Float to Fraction conversion in Python

Question

While doing exercise on the topic of float type to Fraction type conversion in Python 3.52, I found the difference between the two different ways of conversion.

The first method is:

>>> from fractions import Fraction
>>> x = 1232.23
>>> f = Fraction(*x.as_integer_ratio())
>>> print(f)
2709702426188841/2199023255552      #Answer

The second method is:

>>> from fractions import Fraction
>>> x = 1232.23
>>> f = Fraction(str(x))
>>> print(f)
123223/100                          #Answer

I want to know the reason behind these two different answers? Sorry if this is a stupid question , I am new to programming and Python.

Edited: I found a way to convert inaccurate fraction obtained by first method to accurate by limit_denominator method:

>>> from fractions import Fraction
>>> x = 1232.23
>>> f = Fraction(*x.as_integer_ratio())
>>> f = f.limit_denominator(100)     
>>> print(f)
123223/100

Answer 1

Yet again it's because floating point numbers aren't stored in base-10 (decimal), but in base-2 (binary).

A number that is finite length in base-10 might be a repeating decimal in base-2. And because floats are a fixed size, that repeating decimal gets truncated, resulting in inaccuracies.

When you use as_integer_ratio for a number that's a repeating decimal in base-2, you will get you a somewhat silly fraction as a result of the slight inaccuracies in the base-10 to base-2 conversion. If you divide those two numbers, the value will be very close to to your original number.

For instance, while 1/10 = 0.1 in base-10 and is not a repeating decimal, it is in fact a repeating decimal in base-2. Just like 1/3 = 0.333... in base-10.

>>> (0.1).as_integer_ratio()
(3602879701896397, 36028797018963968)

If Python's output was exact, you would see this even when you enter just 0.1 in the prompt, by getting something like 1.00000...01 as the output. But Python hides this inaccuracy from you in the general case, leading to confusion.

Answer 2

This is actually a pretty good question. The reason behind the different results is that x is not truly 1232.23 because there is no exact float representation for 1232.23, so the fractional representation of the nearest float representation of 1232.23 is 2709702426188841/2199023255552, but when you use str(1232.23) it treats it as the exact value 1232.23 and returns the true best fractional representation of the number.