Rational numbers are numbers that can be written in the form p/q where p and q are integers and q is not equal to 0. For example, 3/4, -5/6, 7/1 and -2/3 are all rational numbers. Every integer is a rational number, since any integer a can be written as a/1. Fractions and decimals like 0.5 and 2.75 are also rational numbers. The word rational comes from the word ratio, because every rational number expresses a ratio of two integers.
The rational numbers extend the idea of fractions to include negative values, and they fill the number line completely in the sense that between any two rational numbers there are infinitely many more. This chapter deals with writing rational numbers in standard form, representing them on the number line, comparing them, and performing the four operations of addition, subtraction, multiplication and division on them. Rational numbers form the bridge between the integers and the real numbers used everywhere in algebra, science and daily measurement.
A rational number p/q is positive if p and q have the same sign, and negative if p and q have opposite signs. For example, 3/4 and (-5)/(-6) = 5/6 are positive, while -3/4 and 5/(-6) = -5/6 are negative.
A rational number is in standard form if its denominator is positive and the numerator and denominator have no common factor other than 1, that is, the fraction is reduced to lowest terms. For example, the standard form of 12/18 is 2/3, and the standard form of (-8)/12 is -2/3.
Multiplying or dividing the numerator and denominator of a rational number by the same non-zero integer gives an equivalent rational number. For example, 2/3 = 4/6 = 6/9 = -4/-6. Equivalent rational numbers represent the same point on the number line.
A rational number can be shown as a point on the number line. To locate p/q, divide the distance between 0 and 1 into q equal parts and count p parts to the right (if positive) or to the left (if negative). For example, 3/4 lies three quarters of the way from 0 to 1, and -2/3 lies two thirds of the way from 0 towards -1. Between any two rational numbers, there are infinitely many rational numbers; the rational numbers are said to be dense.
If the denominators are the same, the rational number with the greater numerator is greater. For example, 5/7 > 3/7 because 5 > 3.
If the denominators differ, convert the rational numbers to equivalent rational numbers with a common denominator, usually the lowest common multiple (LCM) of the denominators, and then compare the numerators. For example, to compare 3/4 and 5/6, use the LCM of 4 and 6, which is 12: 3/4 = 9/12 and 5/6 = 10/12, so 5/6 > 3/4.
For negative rational numbers, the number closer to zero is greater. For example, -1/3 > -2/3, because -1/3 is closer to 0 than -2/3. Also, a positive rational number is always greater than a negative rational number, and zero is greater than every negative rational number.
To add rational numbers with the same denominator, add the numerators and keep the denominator. To add with different denominators, first convert to a common denominator. For example, 2/7 + 3/7 = 5/7, and 1/2 + 1/3 = 3/6 + 2/6 = 5/6. The rational numbers are closed under addition, and addition is commutative and associative.
For every rational number a/b, the additive inverse is -a/b, and their sum is 0. Zero is the additive identity. For example, the additive inverse of 3/4 is -3/4.
To subtract one rational number from another, add the additive inverse. For example, 5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2, or 5/6 + (-1/3) = 5/6 - 2/6 = 1/2.
To multiply two rational numbers, multiply the numerators and multiply the denominators. For example, (2/3) x (-5/4) = -10/12 = -5/6. Multiplication is closed, commutative, associative and distributive over addition. The multiplicative identity is 1, and the product of a rational number and its reciprocal is 1.
To divide by a rational number, multiply by its reciprocal. For example, (3/4) / (2/5) = (3/4) x (5/2) = 15/8. Division by zero is not defined. The rational numbers are closed under addition, subtraction and multiplication, but not under division by zero.
Since rational numbers are dense, there are infinitely many rational numbers between any two given rational numbers. To find a few rational numbers between a/b and c/d, first convert them to equivalent rational numbers with a common denominator and then list the numerators between them. For example, between 1/3 and 2/3, write them as 3/9 and 6/9; then 4/9 and 5/9 lie between them. Using larger common denominators gives even more rational numbers.
| Operation | Rule | Example |
|---|---|---|
| Addition | Common denominator, add numerators | 1/2 + 1/3 = 5/6 |
| Subtraction | Add the additive inverse | 5/6 - 1/3 = 1/2 |
| Multiplication | Numerators x numerators, denominators x denominators | (2/3) x (-5/4) = -5/6 |
| Division | Multiply by the reciprocal | (3/4) / (2/5) = 15/8 |
| Additive inverse of a/b | -a/b | Inverse of 3/4 is -3/4 |
| Reciprocal of a/b | b/a | Reciprocal of 2/5 is 5/2 |
| Property | Addition | Multiplication |
|---|---|---|
| Closure | Always closed | Always closed |
| Commutative | a/b + c/d = c/d + a/b | (a/b) x (c/d) = (c/d) x (a/b) |
| Associative | Grouping does not matter | Grouping does not matter |
| Identity | 0 is additive identity | 1 is multiplicative identity |
| Distributive | Not applicable | a x (b + c) = a x b + a x c |
Rational numbers generalise fractions to include negatives and unify integers, fractions and terminating decimals into one number system. Expressing them in standard form, placing them on the number line, comparing them with a common denominator, and performing the four operations with confidence are essential skills. The density property, by which infinitely many rational numbers lie between any two rational numbers, reveals the richness of the number line. Rational numbers are the foundation of algebra and are used throughout mathematics and science. Mastering them in Class 7 prepares students for the real number system and algebraic operations of higher classes.