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You might have come across several applications of** rational numbers** in your daily life without being aware of their importance and value in mathematics. In simpler words, **rational numbers** are any numbers that are involved in various mathematical applications, for example, addition, division, subtraction, and multiplication. You will also learn about different kinds of properties of **rational numbers.**

- They are the set numbers starting from 1 and ending at infinity.
- This set is denoted by ‘N’.

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- They are the set of numbers starting from 0 and ending at infinity.
- The set is denoted by a ‘W’.

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- They are a set of natural numbers with the addition of their negatives.
- The set is denoted by a ‘Z’.

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In this section, a proper** introduction to rational numbers** will be given to you. **Rational numbers **can be represented in a p/q form, where both p and q are integers and q is unequal to zero.

This set of numbers is denoted by ‘Q’.

For example- -6/8, 2/5, -5,and 4

**Properties of Rational Numbers **

Below are some critical** properties of rational numbers:**

**1. Closure Property**

For any two random rational numbers, say ‘a’ and ‘b’, a∗b=c∈Q.

- * is any binary operation (multiplication, addition, subtraction, and division)
- C is the product of the binary operation applied between a and b.
- All three numbers should belong to the set of rational numbers

**Addition**

Closure property pertains to rational numbers under addition.

For example:

3/5+ 1/2= (6+5)/10 = 11/10

**Subtraction**

Closure property pertains to rational numbers under subtraction.

For example:

4/2- 1/2 = (4-1)/2 = 3/2

**Multiplication**

Closure property pertains to rational numbers under multiplication.

For example:

4/7 x -2/5 = -8/35

**Division**

Closure property does not pertain to rational numbers under addition.

For rational number s, s÷0= not defined.

**2. Commutative Property**

For any two random rational numbers, say ‘a’ and ‘b’, a∗b= b*a. Therefore, the result of the equation should be constant regardless of the order of the operands.

**Addition**

Commutative property pertains to rational numbers under addition because for any two rational numbers a and b-

(a+b) = (b+a)

**Subtraction**

The commutative property does not pertain to rational numbers under subtraction because for any two rational numbers a and b-

(a-b) is unequal to (b-a)

**Multiplication**

Commutative property pertains to rational numbers under multiplication because for any two rational numbers a and b-

(a x b) = (b x a)

**Division**

The commutative property does not pertain to rational numbers under addition because for any two rational numbers a and b-

(a÷b) is unequal to (b÷a)

**3. Associative Property**

For any three random rational numbers, say ‘a’, ‘b’ and ‘c’, (a∗b)∗c=a∗(b∗c). Therefore, the result of the equation should be constant regardless of the order of the operands.

**Addition**

Associative property pertains to rational numbers under addition because for any three rational numbers a, b and c-

(a+b)+c = a+(b+c)

**Subtraction**

The associative property does not pertain to rational numbers under subtraction because for any two rational numbers a, b and c-

(a−b)−c≠a−(b−c) because (a-b)-c = a-b-c whereas a-(b-c) = a-b+c.

**Multiplication**

Associative property pertains to rational numbers under multiplication because for any three rational numbers a, b and c-

(a×b)×c=a×(b×c)

**Division**

The associative property does not pertain to rational numbers under addition because for any three rational numbers a, b and c-

(a÷b)÷c≠(a÷b)÷c

**4. Distributive Property**

For any three rational numbers a, b and c-

a(b+c)=ab+aca(b−c)=ab−ac

When zero is added to a random rational number, the result remains constant. Mathematically it is represented as-

For a rational number p/q, p/q+ 0 = p/q

Zero, in this case, is referred to as additive identity.

If (p/q)+(−p/q)=(−p/q)+(p/q)=0 then the additive inverse or the negative of a rational number pq is -pq.

When one is multiplied to a random rational number, the result remains constant. Mathematically it is represented as-

For a rational number p/q, p/q x 1 = p/q

1, in this case, is referred to as multiplicative identity.

If p/q x r/s = 1 then r/s is the multiplicative inverse of p/q. Also, p/q is the reciprocal of the multiplicative inverse r/s.

Representation of** rational numbers on the number line** can be divided into two steps.

Step 1: Equally divide the distance between the two consecutive integers in the ‘n’ number of parts.

Step 2: label the rational numbers on the number line until it contains the number you have to mark.

The number of** rational numbers between rational numbers **is indefinite.

**How to find out rational numbers between rational numbers:**

**Method 1**

Ensure that the two given natural numbers have the same denominator. Once that is settled, you can pick out any rational number that lies between them.

**Method 2**

We can invariably find a rational number between two** rational numbers** by mathematically calculating their midpoint or mean.

**To sum up**

We hope that you understood the nuances of rational numbers and all the properties associated with them.

**What is a rational number in maths?**

Rational numbers are any numbers that are involved in various mathematical applications, for example, addition, division and etc.

**What are five examples of rational numbers?**

-7, 5/6, -7/9, 3, 9.

**Is zero a rational number?**

Yes, zero is a rational number as it can be written as p/q, where p= 0 and q= non zero.

**Is 5 a rational number?**

Yes. 5 can be written as 5/1.

**How can you identify a rational number?**

Any number that can be written as a fraction, where the denominator is unequal to zero.

**Is 2/3 a rational number?**

Yes. 2/3 is written in the p/q format and 3 unequal to zero.

MSVgo is an excellent educational application that has a video library containing a myriad of videos explaining through animations and visualization on any topic from any subject you desire. Head over to the app to learn the concepts of rational numbers and much more.

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