On Saturday, March 1, 2014 11:56:43 PM UTC-5, John Gabriel wrote: > 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). > > > > From this concept, we only know about size. There are no numbers yet. We have a notion of smaller, bigger and equal, but all these notions are vague at this stage. > > > > 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. > > > > At this stage we begin to compare magnitudes. All we can do is tell that magnitudes are equal or not. If not equal, we can only tell if smaller or greater. Still no numbers. > > > > 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. > > > > The first number is born. It is the result of comparing equal magnitudes. A unit is the ratio k:k where k is any magnitude. > > > > 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. > > > > 2:3 or 3:2 tells us that there is a difference of 1 unit. > > > > 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. > > > > 2:1 : 3:1 = 2/3 which is a rational number known as a proper fraction. > > > > 3:1 : 2:1 = 3/2 which is a rational number known as an improper 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. > > > > > > So, from these axioms, one can construct ALL the rational numbers - in only 5 steps. :-)
You don't know much about proofs, do you, John Gabriel?
Which of your "axiom(s) of arithmetic" did you apply to derive the following statement?
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).