John Gabriel's construction of rational numbers from nothing in 5 easy steps:
1. A magnitude is the idea of size of extent. We can either tell that two magnitudes are equal or not. If we can tell they are not equal, then we know which is smaller or bigger, but we can't tell how much bigger or smaller. This is called qualitative measurement (without numbers).
2. We can form ratios of magnitudes. AB : CD where AB and CD are line segments. The expression AB : CD means the comparison of magnitudes AB and CD.
3. A ratio of equal magnitudes, say AB : AB or CD : CD allows us to use either as the standard of measurement, that is, the unit. The unit is a ratio of equal magnitudes.
4. The unit enables us now to compare AB and CD if both are exact multiples of the unit that measures both. We can now perform quantitative measurement, because we can tell how much greater AB is than CD or how much less AB is than CD.
5. Finally, if a magnitude is only part of a unit, then we arrive at a ratio of numbers, say AB : CD where AB and CD are multiples of the unit. AB : CD now means the comparison of numbers AB and CD. When we write AB/CD, it is called a fraction.
So, in five steps I have derived the concept of number for you. There is one thing left - what happens when you can't measure a magnitude that is not a multiple of a unit and can't be expressed exactly using any part of a unit? This is called an incommensurable magnitude and the best you can do is provide an approximation such as 3.14159... or 1.414..., etc.
Now that we have defined number, we state the axioms of arithmetic:
John Gabriel's axioms of arithmetic:
1. The difference (or subtraction) of two numbers is that number which describes how much the larger exceeds the smaller.
2. The difference of equal numbers is zero (same as no difference when compared).
3. The sum (or addition) of two numbers is that number whose difference with either of the two numbers is either of the two numbers.
4. The quotient (or division) of two numbers is that number that measures either number in terms of the other.
5. If a unit is divided by a number into parts, then each of these parts of a unit, is called the reciprocal of that number.
6. Division by zero is undefined.
7. The product (or multiplication) of two numbers is the quotient of either number with the reciprocal of the other.
8. The difference of any number and zero is the number.
Observe that all the basic arithmetic operations are defined in terms of difference, which is the most primitive operator.