Advertisements
Advertisements
Question
Prove that of the numbers `2 -3 sqrt(5)` is irrational:
Solution
Let `2 -3 sqrt(5)` be rational.
Hence 2 and `2 -3 sqrt(5)` are rational.
∴ 2 – ( `2 -3 sqrt(5) )` = 2 – `2 + 3 sqrt(5)` = ` 3 sqrt(5)` = rational [∵Difference of two rational is rational]
∴ `1/3 × 3sqrt(5) = sqrt(5)` = rational [∵Product of two rational is rational]
This contradicts the fact that `sqrt(5)` is irrational.
The contradiction arises by assuming `2 -3 sqrt(5)` is rational.
Hence, `2 -3 sqrt(5)` is irrational.
APPEARS IN
RELATED QUESTIONS
Classify the following number as rational or irrational:
1.101001000100001...
In the following equation, find which variables x, y, z etc. represent rational or irrational number:
`u^2=17/4`
Give an example of two irrationals whose sum is rational.
State whether the following statement is true or false. Justify your answer.
Every real number is an irrational number.
Given universal set =
`{ -6, -5 3/4, -sqrt4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, sqrt8, 3.01, π, 8.47 }`
From the given set, find: set of rational numbers
Check whether the square of the following is rational or irrational:
`3 + sqrt(2)`
Write the following in ascending order:
`7root(3)(5), 6root(3)(4) and 5root(3)(6)`
Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.
Classify the following number as rational or irrational with justification:
1.010010001...
Find three rational numbers between –1 and –2
