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

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

Mathematics
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use google. or this site called wikipedia.
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
exactly. sqrt(3) cannot be expressed in that form.

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