three things a computer does with bits
store them
move them
change them by rules
what can you build
out of switches?
out of switches?
image: ai-generated (gpt-image-2)
the toolbox: and, or, xor
two bits in, one bit out. a rule, nothing else.
and one that flips: not
with a not behind them: nand, nor, xnor
the small circle at the output means: and then flip it.
one gate is enough: nand
every chip is mostly nand. everything else is built from it.
no gate understands anything.
it just switches.
it just switches.
two nor gates, wired into each other
try it: the flip-flop
from one bit to memory
power off, and the loops collapse: RAM forgets.
excursion: what 32-bit and 64-bit mean
16 address lines choose one of 65,536 rows: 64 KB
1111 1111 1111 1111
the highest address a home computer from 1982 could name
32 address lines choose one of 4,294,967,296 rows: 4 GB
1111 1111 1111 1111 1111 1111 1111 1111
the wall of 32-bit windows: a fifth gigabyte of RAM has no address
64 address lines choose one of 18,446,744,073,709,551,616 rows: 16 EB
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111
16 billion gigabytes: your laptop today, with room for a long time
every extra address line doubles the memory a computer can name.
1 + 1 in binary
0 + 0 = 00
0 + 1 = 01
1 + 0 = 01
1 + 1 = 10
the half adder: two gates
eight of them in a row: a byte adder
the carry runs from right to left, like in school.
the rest of arithmetic is built the same way
| subtract | = | add the negative | the negative: flip every bit, then add one |
| multiply | = | add again and again | × 2 is a shift to the left |
| compare | = | subtract, look at the sign | |
| divide | = | subtract again and again |
everything ends at the adder. the adder ends at four gates.
a photo is a grid of bytes
8 × 8 pixels from the parrot's beak
each pixel is one byte: 0 is black, 255 is white.
the numbers are the picture. the computer sees nothing else.
the numbers are the picture. the computer sees nothing else.
brighter: add 40 to every pixel
the yellow pixel:
the same addition as in the byte adder
…and the other 63, then all 16,384
before
after: every pixel + 40
more filters, same toolbox
| darker | new = old − 40 |
| negative | new = 255 − old |
| black & white | new = old ≥ 128 ? 255 : 0 |
| blend two photos | new = (a + b) ÷ 2 |
| blur | nine pixels at a time |
try it: the pixel filter
xor over a byte: a check bit
flip one bit on the way, and the check bit no longer matches.
xor with a key: encryption
four gates, a few billion times a second.
that is all.
that is all.
image: ai-generated (gpt-image-2)