## Proof

We will prove this by contradiction.

Let us add a rational number ( $=yx $), to an irrational number ( = p ). We assume that the result is a rational number ( $=ba $ ).

Putting these in an equation, we get:

$yx +p=ba $

$p=ba −yx $

$p=byay−xb $

We can express p as a rational number. This result contradicts the fact that it is an irrational number. Therefore our assumption (sum of rational and irrational numbers is a rational number) is incorrect. So this sum is an irrational number. This result holds for difference also.

## ✩ Known Irrationals

- Square Root of Prime ($Prime $): √2, √3, √5, √7, √11, √13, √17, √19 …
- Special Numbers: Pi ( π ) , Euler’s number ( e ), Golden Ratio
- Logarithms of primes with prime base: log
_{2}3, log_{3}5…

## Examples of Addition

- 4 + √3 is Irrational, as 4 is rational and √3 is irrational.
- 6 – √2 is Irrational, as 6 is rational and √2 is irrational.
- π – 2 is Irrational, as π is irrational and 2 is rational.
- √7 – 8 is Irrational, as √7 is irrational and 8 is rational.

## ✩ Irrational Result of Operations

Following operations between **rational and irrational** numbers result in an **irrational number**. Whatever the order of operations, the outcome is always an irrational number.

- Rational + Irrational: [ 3 + √2 ], [ 4 + √7 ], …
- Rational − Irrational: [ 5 – √2 ], [ √3 – 6 ], …
- Rational × Irrational: [ 4 × π = 4π ], [ 6 × √3 = 6√3 ], …
- Rational ÷ Irrational: [ 2 ÷ √2 ], [ π ÷ 2 ], …

## Sum of two Irrational numbers

What happens when we add two irrational numbers? In this case, the **resulting number may be rational or irrational**. Let us see some examples.

### Examples

In the following examples a and b are irrational numbers.

#### 1. Example

a = (2 + √5), b = (3 − √5)

We find the sum a + b:

a + b = (2 + √5) + (3 − √5) = 5

Result is a rational number.

#### 2. Example

a = π, b = 6 + π

Let us find the sum of a and b:

a + b = π + (6 + π) = 6 + 2π

Result is an irrational number.

## Related

**Examples of Irrational Numbers (With Lists) ➤**

**Square root of a prime number is Irrational ➤**

**Product of rational & irrational numbers is irrational: Proof, Examples ➤**