Why doesn’t 0.1 plus 0.2 give 0.3? Because 0.1 is a repeating fraction in binary. We convert fractions between bases by successive multiplications, look at the bits of a float with struct, and see two practical ways out: integers for money and math.isclose for comparisons.
To subtract, the processor adds the first number to the complement of the second, so a single adder does both operations. We will see two’s complement, why 8 signed bits range from -128 to +127, and how overflow took down the Ariane 5 and froze Pac-Man on level 256.
The addition and subtraction rules of base 10 hold in any base: only the value that triggers the carry changes. We do arithmetic in binary, octal, and hexadecimal, see why with three addends the carry can be worth 2, and implement both algorithms in Python.
In any base, the value of the base itself is written 10. We start from positional notation to convert integers between decimal, binary, octal, and hexadecimal, using expansion, successive divisions, bit grouping, and Python’s bin, oct, hex, and int functions.
