# Operations on Real Numbers

A rational number is a number which can be written as a ratio. Every rational number has a numerator and a denominator, that is, one integer divided by another integer. Whereas the numbers that cannot be expressed as a ratio of two integers are the irrational numbers, that is, the numbers that are not rational are irrational. The rational and the irrational numbers together form the **real numbers**.

Operations on Real Numbers: Some Rules

The following pointers are to be kept in mind when you deal with real numbers and mathematical operations on them:

- When the addition or subtraction operation is done on a rational and an irrational number, the result is an irrational number.
- When the multiplication or division operation is done on a rational number with an irrational number, the result is an irrational number.
- When two irrational numbers are added, subtracted, multiplied or divided, the result may be a rational or an irrational number.

If *a* and *b *are positive real numbers, then we have,

- √ab = √a √b

- (√a + √b)(√a – √b) = a – b
- (a + √b)(a – √b) = a
^{2}– b - (√a + √b)(√c + √d) = √ac + √ad + √bc + √bd
- (√a + √b)
^{2}= a + 2√ab + b

#### Examples

- Solve (2√2 + 7√7) + (13√2 – 4√7).

**Solution**:

(2√2 + 7√7) + (13√2 – 4√7)

= (2√2 + 13√2) + (7√7 – 4√7)

= (2 + 13)√2 + (7-4)√7

= 15√2 + 3√7

- Solve (7√7 x – 4√7)

**Solution**:

(7√7 x – 4√7)

= 7 x -4 x √7 x √7

= -28 x 7 = -196

- Solve (8√21 / 4√7)

**Solution**:

(8√21 / 4√7)

= (8√7 x √3 / 4√7)

= 2 x √7 x √3 / √7 = 2√3

- Solve (2√2 + 7√7)(2√2 – 7√7).

**Solution**:

(2√2 + 7√7) (2√2 – 7√7)

= (2√2)^{2} – (7√7)^{2}

= 4 x 2 – 49 x 7

= 8 – 343 = -335

- Solve (√2 + √7)(√3 – √11).

**Solution**:

(√2 + √7)(√3 – √11)

= √2√3 – √2√11 + √7√3 – √7√11

= √6 – √22 + √21 – √77