All unnatural square roots are irrational
Proof Ad absurdum: Assume there exists a natural number that is not whole, but is rational.
Therefore can be notated as an irreducible fraction:
Because the fraction is irreducible, none of the factors can cancel each other out.
For any and , .
Now look at :
Because is a natural number, all the denominator factors are supposed to cancel out,
but this is impossible because for any and , .
Therefore must be irrational.
Unfortunately this means that a (non-integer) fraction can never become whole by simply squaring, cubing, etc.
I call this unsatisfying fact my ”Greenfield Lemma”.
|Title||All unnatural square roots are irrational|
|Date of creation||2013-03-22 17:37:03|
|Last modified on||2013-03-22 17:37:03|
|Last modified by||ubershmekel (18723)|
|Synonym||The square of a fraction is always a fraction|