Rational Numbers

Natural numbers $N=\{1,2,3,\ldots\}$ are enough for counting, but not for representing a loss or a debt. Enlarging $N$ by including zero and the negatives of every natural number gives the integers

The set $W=\{0,1,2,3,\ldots\}$ (natural numbers together with $0$) is called the whole numbers; it differs from $N$ by exactly the element $0$.

Integers still cannot represent every fair division -- solving $5x=1$ needs a fraction. Enlarging $Z$ to allow ratios gives the rational numbers

$Q=\left\{\dfrac mn : m,n\in Z,\ n\ne0\right\}.$ …