# Irrational Numbers

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

**Example :**π ( π = 3.1415926535897932384626433832795 (and more...),

√2, √3 ,many cube roots, golden ratio Φ,

**e**Euler’s number etc.

_________________________________________________________________

**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.

_________________________________________________________________

**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.

_________________________________________________________________

**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.

_________________________________________________________________

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

• 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

Home