What are the rational and irrational number give examples!

Recurring decimals such as 0.26262626…, all integers and all finite decimals, such as 0.241, are also rational numbers. Alternatively, an irrational number is any number that is not rational. ... For example, the square root of 2 is an irrational number because it cannot be written as a ratio of two integers.

Worked Examples

1 - recognize Surds

A surd is a square root which cannot be reduced to a whole number.

For example,

4–√=2

is not a surd, because the answer is a whole number.

Alternatively

5–√

is a surd because the answer is not a whole number.

You could use a calculator to find that

5–√=2.236067977...

but instead of this we often leave our answers in the square root form, as a surd.

2 - Simplifying Surds

During your exam, you will be asked to simplify expressions which include surds. In order to correctly simplify surds, you must adhere to the following principles:

ab−−√=a−−√∗b√

a−−√∗a−−√=a

Example

(a) - Simplify

27−−√

Solution

(a) - The surd √27 can be written as:

27−−√=9–√∗3–√

9–√=3

Therefore,

27−−√=33–√

Example

(b) - Simplify

12−−√3–√

Solution

(b) -

12−−√3–√=12−−√∗3–√=(12∗3)−−−−−−√=36−−√

36−−√=6

Therefore,

12−−√3–√=6

Example

(c) - Simplify

45−−√5–√

Solution

(c) -

45−−√5–√=45/5−−−−√=9–√=3

Therefore,

45−−√5–√=3