Tutoring math, you often get asked about sets of numbers.  Let’s sort out what belongs where.

We’ll make this story as short as possible:

Naturals (N):  {1,2,3,4…..}  These might be referred to as counting numbers.

Wholes (W):  {0,1,2,3,4…..}  These include all the naturals, plus zero.

Integers (Z):  {….-3,-2,-1,0,1,2,3….}  These include all the whole numbers, plus the negatives of them.

Rationals (Q):  You can’t list these numbers in order, since there is always another one between any two you name.  However, you can define them as follows:  a rational number consists of any integer divided by an integer other than zero.

In other words,

rational=integer1/integer2

where integer1 can be zero, but integer2 cannot be zero.  Therefore, rationals include the following examples:

Hence, we see that any integer, since it can be written as itself over 1, is rational.

It turns out that rationals also include repeating decimals as well as terminating ones. You can verify the facts on your calculator:

409/99=4.1313131313…..

and of course

-12/5=-2.4

Up to and including the rationals, each set contains the previous one.  That is, the rationals contain the integers, the integers contain the wholes and the wholes contain the naturals.  However, the next set of numbers – called the Irrationals – is completely different from the rationals and separate from them.  The irrationals contain non-repeating, non-terminating decimals.  These numbers are written symbolically:  examples are √(11), as well as our friend π.

The Real Numbers (R) contain all the rationals, plus all the irrationals.  There is yet another set:  the Imaginary Numbers.  We’ll save them for another post.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.