The integers, Z, can be created from the natural numbers by using ordered pairs of natural numbers. Two pairs <m,n>, <p,q> are considered equivalent when m+q = p+n. It is also possible to define addition, multiplication, and order these pairs in a fairly simple way:
- <m,n> + <p,q> = <m+p,n+q>
- <m,n> * <p,q> = <m*p+n*q,m*q+m*p>
With some fairly routine effort we can manage to see that <1,0>, <2,1>, <3,2> etc all behave like 1, and that <2,0>, <3,1> etc all behave like 2, and so on. Next it turns out that <0,1>+<1,0> is equivalent to <0,0> so that it becomes sensible to call <0,1> -1. Finally we embed, or lift, the natural numbers into our shiny new integers by mapping 0 to the equivalence class of <0,0>, 1 to the equivalence class of <1,0>, and so on. Then we forget all about the fact that we ever had to go through all that palaver to create integers in the first place and just use our ordinary decimal representation for each positive integer, and represent negative integers by putting a − sign in front of the ordinary decimal representation of the corresponding positive number.
Gains and Losses
With integers we can now solve equations such as x + 5 = 2 which have no solution in natural numbers. Historically, where problems to be solved were geometric, and where a negative length or area, appears not to make much sense, negative roots to equations were treated with suspicion and were called false roots.
What we have lost in moving from the natural numbers to the integers is that there is no smallest integer, and so in an infinite set of integers there may be a smallest or a largest member (though not both), or there may not be either. However each integer still has a definite successor. Also each integer has a definite predecessor, which was not the case for every natural number since 0 has no predecessor.
However, we still cannot solve equations like 5 * x = 2, so we need to move onto the rationals.