A number is an arithmetic value used to indicate the quantity of an item in mathematics. Our number system contains various types of numbers. In this article, we will look at rational numbers, one of the number types in mathematics, along with some examples, and their properties. 

What is a rational number?

A rational number is a kind of real number that takes the form m/n where n is a non-zero number. In mathematics, the letter “Q” denotes a rational number.

How do you find a rational number?

A rational number has to meet the three conditions listed below:

• It can be represented as a simple fraction m/n (as a numerator divided by a denominator).
• Both the denominator and the numerator must be normal integers. 
• The denominator(n) must be a non-zero integer.

Some Common Examples

m	n	m/n	m/n(decimal)	Is it rational?
3	1	3/1	3	Yes
1	4	1/4	0.25	Yes
2	7	2/7	0.28	Yes
6	17	6/17	0.35	Yes
16	0	16/0	–	No

Types

Positive rational number

A positive rational number is one that has the same sign in both the numerator and the denominator. 

For example, 5/8 is a positive rational number as both 5 and 8 are positive. Similarly, 45, 0.5, 7, 0.75, are some other examples of a positive rational number. Here, 0.5 and 0.75 can also be written as 1/2 and 3/4 respectively.  

Negative rational number

A negative rational number is one in which the numerator and denominator have opposite signs.

For example, -2/7 is a negative rational number. Some other examples include -3/8, -0.25, -9, -17/34 etc. 

Integer form 

As we can write integers in the form of m/n, all integers are rational numbers. 

For example, 9/1, 6/1, 25/1 etc.

Decimal form

Recurring decimal numbers, both terminating and non-terminating, come under rational numbers.

For example, 0.45, -0.98, 1.65 are rational numbers.

Represent rational numbers on a number line. 

Let’s start by drawing a number line because we’ll be learning how to represent a rational number on it.

In the above image, the points to the right of the 0 are denoted by a positive sign and are known as positive numbers. On the other hand, the numbers to the left of 0 are denoted by a negative sign and are called negative numbers.

For example, let us try to represent the numbers 3/4 and -3/4 on the number line. As a rational number, -3/4 is negative. It is marked on the left of 0. Contrary to this, 3/4 is marked on the right of 0 (refer to the image below).

All other rational numbers can be expressed in a similar fashion.

Properties of rational numbers

Closure Property

When two rational numbers, a and b, are subtracted, added, or multiplied, the outcome is always a rational number. The Closure Property does not apply to the division since division by zero is not defined. 

For example,

2/3+ 4/5=22/15

6/7- 8/21=10/21

Commutative Property

When two rational integers, a and b, are considered, multiplication and addition are always commutative. 

Commutative law of addition states that, a+b=b+a 

For example, 2/3+1/2=1/2+2/3=7/6

Commutative law of multiplication states that a*b=b*a 

For example, 2/3*1/2=1/2*2/3=1/3

Associative Property

According to the associative property for addition, a+(b+c)=(a+b)+c.

For example, 1/2+(2/3+2)=(1/2+2/3)+2=19/6

Similarly, according to the associative property for multiplication, a(bc)=(ab)c

For example, 1/2(2/3*2)=1/2*2/3)2=2/3

Distributive Property

For the three rational numbers, a,b, and c; a(b+c)=(ab)+(bc). 

Ex: 1/2(2/3+2/5) =(1/2*2/3)+(1/2*2/5)

8/5=8/5

Thus, LHS=RHS

Identity Property

0 is the additive identity and 1 is known as the multiplicative identity. This implies that the number remains the same after addition with 0 and multiplication with 1 respectively. 

Ex: 1/2+0=1/2 (Additive identity)

4/5*1=4/5 (Multiplicative identity)

Inverse Property

For a rational number m/n, the additive inverse is -m/n and the multiplicative inverse is n/m.

For example, the additive inverse of 1/2 is -1/2 and the multiplicative inverse is 2/1.

Other properties of rational numbers

Property 1:

If m/n is a rational number and x is a non-zero integer then m/n=(m*x)/(n*x).

The rational number remains unchanged if we multiply both the numerator and denominator with the same number.

For example, 1/2=1*6/2*6=6/12

Property 2:

If m/n is a rational number and x is a common divisor then m/n=(m÷x)/(n÷x)

Therefore, when a rational number’s numerator and denominator are divided by a common divisor, the rational number stays unaffected.

For example, 12/14=12÷2/14÷2=6/7

Property 3:

If we let m/n and o/p be two rational numbers, then m/n=o/p ⇒ m*p=n*o

For example, 1/2=2/3

1*3=2*2

Property 4:

For every rational number a, the following conditions hold true. 

• x=0
• x<0
• x>0

For example, 2/3 is greater than 0.

0/3 is equal to 0.

-1/2 is less than 0.

Property 5:

Any one of the following conditions holds true for any two rational integers a, b.

• a=b
• a>b
• a<b

For example, 2/3 and 3/4 are two rational numbers. 3/4 is greater than 2/3.

If 3/6 and 7/114 are two rational numbers, then 3/6=7/14

Property 6:

If a > b and b > c holds true for three rational numbers, then a>c.

For example, for three rational numbers 7/9, 13/30 and -5/16, 7/9 is greater than 13/30, and 13/30 is greater than -5/16. Thus, 7/9 is greater than -5/16.

Relationship between rational and irrational number

We all know that numbers that are not logical are referred to as irrational numbers. Given below is the comparison between a rational and an irrational number:

• A number that can be written as a fraction is called a rational number. For example, 4/6, 6/7 4/8 etc. Irrational numbers are those that cannot be stated as fractions. For example, square root of 2, pi etc.
• A rational number can be represented as non-terminating decimals with repetitive patterns, whereas the same cannot be said about an irrational number.
• Whole numbers, Natural numbers, and integers are all included in the set of rational numbers. The set of irrational numbers is distinct from the other sets of numbers, as it contains none of them.

Final thoughts

Rational numbers is an essential concept to understand in mathematics. This comprehensive article can serve to be perfect for beginner students as it encompasses all the essential information regarding rational numbers. For more information and worksheets on the topic, visit Podium School.