Thinking in binary and hex

Twelve lights, twelve yes-or-no values. So far you have used a register for each, and with sixteen registers that runs out quickly. But a single number holds far more yes-or-no values if you read it as bits. That is why the expert level starts by looking inside the number.

Place values#

A binary number only has the digits 0 and 1. Each place is worth twice as much as the one to its right:

So %1010_0111 is 128 + 32 + 4 + 2 + 1 = 167. Bit 0 is always on the right. The chip ignores the underscores, they make long numbers readable.

Hex as shorthand#

Four bits give 16 possibilities, exactly one hex digit from 0 to F. That is why hex is the shorthand for binary numbers:

You read $A7 as “A = 1010, 7 = 0111”. Each hex digit stands for exactly four bits.

move r0 %1010_0111   # 167
move r1 $A7          # the same number
move r2 $FF_FF       # 65535, underscore here too

Places without bit instructions#

Before the bit instructions arrive tomorrow: with div, floor and mod you can reach single places too. Dividing by 16 moves one hex digit to the right, mod 16 keeps the last one:

div high number 16
floor high high      # upper hex digit
mod low number 16     # lower hex digit

You get bit 3 the same way: divide by 8, round down, mod 2.

53 bits of precision#

Registers store decimal numbers. Whole numbers are exact in them, but only up to 2^53 = 9,007,199,254,740,992. Above that the last digits get lost: 2^53 + 1 cannot be stored and becomes 2^53.