How does a calculator work?

A pocket calculator looks simple, and it is: open one and you find only a handful of parts.

Each key sits on a soft rubber dome. Inside every dome is a black carbon pill. Press a key, the dome squashes, and the pill touches two comb-shaped copper pads on the circuit board, joining them. That's the whole switch.

The pads are wired in a grid of rows and columns, the key matrix. A single chip checks the grid hundreds of times a second to find which key is down. On cheap calculators the chip is glued straight onto the board and covered with a blob of black epoxy.

The chip does all the maths in binary, using thousands of tiny switches called transistors, then lights the digits on the LCD. Pink rubber zebra strips carry the signals to the glass. A strip of solar cell and a tiny button cell power the lot, using about a ten-thousandth of a watt (see chapter 6).

In the next chapters you'll follow one sum all the way: key press, binary, logic gates, the adder and the display. A computer does exactly the same, only much more of it (see ComputerClear, coming soon).

Your calculator has 20 keys but the chip doesn't have 20 wires for them. The keys sit where 5 row wires cross 4 column wires: the key matrix.

The chip scans it. It puts a voltage on row 1 and listens on every column. Nothing? It moves on to row 2, then row 3, and so on, round and round, hundreds of times a second. When a pressed key joins row 3 to column 2, the chip hears the column go high while row 3 is on, and it knows exactly which key it is.

There's a snag. Metal contacts bounce: for a few thousandths of a second they touch, spring apart and touch again. Read too eagerly and one press looks like five. So the chip debounces: it waits until the signal has stayed steady for about 10 to 20 milliseconds.

Then the digit becomes binary. Calculators usually keep each decimal digit in its own 4 bits, called BCD (binary-coded decimal): 7 is 0111, so 47 is 0100 0111. Pure binary would be 101111. BCD wastes a few bit patterns but makes showing digits easy.

Everything a calculator does comes down to one tiny part: the transistor. Think of it as a switch with no finger: a voltage on its gate turns it on or off. A calculator chip has thousands of them; a phone chip has billions.

Wire a few transistors together and you get a logic gate, which looks at 1s and 0s and answers with a 1 or a 0:

NOT flips its input. AND gives 1 only if both inputs are 1. OR gives 1 if either is. XOR ("exclusive or") gives 1 if the inputs are different. A truth table lists every answer.

Look inside (switch to "Inside: CMOS"). Modern chips use CMOS: pairs of opposite transistors. A PMOS switch closes when its input is 0 and connects the output to the supply (1). An NMOS switch closes when its input is 1 and connects the output to ground (0). One of each makes a NOT gate with just 2 transistors. Because one switch is always open, almost no current flows when nothing changes. That's why a calculator can run on a scrap of solar cell (chapter 6).

George Boole wrote down this logic in 1854. In 1937 Claude Shannon showed switches could do it, and every computer since has been built from gates (see CurrentClear for circuits, and ComputerClear, coming soon).

Add two bits and there are only four cases: 0+0 = 0, 0+1 = 1, 1+0 = 1 and 1+1 = 10 (that's 2 in binary: write 0, carry 1). Look closely: the written digit is just XOR and the carry is just AND. Two gates make a half adder.

For longer numbers each column must also add the carry coming in from the right. That's a full adder: two XORs, two ANDs and an OR, 5 gates.

Line up four full adders and you can add two 4-bit numbers (0 to 15). The carry out of each bit feeds the next: a ripple-carry adder. It is slow in one way: the top bit can't be sure of its answer until the carry has rippled through every bit, 2 gate delays per bit.

Subtracting uses the same adder. To do A − B, flip every bit of B (an XOR per bit does it) and add 1 through the first carry. This trick is called two's complement. Multiplying is shift and add: for each 1 in B, add a copy of A shifted left. Your calculator does exactly this, one decimal digit at a time.

Every digit on the calculator is made of just seven bars, segments a to g. Light the right ones and you get any digit from 0 to 9.

The chip holds each digit as 4 bits of BCD (chapter 2). A small circuit called a decoder turns those 4 bits into 7 on/off signals, one per segment. For example segment a is on for 0, 2, 3, 5, 6, 7, 8 and 9. The truth table below lists all of them.

How does a segment go dark? It's a sandwich (on the left). Light passes a polariser, which lets through only light vibrating one way. Between two glass plates, rod-shaped liquid crystal molecules are arranged in a gentle quarter-turn twist. The light follows the twist, turns 90°, and slips through the second polariser, which is turned 90° too. A mirror behind sends it back out: the segment looks pale, like the background.

Put about 3 volts across a segment and the molecules stand up along the electric field. No twist, so the light isn't turned, and the second polariser blocks it: the segment goes dark. No light is made at all, which is why an LCD needs so little power. TVs use the same trick with a backlight and colour filters (see TVClear).

Power. A CMOS chip only uses energy when its gates switch (chapter 3), and the LCD makes no light of its own (chapter 5). So a simple calculator gets by on tens of microwatts: millionths of a watt. Casio lists 100 µW for a scientific model. A little amorphous silicon solar cell in room light makes about that much. The first LED calculators needed hundreds of times more: the 1972 Sinclair Executive used 20 mW.

Precision. A basic calculator has 8 digits and simply cuts off the rest. So 1 ÷ 3 = 0.3333333, and × 3 gives 0.9999999, not 1. That's fixed point: the decimal point sits somewhere in 8 digit places. Scientific calculators use floating point: digits plus a power of ten, like 6.02 × 10²³, and they keep a few hidden guard digits so answers like this round back to 1.

Computers use binary floating point, which can't store 0.1 exactly, so 0.1 + 0.2 comes out as 0.30000000000000004. Mostly harmless. But in 1994 the Pentium FDIV bug, a few missing entries in a table inside Intel's chip, made some divisions wrong in the fifth digit, and it cost Intel $475 million.

sin and cos. Scientific calculators don't store tables of sines. Many use CORDIC (1959): turn an arrow towards the angle you want in smaller and smaller steps, each one an angle whose tangent is ½, ¼, ⅛ … Each step needs only a shift and an add, exactly what the adder in chapter 4 can do. After 16 steps you have about 5 correct digits.