Question

proff that under root 5 is irrational​

Answer 1

Answer:

Let us assume that √5 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

So,

√5 = p/q { where p and q are co- prime}

√5q = p

Now, by squaring both the side

we get,

(√5q)² = p²

5q² = p² ........ ( i )

So,

if 5 is the factor of p²

then, 5 is also a factor of p ..... ( ii )

=> Let p = 5m { where m is any integer }

squaring both sides

p² = (5m)²

p² = 25m²

putting the value of p² in equation ( i )

5q² = p²

5q² = 25m²

q² = 5m²

So,

if 5 is factor of q²

then, 5 is also factor of q

Since

5 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

Hence √5 is an irrational number

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Answer 2

Answer:

let under root 5 is a rational number and its simplest form is p upon q

under root 5= p/q

square of both sides

5=. (p/q)2

5q square= p square

5 divides q square and 5 also divides p square

so, p and q has common factor 5

it is contradict to assume that root 5 is a rational number it is because p and q is a prime number

so root five is a irrational no .

Hence prove