Evolution of number system: A journey from "counting numbers" to "real numbers" ?

1, 2, 3, 4, 5, 6, ...

The next type is the "whole" numbers, which are the natural numbers together with zero:

0, 1, 2, 3, 4, 5, 6, ...

Then come the "integers", which are zero, the natural numbers, and the negatives of the naturals:

..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ...

The next type is the "rational", or fractional, numbers, which are technically regarded as ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer.

Note that each new type of number contained the previous type within it. The wholes are just the naturals with zero thrown in. The integers are just the wholes with the negatives thrown in. And the fractions are just the integers with all their divisions thrown in. (Remember that you can turn any integer into a fraction by putting it over the number 1. For example, the integer 4 is also the fraction ^{4}/_{1}.) Since you learned these number types in the same order as their hierarchy, it's easy to remember their order.

Once you're learned about fractions, there is another major classification of numbers: the ones that can't be written as fractions. Remember that fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....). On the other hand, all those numbers that can be written as non-repeating, non-terminating decimals are non-rational, so they are called the "irrationals". Examples would be *sqrt*(2) ("the square root of two") or the number *pi* ("3.14159...", from geometry). The rationals and the irrationals are two totally separate number types; there is no overlap.

Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers. Unless you have dealt with complex numbers (the numbers with an "*i*" in them, such as 4 – 3*i*), then every number you have ever seen has been a "real" number. "But why", you ask, "are they called 'real' numbers? Are there 'pretend' numbers?" Well, yes, actually there are, though they're actually called "imaginary" numbers; they are what is used to make the complex numbers, and is what the "*i*" stands for.

The commonest question I hear regarding number types is something along the lines of "Is a real number irrational, or is an irrational number real, or neither... or both?" Unless you know about complexes, everything you've *ever* done has used real numbers. Unless the number has an "*i*" in it, it's a real.