Floats

2. Basic Types: Numbers

Floats

You might have noticed that division was the only operator to return a number with a fractional component or decimal, for example #\mathtt{\text{3.0}}# instead of #\mathtt{\text{3}}#. You might think what's the big deal, aren't these numbers the same? In a mathematical sense, yes, but for your computer, no. The difference being that decimals are stored differently in computer memory. We call the datatype that includes decimals #\color{#4271ae} {\mathtt{\text{float}}}#, which is short for floating point number.

Floating point formatThe floating point format is a computer number format that represents a wide dynamic range of real numbers as approximations to support a trade-off between range and precision.

First, let's take a look at the multiple ways of declaring a #\color{#4271ae} {\mathtt{\text{float}}}#.

Declaring a floatThe simplest way to declare a float is to add a fractional component to the desired number, if there's a zero before or after the decimal point you can choose to omit the zero:

>>> 1.0

1.0

>>> 1.

1.0

>>> .1

0.1

Using scientific notation

Scientific notation is the most concise way to declare very large or very small numbers in Python. Using scientific notation will always return a #\color{#4271ae} {\mathtt{\text{float}}}#. Note that #\color{#F5871F} {\mathtt{\text{1e9}}}# and #\color{#F5871F} {\mathtt{\text{1e+9}}}# are both valid notations for the same floating point number.

>>> 1e0

1.0

>>> 1e+9

1000000000.0

>>> 1e-4

0.0001

Using casting

Casting is a way of explicitly converting an object to a different type, accepting the potential data loss in doing so. For casting to and between integers and floats we use the following functions:

Function Definition
\[\color{#4271ae} {\mathtt{int}}(\textit{object}), \hspace{1em} \color{#4271ae}{\mathtt{float}}(\textit{object})\]

Where the #\textit{object}# argument can be of any Python type that supports casting to #\color{#4271ae} {\mathtt{\text{int}}}# or #\color{#4271ae} {\mathtt{\text{float}}}#.

Now, we can cast an #\color{#4271ae} {\mathtt{\text{int}}}# to a #\color{#4271ae} {\mathtt{\text{float}}}# and vice-versa. However, when casting a #\color{#4271ae} {\mathtt{\text{float}}}# to an #\color{#4271ae} {\mathtt{\text{int}}}# the numbers behind the decimal point are forever lost, the casting can't be reversed.

>>> float(1)

1.0

>>> int(1.8)

Since floats are more complicated to store than integers, there's a limit to the numbers the #\color{#4271ae} {\mathtt{\text{float}}}# type can represent. When you enter a number that's bigger than the maximum (or smaller than the minimum) size #\color{#4271ae} {\mathtt{\text{float}}}# can represent, Python will return #\mathtt{\text{inf}}#, which stands for infinity. #\mathtt{\text{inf}}# is a special #\color{#4271ae} {\mathtt{\text{float}}}# that represents the absolute largest value that #\color{#4271ae} {\mathtt{\text{float}}}# can accomodate.

For example, #1\times 10^{999}# is a number that's too big for the #\color{#4271ae} {\mathtt{\text{float}}}# format.

>>> 1e999

inf

This is similar for a value like #-1\times 10^{999}#:

>>> -1e999

-inf

Values that exceed the limits of #\color{#4271ae} {\mathtt{\text{float}}}# can still be used in expressions, but the value that is actually used is truncated to the special #\mathtt{\text{inf}}# value. Which means that all the numbers that too large are equal. This is the same for negative numbers.

>>> 1e999 > 1e998

False

>>> 1e999 == 1e998

True

Floats support the same operators that integers do, but when an equation includes a #\color{#4271ae} {\mathtt{\text{float}}}#, it will return a #\color{#4271ae} {\mathtt{\text{float}}}#.

Exactly the same usage:

>>> 1.0 + 1.0

2.0

When mixing #\color{#4271ae} {\mathtt{\text{float}}}# and #\color{#4271ae} {\mathtt{\text{int}}}# typings in an expression, the expression will always return a #\color{#4271ae} {\mathtt{\text{float}}}#.

>>> 1.0 + 1

2.0