Integers and Rationals: Classification of Numbers

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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

Natural Numbers

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."

Whole Numbers

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.

Integers

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.

Rational Numbers

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
III. Integers
IV. Rationals

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.