prove that √11 is an irrational number using this prove that 3-4/5√11 is an irrational no.
Answers
Firstly we assume that √11 is a rational number.
A rational number can be written in the form of p/q, where q ≠ 0 and p , q are positive number.
√11 = p/q ….( Where p & q are co prime number )
Square both sides, we get
11 = p²/q²
11 q² = p² ….(a)
p² is divisible by 11. So, p will also divisible by 11
Let p = 11 r ( Where r is any positive integer )
Squaring both sides
p² = 121r²
Putting this value in eqn(a)
11 q² = 121 r²
q² = 11 r²
q² is divisible by 11. So, q will also divisible by 11. Since, p and q both are divisible by same number 11. So, they are not co-prime .
Hence Our assumption is Wrong. Therefore, √11 is an irrational number .
Firstly we assume that √11 is a rational number.
A rational number can be written in the form of p/q, where q ≠ 0 and p , q are positive number.
√11 = p/q ….( Where p & q are co prime number )
Square both sides, we get
11 = p²/q²
11 q² = p² ….(a)
p² is divisible by 11. So, p will also divisible by 11
Let p = 11 r ( Where r is any positive integer )
Squaring both sides
p² = 121r²
Putting this value in eqn(a)
11 q² = 121 r²
q² = 11 r²
q² is divisible by 11. So, q will also divisible by 11. Since, p and q both are divisible by same number 11. So, they are not co-prime .
Hence Our assumption is Wrong. Therefore, √11 is an irrational number .Firstly we assume that √11 is a rational number.
A rational number can be written in the form of p/q, where q ≠ 0 and p , q are positive number.
√11 = p/q ….( Where p & q are co prime number )
Square both sides, we get
11 = p²/q²
11 q² = p² ….(a)
p² is divisible by 11. So, p will also divisible by 11
Let p = 11 r ( Where r is any positive integer )
Squaring both sides
p² = 121r²
Putting this value in eqn(a)
11 q² = 121 r²
q² = 11 r²
q² is divisible by 11. So, q will also divisible by 11. Since, p and q both are divisible by same number 11. So, they are not co-prime .
Hence Our assumption is Wrong. Therefore, √11 is an irrational number .Firstly we assume that √11 is a rational number.
A rational number can be written in the form of p/q, where q ≠ 0 and p , q are positive number.
√11 = p/q ….( Where p & q are co prime number )
Square both sides, we get
11 = p²/q²
11 q² = p² ….(a)
p² is divisible by 11. So, p will also divisible by 11
Let p = 11 r ( Where r is any positive integer )
Squaring both sides
p² = 121r²
Putting this value in eqn(a)
11 q² = 121 r²
q² = 11 r²
q² is divisible by 11. So, q will also divisible by 11. Since, p and q both are divisible by same number 11. So, they are not co-prime .
Hence Our assumption is Wrong. Therefore, √11 is an irrational number .
now
let 3-4/5√11 is rational
on subtracting 3
We get -4/5√11 is rational ( as on subtracting rational from rational we get rational)
on dividing by -4/5 we get
√11 as rational (as on dividing rational by rational we get rational)
hence our assumption is wrong we know that √11 is irrational
so this is a contradiction
hence 3-4/5√11 is irrational