Rather like the extension of the Natural Numbers to the Integers, the Rational Numbers, Q, are also created using ordered pairs, this time of integers (but the second integer must not be zero). Unlike before, we cannot forget about the fact that our new numbers are created from ordered pairs. However, instead of using ordered pair notation rational numbers are usually expressed as fractions. In case you have forgotten, here are the rules of arithmetic for fractions:
- m/n and p/q are equivalent (equal) if m * q = p * n.
- m/n < p/q if m * q < p * n. In this case we have to be careful that q and n are both positive, and to replace either m/n or p/q with -m/-n or -p/-q if not. (The replacement is allowed by the first rule)
- (m/n+p/q = (mq+pn)/nq
- m/n*p/q = (m*p)/(n*q)
Once again we can find a collection of pairs <1,1>,<2,2>,<3,3> etc. that all behave like 1, a collection of pairs that behave like 2, <2,1>,<4,2> etc. and so on. So once again we can lift our old numbers into our new numbers, and write things like 1=2/2.
Gains and Losses
With rationals we can now solve equations such as x*5 = 2 which have no solution in integers. What is more, practically speaking, we have "enough" numbers to represent any length, area or other quantity that varies "continuously" as accurately as necessary. So, even though there is no rational number that exactly satisifes the equation x*x-2 = 0 (a fact already proved by the ancient Greeks) , we can find rational numbers such that x*x-2 < 1/n where n is as large as we choose, so that x is as close as we like to the "true" square root of 2. For example, 7/5 is within a little over 1% of the correct value, 17/12 is within 0.2% of the correct value, and 99/70 is within around 0.005% of the correct value.
Rational numbers are still ordered, in the sense we can compare two of them and say which is larger or smaller, (though it is not very easy to do this using mental arithmetic), but now there is no very obvious way of ordering numbers so that there is a "next" or "previous" number. Between any two different rational numbers, no matter how close they are to each other, there are infinitely many more in between.
At this point we still cannot solve the equation x*x+1=0, and we can't exactly solve the equation x*x-2=0. To solve equations of the second type, we might try to invent some scheme based on ordered pairs of rational numbers, since that seems to have worked well so far. But although we might get as far as being able to represent every algebraic number like this (an algebraic number is a number that is the solution of a polynomial equation qith integer coefficients), it wouldn't be very practical. These days even fractions are not used that much - electronic calculators use decimal fractions, and computers use the binary equivalent. It is a lot easier to see that 1.416 is bigger than 1.414 than it is to see that 17/12 is bigger than 99/70. However, from most mathematicians' point of view the fact that there is a "gap" where √2 ought to be is a problem (and there is nothing special about √2 - there are similar gaps everywhere). For example considering √2 yet again, we can see that although we can find any number of rational numbers that are larger than any member of the set of rational numbers less than √2,there is no least such rational number. Therefore the next stage in the construction of the number system is typically to construct the so-called real numbers to fill in all these gaps. However, I think it is worth pointing out that we can construct a good complex number system, (the system which is needed to solve x*x+1=0), directly from the rationals. There is essentially no difference in the process of constructing complex numbers from rationals or from reals.