Advertisements
Advertisements
Question
Prove that the following number is irrational: 3 - √2
Solution
3 - √2
Let 3 - √2 be a rational number.
⇒ 3 - √2 = x
Squaring on both the sides, we get
( 3 - √2 )2 = x2
⇒ 9 + 2 - 2 x 3 x √2 = x2
⇒ 11 - x2 = 6√2
⇒ √2 = `[ 11 - x^2 ]/6`
Here, x is a rational number.
⇒ x2 is a rational number.
⇒ 11 - x2 is a rational number.
⇒ `[ 11 - x^2 ]/6` is also a rational number.
⇒ `sqrt2 = [ 11 - x^2 ]/6` is a rational number.
But √2 is an irrational number.
⇒ `[ 11 - x^2 ]/6 = sqrt2` is an irrational number.
⇒ 11 - x2 is an irrational number.
⇒ x2 is an irrational number.
⇒ x is an irrational number.
But we have assume that x is a rational number.
∴ we arrive at a contradiction.
So, our assumption that 3 - √2 is a rational number is wrong.
∴ 3 - √2 is an irrational number.
APPEARS IN
RELATED QUESTIONS
Classify the following number as rational or irrational:
`sqrt225`
Identify the following as rational or irrational number. Give the decimal representation of rational number:
`sqrt4`
In the following equation, find which variables x, y, z etc. represent rational or irrational number:
v2 = 3
Use method of contradiction to show that √3 and √5 are irrational numbers.
Prove that the following number is irrational: √5 - 2
Find a rational number between `sqrt2` and `sqrt3`
Show that `(2+3√2)/7` is not a rational number, given that √2 is an irrational number.
The product `root(3)(2) * root(4)(2) * root(12)(32)` equals to ______.
Show that x is irrational, if x2 = 27.
Prove that `3 - 2sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.
