Python: Numbers and Arithmetic

Addition and subtraction uses the same characters as used in mathematics i.e. + and – whereas division and multiplication uses / (forward-slash) and * (asterix) respectively. Use ** to multiply a value to the power of (exponent) and % to return the remainder of a value (modulo). If you supply a formula with integers (whole numbers) the result returned will be an integer, if you want to do calculations  with decimal values then supply decimal values for example 20/100 = 0 whereas 20.0/100.0 = 0.2

>>> 1080 + 1920          # Addition
3000
>>> 1080 – 1920          # Subtraction
-840
>>> 1080 * 1920          # Multiplication
2073600
>>> 1080/1920            # Division of integer
0
>>> 1080.0/1920.0        # Division of floating point
0.5625
>>> 2 ** 8               # Power (exponent)
256
>>> 8.75 % 0.5           # Modulo (remainder)
0.25
>>> 65536 / (8 * 8)      # Nested Arithmetic
1024

Python stores decimal values in binary floating-point number format which can have some undesirable side effects.

>>> # Floating Point rounding error
>>> 0.1 + 0.2             # Floating Point Addition
0.300000000000000004      # Should be 0.3 !!!???

Floating-Point Numbers.

To speed up calculations Python stores numerical values as binary floating point numbers and allocates 57 bits. A Floating Point number is a form of scientific notation so 300,000,000 could be represented as 3×10 to the power of 8 and 0.00000015 would be 1.5×10 to the power of minus 7. If we wanted to multiply these numbers together it would be much easier to multiply 3 by 1.5 and then add 10 to the power of 8 + 10 to the power of minues 7 or 8 minus 7 equals 1 therefore the answer = 4.5×10 to the power of 1 or 45. Now for the same reason that ⅓ cannot be represented in decimal without an infinite number of digits i.e. 0.33333… fractions cannot be represented absolutely accurately in binary floating point numbers.

As an example 0.1 * Base10 (decimal) = 1 with no remainder. If we do that same thing with binary we get a recurring number i.e. 0.1 * Base2 (binary) = 0 with remainder 2 using this remainder we 0.2 * Base2 = 0 remainder 4 and so on we will always have a remainder.

0.1 * 2 = 0.2          # 0
0.2 * 2 = 0.4          # 0
0.4 * 2 = 0.8          # 0
0.8 * 2 = 1.6          # 1
0.6 * 2 = 1.2          # 1
0.2 * 2 = 0.4          # 0
0.4 * 2 = 0.8          # 0
0.8 * 2 = 1.6          # 1
...

So let’s see what 0.1 looks like in binary when truncated to 57 significant bits:

0.000110011001100110011001100110011001100110011001100110011001…

Bits 54 and beyond total to greater than half the value of bit position 53 thereby rounding up to:

0.0001100110011001100110011001100110011001100110011001101

In decimal this is:

0.1000000000000000055511151231257827021181583404541015625

Now if we round to 17 significant bits it would be 0.10000000000000001