√2 Proof - Hippasus
Assumption / Hypothesis:
√2 ∈ Q (Rational numbers)
Thus √2 can be expressed as a ratio of two integers / whole numbers.
Proof:
Hippasus labelled the numerator “p” and the denominator “q” of his reduced
fraction separated with a fraction line. These numbers cannot have any common
factor because they are relatively prime numbers. Hence their highest
common factor is 1.
p Î Z
q Î Z \
{0}
√2= p / q / x2
2 = q2 / p2 /*q2
2q2 =
p2
- 2q2 → Multiplying
any number by an even number results in an even number as well. The product of
the multiplier and the multiplicand is always even.
Examples:
2 * 4 = 8
2 * 5 = 10
etc…..
- p2 → Thus, “p” has
to be even, too. Taking the square of any number that results in an even number
has to even, as well.
Examples:
22 = 4
42 =
16
62 =
36
etc…..
- Since p2 is even, it can be expressed as (2a)2 where “a” is an integer
→ a ∈ Z
2q2 = (2a)2 / simplifying
2q2 = 4a2 / divide
both sides by 2
q2 ≠ 2a2 (unequal)
Since both sides are even their smallest
common factor has to be 2
→contradiction 
Conclusion:
√2∉ Q → √2 ∈ Q*
Thus, √2 is an irrational number because it cannot be expressed as a fraction of two integers. Hence, its decimal value will never stop!
q.e.d. – Quod erat demonstrandum OR quite easily done!😉😉
Sources:
Math Joke:
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