Question

prove that √3 is an irrational number and hence prove that 2+√3 is also an irrational number

Answer 1

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Answer 2

Let us assume that √3 is a rational number
So,
√3=a/b
by cross cut the common numbers we got
√3=p/q
so here p and q co-prime
q√3=p
sq. both the sides
3q²=p²..........(1)
As 3 divides p²
so it is also divides p
now
p=3c
for c is any integer
now by sq.both the sides
p²=9c²
from (1)
3q²=9c²
q²=3c²
As 3 divides q²
so it is also divides q
Now 3 is divided by p and q both but we assume that they are co-prime
this had arise a condradiction because our intial assumption is wrong that √3 is rational so √3 is irrational



In second part
take
2+√3=p/q
√3=(p/q)+2
√3=(p+2q)/q
by cross cuit the common numbers we got
√3=a/b
and then follow the above steps
and if it is low number question then you cqan directly state that √3 is irrational
                      HENCE PROVED

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