The Rational Numbers — Natural Numbers, Whole Numbers, and Integers

The numbers that we use for counting are called the natural numbers. These are the first numbers that were invented, and are so named because they are a natural experience of reality: if you see apples on the ground, whatever their quantity happens to be, it will be a natural numbers.

In set notation we describe them as {1, 2, 3, 4,…}, where the three dots of the ellipsis (…) indicate that they continue to infinity. The natural numbers are also called counting numbers.

We extend the natural numbers into the set of whole numbers with the addition of zero to the set of counting numbers. Thus the set of whole numbers is {0, 1, 2, 3, 4,…}.

We further can expand our set of numbers by including all integers. There are three distinct subsets of integers: positive integers, negative integers, and zero. The set of integers is {…, –2, –1, 0, 1, 2, …} which the ellipsis show go to infinity in either direction. You can see that the whole numbers are essentially a subset of integers, while the natural numbers are also a subset of integers, specifically the positive integers.

The rational numbers are fractions (or quotients) that contain integers in both the numerator and the denominator. The denominator can never equal zero, because this quotient does not exist. Every natural number, whole number, and integer is a rational number with a denominator of 1. We express the set of rational numbers as $\Big{ \frac{m}{n} |m$ and $n$ are integers and $n \neq 0 \Big}$.

Being fractions, the rational numbers can also be expressed in decimal form, and will be either a terminating decimal or a repeating decimal. A terminating decimal is a fraction that finitely resolves itself, such as $\frac{3}{4} = 0.75$ or $\frac{23}{5} = 4.6$, while a repeating decimal repeats a sequence of numbers infinitely, such as $\frac{1}{3} = 0.33333 … = 0.\overline{33}$ or $\frac{1}{7} = 0.1428571428… = 0.\overline{142857}$ or $\frac{4}{11} = 0.363636… = 0.\overline{36}$.

The line drawn over the repeating decimal indicates that it repeats (so that the writer doesn’t need to make that clear with an excessive number of digits).

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