Bits and binary

Bits: the smallest piece of data

• A bit is one digit that is either 0 or 1.
• Computers store everything — numbers, text, images — as bits.
• One bit alone is small, so we group many bits together.

Why computers use binary

• A computer chip has tiny switches. Each switch is on or off.
• on = 1 and off = 0. So two states fit a chip perfectly.
• Counting with only 0 and 1 is called binary (base 2).

How binary counts

• Our normal numbers use base 10 (digits 0–9).
• Binary uses base 2. Each place is worth twice the place on its right.
• The places are 8 4 2 1. So 1101 means 8 + 4 + 0 + 1 = 13.

place:   8   4   2   1
bits:    1   1   0   1   ->  8 + 4 + 0 + 1 = 13

Python can convert for you

• bin(13) gives the string '0b1101'. The 0b just means "this is binary".
• int("1101", 2) reads "1101" as base 2 and gives 13.
• These two functions go in opposite directions.

print(bin(13))          # decimal -> binary
print(int("1101", 2))   # binary -> decimal

Fixed bits and overflow

• A computer keeps a number in a fixed number of bits.
• With 4 bits you can store 0 to 15 — that is 2**4 = 16 different values.
• If a result is too big, it wraps around back to 0. This is overflow.

print(2 ** 4)          # how many values 4 bits hold = 16
print((15 + 3) % 16)   # 18 is too big, so it wraps to 2

Common mistakes

• n bits store 2**n different values, from 0 to 2**n - 1.
• Binary is base 2: each place is worth twice the one to its right.

Now you try

• Type your code below and press Run to see the output.
• Press Check answer to test it.