Question

सिद्ध करो कि रूट 5 एक अपरिमेय संख्या है​

Answer 1

Answer:

let √5 be a rational number

then a/c to definition

√5=p/q. ( where p and q are integers and q is not equal to zero and p and q are co-prime )

now;

√5=p/q

squaring both sides

5= p²/q²

5q²=p² ------(i)

q²=p²/5 ------(ii)

from equation (ii) it is clear that 5 divides p²

i.e 5 divides p,-----(a)

let p=5m and put it in equation (i)

5q²=(5m)²

5q²=25m²

q²=5m²

q²/5=m²-------(iii)

from equation (iii) it is clear that 5 divides q²

i.e 5 divides q-------(b)

from equation (a) and (b) 5 divides both p and q but we know that p and q are co-prime so it has no common factor

therefore; our assumption is wrong hence ✓ 5 is an irrational number.