# Irrational Numbers

A number is an Irrational Numbers, if it has a non – terminating and
non – repeating decimal representation.

Example : π ( π = 3.1415926535897932384626433832795 (and more...),
√2, √3 ,many cube roots, golden ratio Φ,
e Euler’s number etc.
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Prove that √ 2 is an ir-rational.

Solution :
√2 is an ir-rational can be prove by negation method
Assume that √ 2 is a rational number.
⇒ √ p / q where p and q are integers having no common factor and q = 0.
Squaring both sides.
⇒ 2 = p
2 / q 2
⇒ p
2 = 2q 2 ---------> (1)
⇒ p
2 is an even integer.
⇒ p is an even integer.
⇒ p = 2m where m is an integer.
Squaring both sides.
⇒ p
2 = 4m 2
2q
2 = 4m 2 ( using (1))
q
2 = 2m 2
⇒ q
2 is an even integer.
⇒ q is an even integer.
So, both p and q are even integers.
∴ p and q have common factor .
But this contradicts to our assumption.
So our assumption is wrong.
∴ √ 2 is an ir-rational number.
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Some useful results on Irrational-numbers:

1) Negative of an ir-rational number is an irrational-number.
2) The sum of rational number and irrational number is an irrational-number.
3) The product of a non-zero rational number and an irrational number is an ir-rational-number.
4) The sum, difference, product and quotient of two ir-rational-numbers need not be an ir-rational-number.
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Some solved examples :

1) Identify √ 45 is a rational or ir-rational-number.
Solution :
√ 45 = √( 9 x 5)
= 3 √5
Since 3 is a rational number and √5 is an ir-rational- number.
∴ The product of 3√5 = √45 is an ir-rational- number.
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2) Insert a rational and ir-rational-number between 2 and 3.

Solution :
If a and b are two positive rational numbers such that ab is not perfect square of a rational number, then √ ab is an irrational-number lying between a and b.
Also, If a and b are rational numbers, then ( a + b ) / 2 is a rational number between them.
Here, a = 2 and b = 3.
So rational number between 2 and 3 is ( 2 + 3) / 2 = 2.5
And ir-rational-number between 2 and 3 is √(2 x 3) = √6

Rational number

Representation of rational number on number line
Comparison of rational number