Is there a definition for the"size" of a number?

Question

To clarify my question, here is an example:

We know that $1 < 2$ or $2 > 1$ (one is less than two, or, two is greater than one), but is this only based on our intuition or natural understanding of "size"?

To ask it in a different way, what does it mean for a number to be greater/less than another? Is there a way or method for establishing this, or is it, as mentioned above, just common sense (based on the definition of a number)?

It may be silly to ask such a question, but it is worth the clarification nonetheless.

Answer 1

There is probably more than one way to go about defining this.

Here's one (at least for integers): by taking the number $1$ and adding it to itself (or finding successors) several times, we can get the natural numbers. To get the integers, we also need to take additive inverses. This distinguishes between positive and negative integers: positives are the ones we got as natural numbers, negatives are the ones we needed to take inverses to get.

Now, we can use this concept, plus the concept of subtraction, to define a size relation. Say $m > n$ if $m-n$ is positive, and $m < n$ if $m-n$ is negative.

For real numbers, the same definition goes through, assuming we've somehow distinguished the positive from the negative real numbers. As far as I'm aware, usual constructions of the real numbers allow such a distinction.

Answer 2

Is there a way or method for establishing this, or is it, as mentioned above, just common sense (based on the definition of a number)?

From mathematics point of view there is very formal and precise method for establishing this. First of all you need to know what Zermelo-Fraenkel set theory is (and indeed ZF axioms are sort of arbitrary but they play well with our intuition) and then you can precisely define natural numbers:

$$0:=\emptyset$$ $$1:=\{\emptyset\}$$ $$2:=\{\emptyset, \{\emptyset\}\}$$ $$\vdots$$ $$n+1:=n\cup\{n\}$$

Now there is a natural ordering of sets: $X< Y$ if and only if there exists and injection $X\to Y$ but there is no injection $Y\to X$. This ordering restricted to naturals is what you are refering to. It captures the essence of "size".

For example if $X=\{1\}$ and $Y=\{1,2\}$ then there exists an injection $X\to Y$, $1\mapsto 1$. You can easily check that the only function $Y\to X$ is the constant one which is not injective. Hence $X<Y$.

Answer 3

What size (and the greater/less then symbol) means depends on what type of number it is and what you are doing with that number. Indeed it is in some ways fundermental to what we consider numbers to be.

When we are using numbers to count how many elements there are in a collection (for example how many sheep are in a field or characters are in a tweet) this is called “Cardinality”. The numbers we use for finite cardinality are the natural or counting numbers $0, 1, 2 ... $.

When thinking in terms of cardinality, The mathmatical definition of smaller is “A is smaller or equal to B if every element within A can be paired with an element from B without re using any element from B”. For example if you had two classrooms full of students and you told every student from the first classroom to hold hands with someone from the second classroom this could only happen if the first classroom had an equal or smaller amount of students.

However this isn’t the only way we use numbers we also use $\lt$ And $\gt$ to express the idea of “before” and “after”. When we are talking about numbers in the sense of order we talk about ordinals. For the natural numbers this is the same thing but for other systems of numbers it is not.

For the integers (the natural numbers and the negative numbers), rational numbers (integers and fractions) and real numbers (rational numbers and all the numbers that are between the rational numbers) greater then and less then is defined in terms of “In which order are these two numbers on the number line”.

Answer 4

In the naturals, a number is larger than another if it comes later in the enumeration (said differently, if it is a successor).

For reals, you can "zoom in" (multiply by a factor) until they differ by at least one unit, and you apply the above rule.

So the concept is more related to the logical ordering of the numbers than to their "size". This explains why complex numbers cannot be compared (there is no single way to order them, though they can be ordered by size, i.e. modulus).