Question

Prove that 3 is an irrational number

Answer 1

Step-by-step explanation:

It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. It is denoted by √3. The square root of 3 is an irrational number. It is also known as Theodorus' constant, named after Theodorus of Cyrene, who proved its irrationality.

Let us assume that √3 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,

√3 = p/q { where p and q are co- prime}

√3q = p

Now, by squaring both the side  

we get,

(√3q)² = p²

3q² = p² ........ ( i )

So,

if 3 is the factor of p²

then, 3 is also a factor of p ..... ( ii )

=> Let p = 3m { where m is any integer }

squaring both sides

p² = (3m)²

p² = 9m²

putting the value of p² in equation ( i )

3q² = p²  

3q² = 9m²

q² = 3m²

So,

if 3 is factor of q²

then, 3 is also factor of q

Since

3 is factor of p & q both

So, our assumption is wrong

hence, √3 is an irrational number.