# Python: Floats, Decimals, Fractions

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In the previous session, we learned how to do arithmetic with integers (whole numbers).

In this session, we will learn that there are other types of numbers.

For a starter it will be good to know that we can determine the type of an expression by using the keyword `type`. The question-answer exchange

>>> type(2)
<class 'int'>
>>> type(2.0)
<class 'float'>

reminds us that `2` is an integer and we learn that `2.0` is a float.

## Floats

For most purposes we can think of a float as a decimal but there are some subtle difference that have to do with how floats are implemented on the hardware as binary numbers.

**Remark:** As a convention, to distinguish integers from floats, one requires floats to contain a decimal point.

**Exercise:** Find out whether it is legal to write
- `2.`
- `1+2.0`

**Activity:**
- What is the difference between `4/2` and `4//2`? Also try `5/2` and `5//2`.
- What does Python know about fractions? (Try `1/3+1/3`.)

**Activity:** To get a better understanding of the difference between integers and floats compute $2^n$ for big enough $n$ for both integers and floats: As Python for the values of `2**n` and `2.0**n` for different values of `n`. Did you try $n=1000$ and $n=10000$?

**Activity:** Find sample calculations that show that, as with a calculator, we need to be aware that the internal presentation of floats is not always accurate. For example,

>>> 0.333333333333333 + 0.333333333333333 + 0.333333333333333
0.9999999999999989

>>> 1.1+2.2
3.3000000000000003

>>> 0.1 + 0.1 + 0.1 - 0.3
5.551115123125783e-17

**Remark:** These results will be less surprising once you know that decimals the $0.1,0.2,\ldots$ (except $0.5$) are infinite in the binary. You can use a [converter from decimal to binary](https://www.rapidtables.com/convert/number/decimal-to-binary.html?x=0.1) for help with this.

**Activity:** Explore how your calculator suffers from similar (but not necessarily indentical) problems. For example, for $n$ big enough, your calculator will compute
$$10^n+1-10^n=0.$$

Find the smallest $n$ for which this happens? Can you think of explanations why this might happen? Compare the mistakes made by your calculator with the mistakes made by the floats of Python.

## Decimals and Fractions

If you are worried about the fact that floats can be inaccurate in unexpected and difficult to understand ways, rerun the experiments above after importing the [decimal](https://docs.python.org/3/library/decimal.html) module:

>>>from decimal import *

For example,

>>> Decimal('1.1')+Decimal('2.2')
Decimal('3.3')

Similarly, there is also a library for [fractions](https://docs.python.org/3/library/fractions.html):

>>> from fractions import *

For example,

>>> Fraction(1,3)+Fraction(1,3)
Fraction(2, 3)
>>> Fraction(Decimal('0.22'))
Fraction(11, 50)