Bits and binary numbers

Eight switches can show 256 different patterns, enough for every number from 0 to 255.

Before reading anything, try the switches below. Press All off. Now turn on only the rightmost switch and read the line under the row: 1. Turn it off and switch on the one to its left: 2. The next one gives 4, and the one after that 8. Keep moving left. By the far end you have met every value a switch in a byte can have, and the last one is worth 128.

Each switch is worth twice the one to its right

What you just saw is the whole idea of binary. In decimal each place is worth ten times the place to its right: ones, tens, hundreds. In binary each place is worth twice as much, so the values run 1, 2, 4, 8, 16, 32, 64, 128. To read a pattern, add the values of the switches that are on. 00100101 has the 32, 4 and 1 switches on, which makes 32 + 4 + 1 = 37. Switch on exactly those three and the sum line should agree. One more rule drops out for free. Only the rightmost switch is worth an odd amount. So a number is even exactly when its last bit is 0.

Why a chip uses two states instead of ten

The switches on the screen stand in for real ones. A computer chip holds billions of tiny electronic switches called transistors, and at any moment each one is either letting current through or blocking it. Engineers write those two states as 1 and 0. One of them is a bit, short for binary digit. The letter you type and the song you play both end up as long rows of bits.

Why not ten states, one for each decimal digit? The voltage on a wire inside a chip is never exact. Heat, neighbouring wires and small differences between transistors push it up and down a little all the time. A circuit that only has to tell high from low can shrug most of that off, because anything above a middle line counts as 1 and anything below counts as 0. Squeeze ten levels into the same range and they sit so close together that a little noise could make one read as its neighbour. Two states also suit a switch. It is either fully on or fully off and never has to hold a careful position in between.

Writing 100 with the switches

Going the other way, from a number to bits, works like paying with the biggest coin that fits. Take 100. 128 is too big. 64 fits, leaving 36. 32 fits, leaving 4. 16 and 8 are both too big for 4. 4 fits, leaving 0. So the 64, 32 and 4 switches are on: 1100100, or 01100100 as a full byte. Pick 100 as the target in the scene and check. Some people prefer a second method. Divide by 2 again and again, write down each remainder, then read the remainders from the bottom up. Both give the same bits.

One more switch doubles the patterns

How many different numbers can eight switches make? Every extra bit doubles the count. One bit gives 2 patterns. Two bits give 4: 00, 01, 10 and 11. Three give 8, and n bits give 2 to the power n. A byte, eight bits, gives 256 patterns, so a byte counts from 0 to 255, and that is also why each of the four numbers in an IPv4 address stops at 255. Need to give 1000 things their own number? 9 bits give only 512. You need 10, which give 1024.

Press +1 a few times starting from 0 and watch the switches. It looks like a car's odometer with only two digits on each wheel. 0, then 1, then the 1s place rolls over and carries into the 2s place: 10. Then 11. Then both roll over at once: 100. With every switch on you have 11111111, which is 255. The real answer to 255 + 1 is 256, and it needs a ninth bit: 100000000. The scene keeps only eight bits, so +1 from 255 shows 0. That is overflow. It comes back in the adder unit.