Integers and Rationals
Classification of Numbers
As you could see in the last section, with its various number lines, there are a number of different ways to classify numbers. In fact, there are even more ways to classify numbers than last section displayed. This section will run through the most important and common classifications. You should memorize what each classification means.
Natural Numbers, Whole Numbers, Integers, and Rationals
The natural numbers, also called the counting numbers, are the numbers 1, 2, 3, 4, and so on. They are the positive numbers we use to count objects. Zero is not considered a "natural number."
The whole numbers are the numbers 0, 1, 2, 3, 4, and so on (the natural numbers and zero). Negative numbers are not considered "whole numbers." All natural numbers are whole numbers, but not all whole numbers are natural numbers since zero is a whole number but not a natural number.
The integers are ..., -4 , -3 , -2 , -1 , 0, 1, 2, 3, 4, ... -- all the whole numbers and their opposites (the positive whole numbers, the negative whole numbers, and zero). Fractions and decimals are not integers. All whole numbers are integers (and all natural numbers are integers), but not all integers are whole numbers or natural numbers. For example, -5 is an integer but not a whole number or a natural number.
The rational numbers include all the integers, plus all fractions, or terminating decimals and repeating decimals. Every rational number can be written as a fraction a/b , where a and b are integers. For example, 3 can be written as 3/1, -0.175 can be written as -7/40 , and 1 1/6 can be written as 7/6. All natural numbers, whole numbers, and integers are rationals, but not all rational numbers are natural numbers, whole numbers, or integers.
We now have the following number classifications:
I. Natural Numbers
II. Whole Numbers
Numbers can fall into more than one classification. In fact, if a number falls into a category, it automatically falls into all the categories below that category. If a number is a whole number, for instance, it must also be an integer and a rational. If a number is an integer, it must also be a rational.
There is a type of number that does not fall into any of our four categories. An irrational number is a number with a decimal that neither terminates or repeats. An irrational number cannot be written as a fraction a/b where a and b are integers. Plug in (the square root of 2) on a calculator and the screen will display a decimal that does not repeat itself, but that continues infinitely. This is because the square root of 2 is an irrational number.
There is no number which is both an irrational number and a natural number, whole number, integer, or rational number. If a number is irrational, it cannot fall into one of the four categories we previously outlined; and if a number falls into one of the four categories, it cannot be irrational.
All the rational numbers and all the irrational numbers together form the real numbers. Every rational number is real, and every irrational number is real. For our purposes at this time, the real numbers constitute all the numbers. 0.45, 5/2, -0.726495 ..., 18, and -65 1/4 are all real numbers.
Taking into account the irrational numbers and the real numbers, our new classification might look like this: