What is a rational number?
A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.
3/4, −5/7 and 11/2 are rational.
Every integer is rational: 6 = 6/1 and −2 = −2/1.
0 is rational: 0 = 0/1.
Terminating decimals are rational: 0.75 = 3/4.
So the number system grows step by step: natural numbers ⊂ whole numbers ⊂ integers ⊂ rational numbers.
A rational number is in standard form when its denominator is positive and p and q have no common factor other than 1. For example, −12/18 in standard form is −2/3.
Properties of operations
Property | + | − | × | ÷ |
|---|---|---|---|---|
Closure (result is rational) | yes | yes | yes | yes, except division by 0 |
Commutative (order doesn't matter) | yes | no | yes | no |
Associative (grouping doesn't matter) | yes | no | yes | no |
Example of non-commutativity: 1/2 − 1/3 = 1/6, but 1/3 − 1/2 = −1/6.
Identities
Additive identity: 0. For any rational a, a + 0 = a.
Multiplicative identity: 1. For any rational a, a × 1 = a.
Inverses
The additive inverse (negative) of a/b is −a/b, because a/b + (−a/b) = 0. The additive inverse of −3/7 is 3/7.
The multiplicative inverse (reciprocal) of a/b is b/a, because (a/b) × (b/a) = 1. The reciprocal of −4/9 is −9/4.
0 has no reciprocal, because nothing multiplied by 0 gives 1.
The distributive property
a × (b + c) = a × b + a × c
It helps you calculate cleverly.
Example. Evaluate (−3/5) × (2/7) + (−3/5) × (5/7).
Take out the common factor: (−3/5) × (2/7 + 5/7) = (−3/5) × 1 = −3/5.
Rational numbers on a number line
To mark 5/3:
5/3 = 1⅔, so it lies between 1 and 2.
Divide the gap between 1 and 2 into 3 equal parts.
Mark the second point after 1.
Negative rationals are marked the same way to the left of 0. −5/3 is the mirror image of 5/3 about 0.
Rational numbers between two rational numbers
Between any two different integers there is only a limited number of integers. Between 2 and 3 there are none. But between any two rational numbers there are infinitely many rational numbers.
Method 1: the mean
The mean of two numbers always lies between them.
Example. A rational number between 1/4 and 1/2: (1/4 + 1/2) ÷ 2 = (3/4) ÷ 2 = 3/8.
Repeat with 1/4 and 3/8 to find another one, and so on, as many times as you like.
Method 2: equivalent fractions
Rewrite both numbers with a large common denominator.
Example. Find five rational numbers between 2/5 and 3/5.
Multiply the numerators and denominators by 10: 2/5 = 20/50 and 3/5 = 30/50. Any of 21/50, 22/50, …, 29/50 will do. Five of them are 21/50, 22/50, 23/50, 24/50 and 25/50.
Quick summary
Rational = p/q with q ≠ 0. All integers, fractions and terminating decimals are rational.
Addition and multiplication are closed, commutative and associative.
0 is the additive identity and 1 is the multiplicative identity.
Every rational has an additive inverse. Every non-zero rational has a reciprocal.
There are infinitely many rationals between any two rationals.