A funny thing I saw the other day…
(1 – 0)2 = (0 – 1)2 //which is true, then adds square roots on both sides
√(1 – 0)2 = √(0 – 1)2 //which again is true… extracting the square roots
1 – 0 = 0 – 1 //forgetting the modulus
1 = –1
sophism
Math sophism on Pythagorean theorem
A good math sophism is like an old wine; and if you are like me, then, you would enjoy a “sophism” from time to time. Maths less obvious mistakes are like miss prints on stamps.
An elegant example of such mathematical fallacy is good to keep an idiot busy for couple of days. The one I’ll describe next is just what you need.
It starts from the cos2x + sin2x = 1 which is the trigonometric form of the Pythagorean theorem, and it goes like this:
cos2x + sin2x = 1
cos2x = 1 – sin2x
cos x = (1 – sin2x)1/2
1 + cos x = 1 + (1 – sin2x)1/2
(1 + cos x)2 = (1 + (1 – sin2x)1/2)2
for x = π (pi = 3,14159265…), we have,
(1 – 1)2 = (1 + (1 – 0)1/2)2
0 = (1 + 1)2
0 = 4 //interesting ?!
The mistake is a little hard to notice, but the result should tell you that something is wrong.