The Adder
Stibitz built the first one at home in 1937, out of scrap relays from the Bell Labs pile, strips cut from a tobacco tin, two dry cells and two torch bulbs. It added one bit, and one bit is genuinely complete: give it two inputs and it gives you a sum and a carry, needing nothing else. The trouble starts at two bits, because the second cannot finish until it knows what the first one carried.
New to logic gates? Start here
Everything is a switch
Underneath all of it is one part: a switch that is either on or off, and that can be operated by another switch rather than by a finger. That is the whole of the hardware vocabulary. AND is two switches in a row, so both must be on. OR is two side by side, so either will do. NOT turns the answer around.
There is nothing else in the box. Adding, remembering, choosing and counting are all arrangements of those three, and the machines in this topic are those arrangements, in the order somebody had to think of them.
The machine for this idea on its own is Flip-Flop, if you would rather press it than read about it.
Numbers in a fixed-size box
A number on paper can be as long as it needs to be. A number in a computer gets a fixed number of digits and no more, so there is always a largest one it can hold and always a smallest difference it can tell apart.
Almost everything surprising about computer arithmetic follows from that. What happens when you count past the largest value depends on which kind of number it is: a fixed-width integer wraps round to the smallest, and a floating-point number goes to infinity instead. A value too fine for the gap is not rejected, it is rounded to the nearest one that fits, and then the rounding is added to the next rounding. These are not faults in the machines here. They are the edges of the box, and the machines are what people built to live inside it.
The machine for this idea on its own is Two's Complement, if you would rather press it than read about it.
One bit adds, and then the carry ties them all together
1 One bit, added, and the carry it produces
Three inputs, two outputs, no memory and nothing to wait for. Press any of them.
sum 0, carry out 0
Two ones give a sum of 0 and a carry of 1, which is the only interesting row: the answer does not fit, so part of it has to go somewhere else.
2 The carry, taken as an input to the next bit
That carry is an input to the bit on its left. It is the same signal, arriving one stage later, and it is the only thing connecting the bits to each other.
3 Four of them chained, and the answer that is not ready yet
Eight of them, most significant first. A stage is marked when it received a carry, which is the same as saying it could not start until the stage to its right had finished.
4 The worst case, counted rather than asserted
The worst case is not an argument. Every pair of eight-bit numbers is 65,536 additions, which this browser can simply do.
This is why carry-lookahead exists: not to make addition cleverer, but to stop the answer to the left waiting on the answer to the right.
These ran in this browser when the page loaded. The second row is the expensive one: it adds every eight-bit pair there is and compares the result with ordinary arithmetic.
| claim | held | measured |
|---|---|---|
| each row is the parity of its inputs and a carry when two or more are set | yes | all 8 rows |
| all 65,536 eight-bit pairs match ordinary addition | yes | 65,536 pairs, 0 wrong |
| one bit added alone needs nothing to its right | yes | 1+1 gives sum 0 carry 1 |
| a carry can travel the whole width, so the last bit waits for the first | yes | 255+1 travels 8 stages, 1+0 travels 0 |
| the worst case was found by trying every pair, not argued | yes | 1 + 127 travels 8 of 8 |
What is real here, and what is not
The delays are stages, not seconds
This page counts how many stages a carry passes through, which is a real property of the circuit. It does not model time: no gate delay in picoseconds, no wire capacitance, no clock. A real ripple adder's speed depends on all three, and the number here is the thing they multiply, not the answer.
The full adder is a table, not a gate diagram
Addition here is eight rows looked up, not XOR and AND gates wired together. The rows are the same rows those gates produce, and the page checks that each one is the parity of its inputs with a carry when two or more are set. What you cannot see here is the gate count, which is where a real design spends its area.
The year is sourced; the month is not, and this page says so
The IEEE Computer Society's account of Stibitz places the kitchen-table adder in 1937 and describes the relays, the tobacco tin and the bulbs, but gives no month. Secondary accounts almost all say November, and the chronology sorts it there so that it sits before the 1945 entries rather than after them. That month is repetition, not evidence, and it is not printed anywhere on this page as though it were.
One bit is the Model K, and eight bits is not
Stibitz's machine added one bit and lit a bulb. The eight-bit chain on this page is the idea extended, not a model of anything he built, and the ripple arrangement it uses is the simplest of several. Carry-lookahead, carry-select and carry-save all exist to avoid exactly the walk this page makes you watch.
No sound
A relay clack per stage would be evocative and would measure nothing, because nothing here has a duration. The rule on this site is that a sound has to carry the measurement.
Sources
- IEEE Computer Society, Computer Pioneers: George Robert Stibitz. The 1937 kitchen-table adder, the scrap relays and what it was built from. Fetched and checked: it dates the work to 1937 and names no month.
- G. R. Stibitz, Complex computer, US patent 2,668,661. Bell Labs relay arithmetic described by the man who built it.
- Logical Art, the studio this belongs to.