Question

Define rational numbers. show that root 3 is not a rational number.

Answer 1

use google. or this site called wikipedia.

Answer 2

Rational numbers are those numbers which can be expressed in the form p/q where p and q are integers and q is not zero

Answer 3

exactly. sqrt(3) cannot be expressed in that form.

Answer 4

Let us assume, to the contrary, that \[\sqrt{3}\] is a rational no

Answer 5

then root3 = p/q
where p and q are coprime integers and q is not zero
Squaring both sides of (I), we get
3= p^2/q^2 or 3q^2 = p^2 ---- (I)
Now 3q^2 is a multiple of 3 which means that it is divisible by 3
but p^2 is equal to 3q^2
Hence p^2 is also divisible by 3
This means that p is divisible by 3 ----(A)
So let p = 3r where r is an integer
Squaring this we get
p^2 = 9 r^2 ---- (II)
From (I) and (II) we get
3q^2 = 9r^2 or q^2 = 3r^2
This means that 3r^2 is a multiple of 3 which means it is divisible by 3
This means q^2 is also divisible by 3
This means that q is also divisible by 3 ---- (B)
From (A) and (B) we find that p and q have a common factor 3.
But in the beginning we said that p and q are coprime that is their HCF is 1.
This contradiction has arisen due to our wrong assumption that root3 is rational.
Hence root3 is irrational.

Answer 6

Hope this clears it for u Gilbert

Answer 7

Thank you so much, you are awesome.

Answer 8

U r welcome

Answer 9

don't forget to click on the medal tab

Answer 10

thanks