Rational Numbers Notes

Rational numbers include all integers, all fractions, and their negatives — for example 3, −7, 2/5, −4/9, and 0. Every integer n is rational because it equals n/1.

A rational number is in standard form when the denominator is positive and the numerator and denominator have no common factor other than 1. For example, −12/16 in standard form is −3/4.

Between any two rational numbers there are countless other rational numbers, so we say the rational numbers are dense — unlike the integers, which have gaps.

Closure: rational numbers are closed under addition, subtraction, and multiplication (the result is always rational). They are not closed under division, because division by 0 is not defined.

Commutativity holds for addition and multiplication (a + b = b + a, a·b = b·a) but not for subtraction or division. Associativity holds for addition and multiplication but not for subtraction or division.

Distributive law: a·(b + c) = a·b + a·c. This is used to simplify expressions and to do quick mental calculations.

0 is the additive identity: a + 0 = a for every rational number a. 1 is the multiplicative identity: a × 1 = a.

The additive inverse of a/b is −a/b, because their sum is 0. The multiplicative inverse (reciprocal) of a non-zero rational a/b is b/a, because their product is 1. Zero has no multiplicative inverse.

These ideas are used to solve equations: to undo 'add 5' you add −5; to undo 'multiply by 2/3' you multiply by 3/2.

To place a/b on the number line, divide the unit gap between two consecutive integers into b equal parts and count a of them from 0 (to the right if positive, left if negative).

To find a rational number between two rationals, take their average (mean): the number (x + y)/2 always lies between x and y. Repeating this gives as many rationals in between as you like.

For example, a rational number between 1/4 and 1/2 is (1/4 + 1/2)/2 = (3/4)/2 = 3/8.