non – repeating decimal representation.

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

_________________________________________________________________

√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

⇒ p

⇒ p

⇒ p is an even integer.

⇒ p = 2m where m is an integer.

Squaring both sides.

⇒ p

2q

q

⇒ q

⇒ 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.

_________________________________________________________________

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.

_________________________________________________________________

1) Identify √ 45 is a rational or ir-rational-number.

√ 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.

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

• 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

GMAT

GRE

1st Grade

2nd Grade

3rd Grade

4th Grade

5th Grade

6th Grade

7th grade math

8th grade math

9th grade math

10th grade math

11th grade math

Precalculus

Worksheets

Chapter wise Test

MCQ's

Math Dictionary

Graph Dictionary

Multiplicative tables

Math Teasers

NTSE

Chinese Numbers

CBSE Sample Papers