# Calculators

by Chris Woodford. Last updated: March 11, 2017.

Can you remember Avogadro's constant to six decimal places? Can you figure out the square root of 747 in less than a second? Can you add up hundreds of numbers, one after another, without ever making a mistake? Pocket calculators can do all these things and more using tiny electronic switches called transistors. Let's take a peek inside a calculator and find out how it works!

Photo: Left: This Casio fx-570 calculator has given
me sterling service since 1984 and is still going strong today.
In case you're wondering, Avogadro's constant (one of many
constants stored in this calculator and available at the touch of a
button) used to be quoted as 6.022045 × 10^{23} (since 2011, newer
sources have given a more accurately calculated value of 6.022141 × 10^{23}).

## What is a calculator?

Photo: Right: My newer Casio Calculator, an fx-991ES, has a much larger "natural display" that can show entire equations and even perform calculus! The larger dark gray keys at the bottom are the numbers and the main "operators" (+, −, ×, ÷, = etc). The lighter gray keys above them carry out a whole range of scientific calculations with a single button click. The brown-colored square in the extreme top right is a solar cell that powers the machine along with a small button battery.

Our brains are amazingly versatile, but we find it hard to calculate in our heads because they can store only so many numbers. According to a famous bit of 1950s research by psychologist George Miller, we can remember typically 5–9 digits (or, as Miller put it: "the magical number seven, plus or minus two") before our brains start to ache and forget. That's why people have been using aids to help them calculate since ancient times. Indeed, the word calculator comes from the Latin calculare, which means to count up using stones.

Photo: This is what calculators looked like in the 1970s. Note the very basic 8-digit green display (it's called a vacuum fluorescent display) and the relatively small number of mathematical functions (all you could really do was +, −, ×, ÷, square roots, and percentages). What you can't see from this photo is how thick and chunky this calculator was and how big its batteries were. Modern calculators are far more advanced, much cheaper, and use a fraction as much battery power.

Mechanical calculators (ones made from gears and levers) were in widespread use from the late-19th to the late-20th century. That's when the first affordable, pocket, electronic calculators started to appear, thanks to the development of silicon microchips in the late 1960s and early 1970s. Calculators have much in common with computers: they share much of the same history and work in a similar way, but there's one crucial difference: a calculator is an entirely human-operated machine for processing math, whereas a computer can be programmed to operate itself and do a whole range of more general-purpose jobs. In short, a computer is programmable and a calculator is not. (A programmable calculator sits somewhere between the two: you can program it, but only to do relatively simple mathematical calculations.)

## What's inside a calculator?

If you'd taken apart a 19th-century calculator, you'd have found hundreds of parts inside: lots of precision gears, axles, rods, and levers, greased to high heaven, and clicking and whirring away every time you keyed in a number. But take apart a modern electronic calculator (I just can't resist undoing a screw when I see one!) and you might be disappointed at how little you find. I don't recommend you do this with your brand-new school calculator if you want to stay on speaking terms with your parents, so I've saved you the bother. Here's what you'll find inside:

Caption: Inside the fx-570, which is face-down
here. We're effectively looking up into the machine from below.

Don't worry, I managed to put it all back together again just fine!

- Input: Keyboard: About 40 tiny plastic keys with a rubber membrane underneath and a touch-sensitive circuit underneath that.
- Processor: A microchip that does all the hard work. This does the same job as all the hundreds of gears in an early calculator.
- Output: A liquid crystal display (LCD) for showing you the numbers you type in and the results of your calculations.
- Power source: A long-life battery (mine has a thin lithium "button" cell that lasts several years). Some calculators also have a solar cell to provide free power in the daylight.

And that's about it!

## What happens when you press a key?

Press down on one of the number keys on your calculator and a series of things will happen in quick succession:

- As you press on the hard plastic, you compress the rubber
membrane underneath it. This is a kind of a miniature trampoline that
has a small rubber button positioned directly underneath each key and a
hollow space underneath that. When you press a key, you squash flat the
rubber button on the membrane directly underneath
it.

Photo: The keyboard membrane. I've left one of the keys on the membrane to give you an idea of the scale. There's one rubber button directly beneath each key. Read more in our article about computer keyboards. - The rubber button pushes down making an electrical contact between two layers in the keyboard sensor underneath and the keyboard circuit detects this.
- The processor chip figures out which key you have pressed.
- A circuit in the processor chip activates the appropriate segments on the display corresponding to the number you've pressed.
- If you press more numbers, the processor chip will show them up on the display as well—and it will keep doing this until you press one of the operations keys (such as +, −, ×, ÷) to make it do something different. Suppose you press the + key. The calculator will store the number you just entered in a small memory called a register. Then it will wipe the display and wait for you to enter another number. As you enter this second number, the processor chip will display it digit-by-digit as before and store it in another register. Finally, when you hit the = key, the calculator will add the contents of the two registers together and display the result. There's a little more to it than that—and I'll go into a few more details down below.

Artwork: A seven-segment display can show all the numbers from 0-9.

## How does the display work?

You're probably used to the idea that your computer screen makes letters and numbers using a tiny grid of dots called pixels. Early computers used just a few pixels and looked very dotty and grainy, but a modern LCD screen uses millions of pixels and is almost as clear and sharp as a printed book. Calculators, however, remain stuck in the dark ages—or the early 1970s, to be precise. Look closely at the digits on a calculator and you'll see each one is made from a different pattern of seven bars or segments. The processor chip knows it can display any of the numbers 0-9 by activating a different combination of these seven segments. It can't easily display letters, though some scientific calculators (more advanced electronic calculators with lots of built into mathematical and scientific formulae) do have a go.