Every one of you is already aware of Rational Numbers and Irrational Numbers. Do you know there are some operations that you can carry out with these real numbers? Let us now study in detail about the operations such as addition, multiplication, subtraction, and division of the rational and irrational numbers.

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## Rational and Irrational Numbers

An **irrational number** is a number that is not rational that means it is a number that cannot be written in the form \( \frac{p}{q} \). An irrational Number is a number on the Real number line that cannot be written as the ratio of two integers. They cannot be expressed as terminating or repeating decimals. For example.

- Pi = 3.14…..It continues forever and never repeats. The few digits of this pattern look like 3.1415926535897932384626433832795
- √
*3*= 1.732050807. So one of the most important thing about irrational numbers is that it never repeats and never terminates.

**Browse more Topics under Number Systems**

- Rational Numbers
- Irrational Numbers
- Irrational Numbers Between two Numbers
- Laws for Exponents for Real Numbers

## Operations on Irrational Numbers

Let us now see the operations of the irrational numbers and the pattern they follow.

### Addition of the Irrational Numbers

Irrational Number + Irrational Number = May or may not be an Irrational Number

Example: √2 = 1.414… , √3 = 1.732… , √5 = 2.236…

Let us add these irrational numbers √2 + √3

= 1.414… + 1.732…

= 3.146……

We can see the pattern which we get from adding these numbers is non-repeating and non terminating. So this makes the entire number an irrational number. This is not true in all cases. Let us see another example.

( 5 – √2 ) + √2

= 5 – √2 + √2 = 5

We know 5 is definitely an irrational number.

You can see that in the first example when we add two irrational numbers, the result is an irrational number. But in the second example, the addition of two irrational numbers gives us a rational number. Because of this, we say that the addition of two irrational numbers may or may not be an irrational number.

### Subtraction of the Irrational Numbers

Irrational Number – Irrational Number = May or may not be an Irrational Number

Again we take the same roots as above.

√2 = 1.414… , √3 = 1.732… , √5 = 2.236…

Let us subtract these irrational numbers

√3 – √2

= 1.732… – 1.414… = 0.318…

Again we see the pattern which we get from subtracting these numbers is non-repeating and non terminating. So this makes the entire number an irrational number. But,

( 5 + √2 ) – √2

So we get = 5 + √2 – √2 = 5

5 is an irrational number. So this example makes it clear that subtraction of two irrational numbers may or may not be an irrational number.

### Multiplication of the Irrational Numbers

Irrational Number × Irrational Number = May or may not be an Irrational Number

√2 = 1.414… , √3 = 1.732… , √5 = 2.236…

Let us multiply these irrational numbers

√2 – √5

= 1.414… × 2.236… = 3.162….

So again this number is non-repeating and non terminating. So this makes the entire number an irrational number. Let us see another example

( 5 √3 ) × √3

= 5 × √3 × √3 = 15

Here 15 clearly is a rational number. Because of this, we say that the multiplication of two irrational numbers may or may not be an irrational number.

### Division of the Irrational Numbers

Irrational Number / Irrational Number = May or may not be an Irrational Number

√2 = 1.414… , √3 = 1.732… , √5 = 2.236…

\( \frac{√2}{√3} \) = \( \frac{ 1.732…}{1.414…} \) = 1.2234…

We see the pattern which we get from dividing these numbers is non-repeating and non terminating. So this makes the entire number an irrational number. But,

\( \frac{5√5}{√5} \) = 5

Here 5 is a rational number. Because of this, we say that when we divide two irrational numbers we may or may not get an irrational number.

## Solved Examples for You

**Question 1: (16) − (14√2 )**

**2√2****2****30√2****16 – 14√2**

**Answer:** The correct option is “D”. (16) − (14√2 ) is a rational number and 14√2 is an irrational number. We cannot add rational and irrational numbers directly. So we cannot solve the given numbers directly. Therefore, 16 – 14√2 is the answer.

**Question 2: From the pairs of the number given below, whose product is a Rational and Irrational Numbers?**

**√12, √3****√4, √3****√10, √3****√2, √3**

**Answer **: Products of the numbers are as follows

√12 × √3 = 6 ⇒ natural number

√4 × √3 = √12 ⇒irrational number

√10 × √3 = √30 ⇒irrational number

√2 × √3 = √6 ⇒irrational number

**Question 3: What is the difference between rational and irrational numbers?**

**Answer:** The difference between rational and irrational numbers is that an irrational number refers to a number that can’t be articulated in a ratio of two integers. On the other hand, in rational numbers, both numerator and denominator are whole numbers, where the denominator does not equal to zero. Whereas, you can’t write an irrational number in a fraction.

**Question 4: Are numbers irrational?**

**Answer:** Irrational numbers refer to real numbers that, when articulated as a decimal, go on forever after the decimal and also do not ever repeat. In other words, they are numbers which we cannot express as the ratio of two whole numbers. When we express it as a decimal, irrational numbers go on forever after the decimal point and do not repeat ever.

**Question 5: Who invented irrational numbers?**

**Answer:** Hippasus is one who invented these numbers. He is occasionally attributed with the invention of the irrational numbers, after which he was drowned at sea. Pythagoreans stated that all numbers can be articulated as the ratio of integers, thus the invention of irrational numbers did shock them.

**Question 6: Is 34 a rational number?**

**Answer:** The number 34 is a rational number if we can be express 34 as a ratio, as in rational. A quotient is an outcome we will get when we will divide one number by the other number. Thus, you see that for 34 to be a rational number, the quotient of the two integers must be equal to 34. Therefore, yes, 34 is a rational number.