Square root Python 2.7.12

Question

Why does the math module return the wrong result?

First test

A = 12345678917
print \'A =\',A
B = sqrt(A**2)
print \'B =\',int(B)
         


        

Answer 1

Because math.sqrt(..) first casts the number to a floating point and floating points have a limited mantisse: it can only represent part of the number correctly. So float(A**2) is not equal to A**2. Next it calculates the math.sqrt which is also approximately correct.

Most functions working with floating points will never be fully correct to their integer counterparts. Floating point calculations are almost inherently approximative.

If one calculates A**2 one gets:

>>> 12345678917**2
152415787921658292889L

Now if one converts it to a float(..), one gets:

>>> float(12345678917**2)
1.5241578792165828e+20

But if you now ask whether the two are equal:

>>> float(12345678917**2) == 12345678917**2
False

So information has been lost while converting it to a float.

You can read more about how floats work and why these are approximative in the Wikipedia article about IEEE-754, the formal definition on how floating points work.