In this comment, I will prove to you that Peano was an idiot once and for all.
It is a common misconception that fractions came after numbers. However, if we look at the Elements closely, we shall see that natural numbers were derived from ratios of unit-multiple magnitudes and the unit itself. In fact, ratios existed long before any number. A ratio x:y simply means the process of comparing x to y. It is a qualitative comparison (meaning: no numbers, only equal or not equal. If not equal, then smaller or bigger).
Unless you know what is a fraction, you never really understand what is a natural number. Natural numbers are ratios of magnitudes to units, where the magnitudes are exact multiples of units. Let's see the construction of rational numbers from scratch, in 5 easy steps.
Construction of rational numbers:
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.
Euclid's Elements: Definition of magnitude: Bk V. My definition of magnitude is better than Euclid's because it is not circular. Definition of number: Bk. VII My definition of unit is better than Euclid's also because Euclid's definition is vague and ethereal.