Advertisements
Advertisements
प्रश्न
Prove that the following is irrational:
`1/sqrt2`
उत्तर
`1/sqrt2`
`1/sqrt2 xx sqrt2/sqrt2 = sqrt2/2`
Let a = `(1/2)sqrt2` be a rational number.
∴ `1/2 (sqrt2)` is rational.
Let `1/2 (sqrt2) = a/b`, such that a and b are co-prime integer and b ≠ 0.
∴ `sqrt2 = (2a)/b` ...(1)
Since the division of two integers is rational.
∴ `(2a)/b` is rational.
From (1), `sqrt2` is rational, which contradicts the fact that `sqrt2` is irrational.
∴ Our assumption is wrong.
Thus, `1/sqrt2` is irrational.
APPEARS IN
संबंधित प्रश्न
Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
In the following equation, find which variables x, y, z etc. represent rational or irrational number:
v2 = 3
Find one irrational number between 0.2101 and 0.222... = `0.bar2`
Classify the numbers π as rational or irrational:
Prove that of the numbers ` 2 - sqrt(3)` is irrational:
Prove that of the numbers `3 + sqrt (2)` is irrational:
Prove that of the numbers `5 + 3 sqrt (2)` is irrational:
Prove that of the numbers `2 -3 sqrt(5)` is irrational:
Give an example of two irrationals whose product is rational.
State whether the given statement is true or false:
1 .The sum of two irrationals is an irrational
Prove that 5`sqrt(2)` is irrational.
Find the square of : √5 - 2
Find a rational number between `sqrt2` and `sqrt3`
Without using division method show that `sqrt(7)` is an irrational numbers.
Write a pair of irrational numbers whose sum is irrational.
Compare the following:
`root(3)(48) and sqrt(36)`
Write the following in ascending order:
`2root(3)(3), 4root(3)(3) and 3root(3)(3)`
Write the following in ascending order:
`7root(3)(5), 6root(3)(4) and 5root(3)(6)`
Write the following in descending order:
`sqrt(6), root(3)(8) and root(4)(3)`
Number of rational numbers between 15 and 18 is finite.
