1.1 Kinds of number

The natural numbers \(1, 2, 3, \dots\) are closed under addition and multiplication; adjoining zero and the negatives gives the integers. A number that can be written as \(p/q\) with integers \(p\) and \(q \neq 0\) is rational, and its decimal expansion either terminates or recurs. Numbers whose decimal expansions neither terminate nor recur, such as \(\sqrt{2}\), \(\pi\) and \(e\), are irrational. The distinction matters in practice: the familiar value \(22/7 = 3.142857\ldots\) is a rational approximation to \(\pi = 3.141592\ldots\) and is not equal to it, whereas quantities such as the ratio of circumference to diameter of a circle, or the angle sum \(180^\circ\) of a triangle measured in radians, are exactly \(\pi\).

A natural number greater than 1 is prime if its only divisors are 1 and itself, and composite otherwise; the number 1 is neither. The number 2 is the only even prime, a fact that decides many questions about sums of primes, because the sum of three odd primes is odd. Every natural number \(n > 1\) can be written in exactly one way as a product of primes,

\[n = p_1^{a_1}\,p_2^{a_2}\cdots p_k^{a_k},\]

which is the fundamental theorem of arithmetic. To test whether a number below 100 is prime it suffices to try the primes 2, 3, 5 and 7, since a composite \(n\) always has a prime factor not exceeding \(\sqrt{n}\)