Question

17] Prove that root 3 is an irrational number​

Answer 1

The value of √3= 1.73205080757

The definition of an irrational number is that it is a number which doesn't end and it doesn't have a group which is repeating itself.

√3 value is the same

Hope It helps.

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Answer 2

Answer:

let us prove this by the method of contradiction

let √3 be a rational number

therefore √3 = p/q

squaring both sides

3 = p²/q²

=> 3 q² = p² (1)

since p² is a multiple of 3, p² is divisible by 3 and p is also divisible by 3 ............(2)

hence we can write p = 3r where r is any integer

substituting p = 3r in equation (1)

we get 3 q² = (3r)² = 9r²

=> 3q² = 9r²

=> q² = 3 r² (3)

now from equation (2) we can say that q² is a multiple of 3 so q² is divisible by 3 hence

q is also divisible by 3..............(4)

thus we can conclude from the statements (2) and (4) that both p and q are divisible by 3 hence 3 is a common factor of p and q.

but for the number to be rational, p and q are to be co - prime numbers, means they can have only 1 as a common factor.

so our assumption is wrong, and √3 is an irrational number