## Education

# Know the Differences between Rational and Irrational Numbers

If the concepts of rational and irrational numbers confuse you, this article will help you clear all your doubts and confusion.

**Topperlearning Expert**10th Aug, 2020 08:06 pm

There are different kinds of numbers which can be grouped into sets. You may have already come across and even know the sets mentioned below.

**1. Natural Numbers:** The set of natural numbers starts from 1.

N = {1, 2, 3, 4, 5,…}

**2. Whole Numbers:** The set of whole numbers starts from 0.

W = {0, 1, 2, 3,…}

**3. Integers:** The set of integers are nothing but a collection of whole numbers and negative numbers (natural numbers with the minus sign).

I or Z = {…, –4, –3, –2, –1, 0, 1, 2, 3, 4,…}

Now, answer this question:

In which set does lie?

Think!!!

The answer: In the set of **rational numbers**.

Any number of the form , where *p* and *q* are both integers and *q* is not equal to zero, is called a rational number.

It can be written as Q =

Now, answer this:

will lie in which set?

Confused?

The answer: In the set of **irrational numbers.**

Any number which cannot be expressed in the form, where *p* and q are both integers and q is not equal to zero, is called an irrational number.

A set of irrational numbers is denoted by .

Now, we will see some **important properties **of rational and irrational numbers.

- The sum of two or more rational numbers is always a rational number. This means that the set of rational numbers is
**closed under addition.**If x and y are any two rational numbers, then x + y and y + x are also rational numbers. - The difference of two rational numbers is always a rational number. This means that the set of rational numbers is
**closed under subtraction.**If x and y are any two rational numbers, then x – y and y – x are also rational numbers. - The product of two or more rational numbers is always a rational number. This means that the set of rational numbers is
**closed under multiplication.**If x and y are any two rational numbers, then xy and yx are also rational numbers. - The division of a rational number by a non-zero rational number is always a rational number. This means that the set of rational numbers is
**closed under division if the divisor****≠****0.**If x and y are any two rational numbers, then when y ≠ 0 and when x ≠ 0 are also rational numbers. **Addition, subtraction, multiplication and division of any two irrational numbers is not necessarily an irrational number.**

Here are some pointers which will definitely make the concepts of rational and irrational numbers easier for you to understand.

**The decimal expansion of**

- Rational numbers are either terminating or non-terminating and recurring.

**Examples: **

Terminating decimal expansion

Non-terminating recurring decimal expansion

- Irrational numbers are non-terminating and non-recurring.

**Example: **Non-terminating, non-recurring decimal expansion

** **

**Terminating:** A **terminating decimal** doesn't keep going. A **terminating decimal** will have a finite number of digits after the **decimal** point. ** **

**Non-terminating and recurring (repeating):** A **non**-**terminating** and **repeating** decimal is a decimal number which continues endlessly, with the group of digits **repeating** endlessly. Decimals of this type can be represented as fractions.

**Non-terminating and non-recurring (non-repeating):** A **non**-**terminating** and **non**-**repeating** decimal is a decimal number which continues endlessly, with no group of digits **repeating** endlessly. Decimals of this type cannot be represented as fractions.** **

** **

If you want to find out whether the decimal expansion of a given rational number is terminating or non-terminating and repeating without actually performing the division, you need to follow the steps below:

If x = is a rational number.

1. If the prime factorisation of* q *is of the form 2^{n}5^{m}, where *m* and* n *are non-negative integers, then x has a decimal expansion which **terminates**.

**Example**:

2. If the prime factorisation of* q *is **not** of the form 2^{n}5^{m}, where *m* and* n *are non-negative integers, then x has a decimal expansion which is **non-terminating repeating.**

**Example**:

Now, we will solve some problems based on the concepts of rational and irrational numbers.

- Rationalise the denominator of .

**Solution: **

** **2. If x = , find the value of

**Solution:**

** **

We know that

(a + b)^{2} = a^{2} + b^{2 }+ 2ab

3. Find the two irrational numbers between 3 and 5.

**Solution: **

If a and b are two rational numbers, then the irrational number between them =

The concepts of rational and irrational numbers are very easy to understand and are a part of both CBSE and ICSE board. Get Selina solutions for Class 9, NCERT solutions for Class 7 and NCERT solutions for Class 8.

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