上QQ阅读APP看书，第一时间看更新

Real numbers

Real numbers, or floating point numbers, are represented in Python according to the IEEE 754 double-precision binary floating-point format, which is stored in 64 bits of information divided into three sections: sign, exponent, and mantissa.

Quench your thirst for knowledge about this format on Wikipedia: http://en.wikipedia.org/wiki/Double-precision_floating-point_format.

Usually, programming languages give coders two different formats: single and double precision. The former takes up 32 bits of memory, and the latter 64. Python supports only the double format. Let's see a simple example:

>>> pi = 3.1415926536  # how many digits of PI can you remember?
>>> radius = 4.5
>>> area = pi * (radius ** 2)
>>> area
63.617251235400005

In the calculation of the area, I wrapped the radius ** 2 within braces. Even though that wasn't necessary because the power operator has higher precedence than the multiplication one, I think the formula reads more easily like that. Moreover, should you get a slightly different result for the area, don't worry. It might depend on your OS, how Python was compiled, and so on. As long as the first few decimal digits are correct, you know it's a correct result.

The sys.float_info struct sequence holds information about how floating point numbers will behave on your system. This is what I see on my box:

>>> import sys
>>> sys.float_info
sys.float_info(max=1.7976931348623157e+308, max_exp=1024, max_10_exp=308, min=2.2250738585072014e-308, min_exp=-1021, min_10_exp=-307, dig=15, mant_dig=53, epsilon=2.220446049250313e-16, radix=2, rounds=1)

Let's make a few considerations here: we have 64 bits to represent float numbers. This means we can represent at most 2 ** 64 == 18,446,744,073,709,551,616 numbers with that amount of bits. Take a look at the max and epsilon values for the float numbers, and you'll realize it's impossible to represent them all. There is just not enough space, so they are approximated to the closest representable number. You probably think that only extremely big or extremely small numbers suffer from this issue. Well, think again and try the following in your console:

>>> 0.3 - 0.1 * 3  # this should be 0!!!
-5.551115123125783e-17

What does this tell you? It tells you that double precision numbers suffer from approximation issues even when it comes to simple numbers like 0.1 or 0.3. Why is this important? It can be a big problem if you're handling prices, or financial calculations, or any kind of data that needs not to be approximated. Don't worry, Python gives you the decimal type, which doesn't suffer from these issues; we'll see them in a moment.