Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

SamuelALDEN

  • 3 years ago

The square root of two is a rational number.

  • This Question is Closed
  1. UnkleRhaukus
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    False

  2. SamuelALDEN
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    True or false?

  3. Boblovesmath
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Sorry I meant it is an irrational number. So it is false.

  4. SamuelALDEN
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Thank you!

  5. UnkleRhaukus
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    prove by contradiction , assuming the square root of two is rational , it can be written as a ratio of natural numbers that have no common factors p,q say \[√2=p/q\] \[2=p^2/q^2\]\[2q^2=p^2\] this implies \(p^2\) is even if \(p^2\) is even then \(p\) is even if \(p \) is even it can be written as two times another natural number \(r\) \[p=2r\] \[2q^2=p^2=(2r)^2=4r^2\]\[q^2=2r^2\] which similarly implies that \(q\) is even With \(p,q\) both even the natural numbers necessarily have a common factor of two , which contradicts the statement of rationality, \[\sqrt2\not\in\text{rational numbers}\] \[\square\]

  6. Boblovesmath
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So in order to be rational the none of them should be even or at least one of them should be even and the other not even in order not to have a common factor of 2? Can you please explain this situation to me if you may?

  7. UnkleRhaukus
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    if the numer is rational there must be a way to express it as a ratio of (relatively prime) natural numbers, (ie no common factors)

  8. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy