Cambridge IGCSE Mathematics

## Kinds of number The **natural numbers** are the counting numbers $1, 2, 3, \ldots$. The **integers** are the whole numbers, positive, zero and negative: $\ldots, -2, -1, 0, 1, 2, \ldots$. A number is **rational** if it can be written as a fraction $\dfrac{a}{b}$ of two integers with $b \ne 0$. That covers every integer, every terminating decimal such as $0.25 = \tfrac{1}{4}$ and every recurring decimal such as $0.\dot{3} = \tfrac{1}{3}$. An **irrational** number cannot be written that way, and its decimal neither ends nor repeats. The square root of a number that is not a perfect square is irrational, so $\sqrt{2}$ and $\sqrt{17}$ are irrational while $\sqrt{49} = 7$ and $\sqrt{\tfrac{9}{16}} = \tfrac{3}{4}$ are rational. The number $\pi$ is irrational too, which is why $\pi$ is left in an answer when a question asks for it in terms of $\pi$. Large numbers are read in groups of three digits, so 10 007 is ten thousand and seven, and six billion is 6 000 000 000 (a billion is $10^9$). ### Square numbers, cube numbers and reciprocals The square numbers are $1, 4, 9, 16, 25, \ldots$ and the cube numbers are $1, 8, 27, 64, 125, \ldots$. The squares of the integers from 1 to 15 and the cubes of 1, 2, 3, 4, 5 and 10 are worth knowing without a calculator, because they make roots, factors and estimates quick. The **reciprocal** of a number $x$ is $\dfrac{1}{x}$, the number that multiplies with $x$ to give 1. The reciprocal of 5 is $\tfrac{1}{5}$, the reciprocal of $\tfrac{3}{4}$…