√2 Proof - Hippasus

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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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