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Proof That The Square Root Of 2 Is Not Equal To A Decimal Fraction

by Steven Wagenheim
4 replies
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Okay, for those who are interested (probably nobody) the square root of 2
is not equal to a decimal fraction.

In other words...

square root of 2 is not equal to p/q

If we square both sides, we get 2=p squared over q squared.

If we multiply both side by q squared, we get 2q squared = p squared

Okay, here are our possibilities for p and q

p is odd and q is odd
p is even and q is even
p is odd and q is even
p is even and q is odd

For the statement, 2q squared = p squared to be true, it MUST be true
for all the above situations.

Therefore, if we can prove it's not true for just ONE of the situations,
then we have proven that the square root of two is not equal to a
decimal fraction.

Let's take this one.

p is odd and q is even

an odd number squared is always odd. So p squared for p being odd results
in an odd number.

Now let's go to q being even.

An even number squared is always even.

Any number multiplied by 2 is always even.

Therefore, 2q squared is an even number.

p squared = odd
q squared = even

odd does NOT equal even.

Therefore, the square root of 2 is NOT equal to a decimal fraction.

I hope you found this educational and entertaining.

Yes, math CAN be fun.
  • Profile picture of the author KimW
    I much prefer music trivia :-)
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  • Profile picture of the author Michael Ellis
    A week gone and this?!?

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  • Profile picture of the author HeySal
    Gee, Steve, that's such a relief. You have no clue how worried I was about that one.
    Whew.
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    Sal
    When the Roads and Paths end, learn to guide yourself through the wilderness
    Beyond the Path

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  • Profile picture of the author artwebster
    Therefore, the square root of 2 is NOT equal to a decimal fraction.

    Fascinating. What, pray, is a decimal fraction?

    The square root of 2 is akin to mathematical pi where the number of decimal places of the result just keeps on growing. I tried to follow your logic but it simply doesn't make any sense.

    To four decimal places the square root of 2 is 1.4142 and that can be a negative or positive figure, but it is still a decimal or a fraction
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    • Profile picture of the author Steven Wagenheim
      Originally Posted by artwebster View Post

      Therefore, the square root of 2 is NOT equal to a decimal fraction.

      Fascinating. What, pray, is a decimal fraction?

      The square root of 2 is akin to mathematical pi where the number of decimal places of the result just keeps on growing. I tried to follow your logic but it simply doesn't make any sense.

      To four decimal places the square root of 2 is 1.4142 and that can be a negative or positive figure, but it is still a decimal or a fraction
      Art, in simpler terms, the square root of 2 does not equal a specific
      number that can be calculated to a finite result.

      For example, the square root of 9 is 3. It is a finite number. The square
      root of 2 is not.

      Here are the definitions of a decimal fraction.

      Please note the last one. 0.25 is finite. It does not go in into infinity, which essentially is what
      the square root of 2 does. It does not have a finite result, thus it is not a decimal fraction.

      And please don't tell a math major that he doesn't make any sense.
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