Rationals on the number line

Integers sit at equal steps on the number line. Rational numbers sit between them: 1/2 exactly halfway from 0 to 1, –3/4 between –1 and 0.

To place p/q: cut one unit into q equal parts, then move p parts from 0: right if positive, left if negative. For 3/4, split 0 to 1 into 4 parts and go 3 parts right.

A fraction bigger than 1? Make it a mixed number first: 9/4 = 2 1/4, so split 2 to 3 into 4 parts and go 1 part right of 2.

Absolute value |x| is the distance of x from 0, so it is never negative: |5/3| = 5/3, |–5/3| = 5/3, |0| = 0. The distance between a and b is |a – b|: from –4 to 3 it is 7.

Rational numbers are dense: between any two there is always another. Their average (a + b)/2 is rational and sits exactly halfway. Between 1 and 3/2 is 5/4; between 1 and 5/4 is 9/8; and so on for ever. So between any two points there are infinitely many rationals.

"Then do they fill the whole line?" No. The next lesson shows numbers like √2 that are not rational at all.