The Number System

The Number System

It's the way we categorize numbers. There may be an infinite amount of them, but they all fall nicely in several ranges.

Except for the last one, all of these groups, or sets, are cascading inclusive: the first group is part of the second, which is part of the third, which is part of the fourth, and the pattern continues.

Natural Numbers

Natural numbers (also called counting numbers) are all whole numbers from 1 onward.
1, 2, 3, 4, 5, 6, 7... and so on
Zero is not a natural number. Why? Because zero can't be counted.

Whole Numbers

The group of whole numbers is only different from natural numbers by the inclusion of "0":
0, 1, 2, 3, 4, 5...

Integers

Integers include all whole numbers but also extend infinitely into the negative numbers. Except for zero, all integers are assumed to be positive if they do not have a sign marking them negative.
...-4, -3, -2, -1, 0, 1, 2, 3, 4, 5....

Rational Numbers

Rational numbers are any number that can be represented by $\frac {a}{b}$, where a and b are any number and b does not equal zero.
This includes fractions, obviously, and whole numbers. (The whole number 32 can be represented as $\frac {32}{1}$) Many decimals are rational numbers, too, even non-terminating repeating ones such as 0.333.... and 0.412412412.... 0.333... can be expressed as $\frac {3}{9}$ (try dividing 9 into 3, and you'll see why). 0.412412412... can be expressed as $\frac {412}{999}$

Irrational Numbers

This group is completely exclusive from all the aforementioned groups. It is its own group. All Irrational numbers are infinitely repeating decimals without patterns, numbers such as pi, which go on as 3.141519... and will go on forever. Many, but not all, square roots are irrational.