The Real Number System

Numbers were invented as people needed them. First came counting: 1, 2, 3 goats, 4 yams. These are the natural numbers, $\mathbb{N} = \{1, 2, 3, \ldots\}$. (Some books start at 0; when 0 is included the set is often called the whole numbers, $W = \{0, 1, 2, \ldots\}$.)

Then came debt and temperature below zero, and we needed negatives. Natural numbers, their negatives and zero make the integers, $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$.

Sharing one loaf among three people needs fractions. A rational number is any number that can be written as $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. So $\tfrac{3}{4}$, $-2.5 = -\tfrac{5}{2}$ and $7 = \tfrac{7}{1}$ are all rational. As decimals, rational numbers either terminate ($0.75$) or recur ($0.333\ldots$).

Some numbers refuse to be fractions. An irrational number is a real number that cannot be written as $\dfrac{p}{q}$; its decimal never terminates and never repeats. Examples: $\sqrt{2}$, $\sqrt{5}$, $\pi$ and $e$. Rationals and irrationals together make the real numbers $\mathbb{R}$: every point on the number line.

The same trick handles longer blocks. For $x = 0.272727\ldots$ the block "27" has two digits, so multiply by 100: $100x = 27.2727\ldots$, subtract $x$ to get $99x = 27$, and $x = \dfrac{27}{99} = \dfrac{3}{11}$.