## anonymous 5 years ago Define rational numbers. show that root 3 is not a rational number.

1. anonymous

use google. or this site called wikipedia.

2. anonymous

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

3. anonymous

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

4. anonymous

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

5. anonymous

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.

6. anonymous

Hope this clears it for u Gilbert

7. anonymous

Thank you so much, you are awesome.

8. anonymous

U r welcome

9. anonymous

don't forget to click on the medal tab

10. anonymous

thanks