Two old witches brew a magical rejuvenation potion in their kitchen. Before they drink the mysterious cocktail, one of the witches recites a spell. It begins with the puzzling verses of the witch`s one-times-one of Goethe's famous "Faust".
"This you must ken!
From one make ten,
And two let be,
Make even three,
Then rich you'll be.
With the witch's one-time-one, Johann Wolfgang von Goethe swirled the world of numbers properly. But who knows, maybe the famous poet was inspired by the equation below. This seems to prove that 2 = 1 – is that right?
a = b | • a We multiply both sides by “a”
a² = a b | + a² Now we add on both sides “a²”
a² + a² = a b + a² | - 2ab Then we subtract ”2ab”
a² + a² - 2ab = a b + a² - 2ab … and calculate (simplify)
2a² - 2ab = a² - ab … and exclude
2 • (a² - ab) = 1 • (a² - ab) | : (a² - ab) Finally, both sides are divided by “(a² - ab)”
2 = 1
The calculation begins with the equation a = b.
However, if a = b, then a² = ab and thus the expression (a ² - ab) is equal to zero. In the last step of the calculation it is therefore divided by zero, which is generally known to be prohibited in mathematics. Thus, the supposed proof that 2 = 1, is wrong and with the witch`s one-times-one Goethe cannot rely on mathematics but only to his poetic license.