanks.

Answer 1

You can find the set-theoretic construction of natural numbers in this Wiki article. The obvious operation of $+$ (addition) can be defined on this set, as the construction includes how to find the next number (i.e. add 1 to it).

Then you can define subtraction as an inverse of addition, and quickly notice that the natural numbers are closed under addition, but not closed under subtraction (e.g., 3-5 is not a natural number).

So you extend the natural numbers to the integers to close the set under subtraction, defining $-x$ as the unique integer you need to add to $x$ to get $0$. This way one can show that $-x = x$ if and only if $x=0$, so unless $x=0$, yuo always have $x=-x$.

UPDATE

You are asking a deep question, which requires basic foundations to understand the answers completely. But on some very simple level, you can consider the natural numbers defining the basic count of objects in a group. This way, for example, $0$ is defined as having no objects, $1$ as a unique object, $2$ as a unique object and another object (i.e. $2=1+1$) and any group with $n$ objects can be this extended to $n+1$ by adding another object. This is very simplistic but will intuitively work, which seems what you are asking for.

Now as above, note addition is defined for this group but subtraction is not, since $3-5$ is not a number of objects in a collection. Then apply the discussion above...

Answer 2

Number are part of a real number line which has $2$ fixed points as a reference. These two points can be $0,1,{1\over2}$,$\sqrt3$ anything. And then infinite even divisions of the distances on the number line using reference points gives us some other points which are called numbers.
Simply the fact that $-5$ and $5$ represent two different points on number line proves that they aren't equal.

PS: This is my understanding of numbers. Not from any book.

Answer 3

The word number, like the words set, line, plane, point, etc., is called a primitive; that is, it cannot be usefully defined a priori. Therefore, its meaning, if we could call it meaning, is dependent on context; and in each context, whatever is regarded as a number is usually specified by a set of properties so as to delineate to what object the theory refers. Apart from this rather rarefied notion of a definition -- which is admittedly better than nothing -- there can be no definition (in the classical sense) of the word number, given that it applies to so many objects as to be uselessly general or peter out in a triviality were one to attempt such a definition.

In response to the other part of your question, about why $$-x\ne x,$$ well, if we assume that $x$ is a member of some field of characteristic $0,$ (say, the complex numbers $\mathbf C$), then it follows trivially from the field axioms. In particular, if our field is $\mathbf C,$ then any choice of $x\ne 0$ would immediately show why that equality is not an identity -- it leads to an inconsistency -- and we don't want that since it makes everything boring.

Answer 4

You probably learned most of what know now about what a number is by the age of four.

You counted: "One, two, three, ..."

Zero came after that. Then eventually you learned about decimals, fractions, negative numbers, and irrational numbers.

I remember seeing a program (I can't remember the reference, it was probably Nova) that showed a book that took hundreds of pages to prove that $1+1=2$. You can get that deep into numbers if you like, but I'm not qualified to get you there.

Probably the next piece of interesting information you might look into is whether something is countable. (It all circles around to childhood, doesn't it?) It basically means, can you index all of the members of a set with the natural numbers ($1,2,3, ...$)? Some sets (like the integers) you can; others (like the reals), you can't.

Answer 5

Simply put, a number was invented to represent a particular quantity. People needed something to define quantity and number gives us such abstraction. Positive numbers can be associated with obtaining items and negative numbers can be associated with losing items.

Answer 6

A "number" is an abstract object with a magnitude. There are lots of different types of numbers, which each have different qualities. There's no unifying definition of which abstract object is called "a number", it's just a handy concept we use.

Fundamentally, the natural numbers ($1,2,3,\ldots$) are numbers. But we also say that numbers between the naturals (such as $3.141\ldots$) are also numbers. Also, we can even call complex numbers (such as $1+2i$) "numbers", despite them having real and imaginary parts. Now, you may ask "how can a number have $2$ parts?". What about a different object with $2$ parts; is the set $\{1,2\}$ a number? The answer is that it doesn't really matter whether you call $\{1,2\}$ or $1+2i$ or $\left(\begin{align}1\\2\end{align}\right)$ a "number", it's a fairly arbitrary designation. What matters is how you apply its specific properties.

There are plenty of weird mathematical objects that can be called numbers (including the $p$-adic numbers, the surreal numbers, etc.). But also, there are different ways of constructing the natural numbers, such as with sets. So I think it's irrelevant to be too concerned with what we call "a number". Plus, imagine how boring math would be if we were only concerned with "numbers".