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anonymous

  • 5 years ago

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

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  1. anonymous
    • 5 years ago
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    use google. or this site called wikipedia.

  2. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    exactly. sqrt(3) cannot be expressed in that form.

  4. anonymous
    • 5 years ago
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    Let us assume, to the contrary, that \[\sqrt{3}\] is a rational no

  5. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    Hope this clears it for u Gilbert

  7. anonymous
    • 5 years ago
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    Thank you so much, you are awesome.

  8. anonymous
    • 5 years ago
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    U r welcome

  9. anonymous
    • 5 years ago
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    don't forget to click on the medal tab

  10. anonymous
    • 5 years ago
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    thanks

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