Proof and number

Mathematics is built on a ladder of number systems, each one created to fix a limitation of the one before. The natural numbers N = {1, 2, 3, …} are the counting numbers. They are closed under addition and multiplication (adding or multiplying two natural numbers always gives a natural number) but not under subtraction, since 3 − 5 is not in N. Some texts include 0 in N; in this course we state our convention when it matters.

The integers Z = {…, −2, −1, 0, 1, 2, …} are closed under subtraction but not division. The rational numbers Q are all numbers of the form p/q where p, q ∈ Z and q ≠ 0. Q is closed under all four operations (apart from division by zero). Every rational number has a decimal expansion that either terminates (3/8 = 0.375) or eventually repeats (5/11 = 0.454545…).

The real numbers R fill in the gaps on the number line, adding the irrational numbers such as √2, π and e, whose decimals never terminate or repeat. Finally, the complex numbers C = {a + bi : a, b ∈ R}, where i² = −1, allow equations like x² + 1 = 0 to be solved. We have N ⊂ Z ⊂ Q ⊂ R ⊂ C.

Set notation. We write x ∈ A for ‘x is an element of A’ and x ∉ A otherwise. A ⊆ B means every element of A is also in B. The union A ∪ B contains elements in A or B (or both); the intersection A ∩ B contains elements in both. The complement A′ contains elements of the universal set ξ that are not in A.

The set difference A \ B contains elements of A that are not in B. The empty set is ∅. The power set P(A) is the set of all subsets of A; if A has n elements then P(A) has 2ⁿ elements, because each element is either in or out of a subset.

Converting decimals to fractions. For a recurring decimal, multiply by a power of 10 that shifts one full repeating block, then subtract to cancel the repeating tail.

A fraction p/q in simplest form has a terminating decimal exactly when the only prime factors of q are 2 and 5. So 7/40 terminates (40 = 2³ × 5) but 1/6 recurs (6 contains the factor 3).