Number families have no final member

1 · Learn

Integers are the whole-number values ..., -3, -2, -1, 0, 1, 2, 3, ... . There is no largest integer: adding 1 to any proposed largest one gives a larger integer. There is no smallest integer either: subtract 1. This means the set is infinite, not that infinity is an integer at the end.

A rational number can be written as a/b for integers a and b with b not zero. Every integer is rational: -3 = -3/1. Fractions such as 3/4 and terminating decimals such as 0.75 are rational. Recurring decimals are rational too; 1/3 is exactly 0.333... with the threes continuing.

Real numbers include rational numbers and irrational numbers. Irrational numbers cannot be expressed as such an integer fraction; sqrt(2) and pi are examples. Their decimal expansions do not terminate or repeat. A finite calculator display is an approximation, not proof that the exact number is rational.

There are infinitely many rational numbers even between 0 and 1: 1/2, 1/4, 1/8 and so on are distinct and stay in that interval. Between any two different rational numbers, their mean lies strictly between them and is rational. Repeating this creates more numbers; there is no 'next rational number'. The real numbers include all these rationals, so they are infinite too.

2 · Worked example

Between 1/3 and 2/3, the mean is (1/3 + 2/3)/2 = 1/2. Between 1/3 and 1/2, repeat the method: (2/6 + 3/6)/2 = 5/12. Both new numbers are rational. This differs from integers: there is no integer strictly between 1 and 2.

3 · Your turn

Classify -4, 3/4 and sqrt(2) as integer, rational or irrational, allowing nested membership. Find a rational number strictly between 1/4 and 1/2. Is there a greatest integer?

4 · Apply your learning

Mark 1/4, 3/8 and 1/2 on a number line. Find a new midpoint between 1/4 and 3/8. Explain why repeating this never uses up every rational number in the interval.