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
( contradiction 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
Addition rational numbers
Subtraction of rational numbers
Conversion of rational numbers to decimal
Irrational Numbers

Number system Page

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