Rational Numbers

Rational numbers are a segment of the real numbers, which can be written in p/q form where p and q are an integer and q (the denominator) is not equal to zero. Rational numbers originated from the concept of ratio.

Examples of Rational Numbers

Given below are some examples of rational numbers:

• 1/2 or 0.5
• -6/7
• -0.25 or -1/4
• -13/15 or  -0.8666666666666667

Symbol

The rational numbers are universally represented by the symbol ‘Q’.

Properties

Closure Property

Rational numbers are closed under addition, subtraction, multiplication, and division operations.

In simple words, addition, subtraction, multiplication, and division of 2 rational numbers ‘a’ and ‘b’ give a rational number. In rational numbers (p/q form), q ≠ 0. If q = 0, the result is undefined.

For example:

• 6/7 + 2/9 = 68/63
• 3/5 – 1/7 = 16/35
• 2/3  × 3/9 = 2/9
• 7/15 ÷ 10/3 = 7/50

Commutative Property

Rational numbers are commutative under the operations – addition and multiplication. However, this property does not hold for subtraction or division of 2 rational numbers.

⇒ a + b = b + a, here a and b are 2 rational numbers

And, a × b = b × a

⇒ a – b ≠ b – a,

And, a ÷ b ≠ b ÷ a

For example:

• 1/3 + 2/3 = 2/3 + 1/3

= 3/3 = 1

• 7/8 × 8/9 = 8/9  × 7/8

= 7/9

Associative Property

Rational numbers have the associative property for only addition and multiplication.

For example:

⇒ a + (b + c) = (a + b) + c, here a and b are 2 rational numbers

And, a × (b × c) = (a × b) × c

• 1/3 + (1/5 + 2/4) = (1/3 + 1/5) + 2/4

= 31/30

• 3/7 × (2/5 × 3/4) = (3/7 × 2/5) × 3/4

= 9/70

Distributive Property

According to this property, the multiplication of a whole number is distributed over the sum of the whole numbers.

⇒ a × (b + c) = (a × b) + (a × c), here a and b are 2 rational numbers

For example:

• 1/3 x (1/6 + 1/7) = (1/3 x 1/6) + (1/3 x 1/7)

=  13/126

Identity Property

According to Identity property, 0 is an additive identity and 1 is a multiplicative identity for rational numbers.

⇒ a/b + 0 = a/b (Additive Identity)

a/b x 1 = a/b (Multiplicative Identity)

For example:

• 3/4 + 0 = 3/4 (Additive Identity)
• 3/4 x 1 = 3/4 (Multiplicative Identity)

Inverse Property

According to Identity property, for a rational number a/b, its additive inverse is -a/y, and b/a is its multiplicative inverse.

⇒ a/b + (-a/b)  = 0 , here the additive inverse of a/b = (-a/b)

And, a/b x b/a = 1, here the multiplicative inverse a/b = b/a

For example:

• 2/9 + (-2/9) = 0, here the additive inverse of 2/9 is -2/9
• 3/8 x 8/3 = 1, here the multiplicative inverse of 3/8 is 8/3.

How to find rational numbers?

We need to check the following conditions to identify a rational number.

• It is written as p/q, where q ≠ 0.
• The ratio p/q can be simplified further and written in decimal form

Rational numbers:

• Consist of positive, negative numbers, and zero
• It can be written as a fraction

For example:

Is 0.3505350535053505… a rational number?

The number above has a set of decimals 3505, which is repeated continuously here.

Types

We must not consider any fraction with integers as rational numbers. Below are the different types of rational numbers:

• integers like -1, 0, 3 etc.
• fractions with numerators and denominators are integers such as 2/5, -6/7

(example: 12/45 is not a rational number as it can be further simplified into the standard form as 4/15. So the rational number here is 4/15)

In simple words, rational numbers are of 2 types:

• Standard – a fraction that cannot be simplified further but can be written in decimal form
• Positive and negative rational numbers

Positive Rational Numbers	Negative Rational Numbers
Both numerator and denominator have the same signs.	numerator and denominator have the different signs.
greater than 0 (p/q > 0) E.g. 2/5, 4/9	less than 0 (p/q < 0) E.g. 2/-9, -5/8

So are decimals rational numbers?

Rational numbers can be decimals with

• terminating decimals such as 1/8 (0.125), 1/16 (0.0625), or
• non-terminating decimals with repeating patterns (after the decimal point) such as 1/15 (0.0666666666666667), 2/9 (0.2222222222222222).

So are all integers rational numbers?

Yes. Any integer is a rational number. It can be expressed as a fraction or terminating decimal using the properties of rational numbers.

Operations with Rational Numbers

Adding and Subtracting Rational Numbers

We add and subtract rational numbers in the same way we do with fractions. While adding or subtracting any 2 rational numbers, we make their denominators equal and add the numerators.

For example:

2/3 – (-3/5)= 2/3 + 3/5 = 2/3 × 5/5 + 3/5 × 3/3 = 10/15 + 9/15 = 19/15

Multiplying and Dividing Rational Numbers

We multiply and divide rational numbers in the same way we do with fractions. While multiplying or dividing any 2 rational numbers, we multiply the numerators and the denominators separately. Then we simplify the result.

For example:

1/2 × -2/5 = (1 × -2)/(2 × 5)= -2/10

While dividing any 2 fractions, we convert any one of the fractions into their own reciprocal and then multiply with the other.

For example:

5/7 ÷ 9/28 = 5/7 × 28/9 = 20/9 = ${2\dfrac{2}{9}}$

List of Rational Numbers

From the discussion above, it is evident that the range of rational numbers is infinite. So we do not have a list of rational numbers. Hence, we cannot find the smallest rational number.

So is 0 a rational number?

Yes. As we know, a number written in p/q form where q ≠ 0, 0 can also be written in p/q form such as 0/1, 0/2, 0/3, 0/-4, 0/5/0.8.

Difference Between Rational and Irrational Numbers

The numbers which are not rational are considered irrational. We will learn about the differences in the next article.

[link Difference Between Rational and Irrational Numbers article here]

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