Question

prove that 2 root 3 -1 is an irrational number

Answer 1

Answer: let us assume to the contrary that 2 root 3-1 is rational no. so, there exits two positive co-prime integers a and b such that:

Step-by-step explanation:2 root 3 -1=a/b

2 root 3=a/b+1

2 root 3 =a+b/b

root 3=a+b/2b

here a, b , 2 are are integers and hence a+b/2b is rational.

so, root 3 will also be rational(as LHS=RHS).

BUT THIS CONTRADICTS THE FACT THAT ROOT 3 IS IRRATIONAL. THIS CONTRADICTION HAS ARISEN BECAUSE OF OUR INCORRECT ASSUMPTION THAT 2 root 3 -1 IS RATIONAL.

HENCE WE MAY CONCLUDE THAT 2 root 3-1 IS IRRATIONAL.

Answer 2

Answer:

Step-by-step explanation:let2√3-1 =a/b,(where and b are Co prime )

So ,√3-1=a/2b

√3=a/2b+1

So, a/2b+1 is rational number

So √3 is also rational but that contradict fact that √3 is an irrational number

So that this was happen due to our wrong assumption that 2√3-1 is rational number

So,we concluded that 2√3-1 is irrational