Computers

This is an historical question/answer.

In the old days, mainframe computers (like the ENIAC) used telephone relays as computational elements, relays are electronic-magnetic switches: a magnet can be turned on by driving a current through the primary side of the relay, this magnet pulls a little switch on the secondary side of the relay from one position to another. Defining the relaxed position as zero and the attracted position as one, two values are represented.

By combining multiple relays to form a larger value is an easy trick, it's like writing numbers using multiple digits:

in decimal (ten-valued numbering) the number ten is represented by one ten-valued digit and zero one-valued digits:

1 * ten + 0 * one = 10 = ten

This very same trick is used in binary (two-valued numbering) where the number two is represented by one two-valued digit and zero one-valued digits:

1 * two + 0 * one = 10 = two

As you can see the meaning of 10 differs by the numbering system, when we want to write ten in binary, we have to chain multiple digits together. First again in the decimal system:

Hundred is represented by one ten-times-ten-valued digit, zero ten-valued digits and zero one-valued digits:

1 * ten-times-ten + 0 * ten + 0 * one = 100 = hundred

Note that I wrote hundred as ten-times-ten, this is the whole idea behind numbering systems!

Writing hundred in binary, we need

one two*two*two*two*two*two-valued (sixty-four-valued) digit,

one two*two*two*two*two-valued (thirty-two-valued) digit,

zero two*two*two*two-valued (sixteen-valued) digits,

zero two*two*two-valued (eight-valued) digits,

one two*two-valued (four-valued) digits,

zero two-valued digits and

zero one-valued digits:

1 * sixty-four + 1 * thirty-two + 0 * sixteen + 0 * eight + 1 * four + 0 * two + 0 * one = 1100100 = hundred

This can be done easily with long division, but instead of checking how many times the value can be subtracted, you only have to check whether it can be subtracted (being either once or not at all):

hundred - one-hundred-twenty-eight is less than zero (0 * one-hundred-twenty-eight)

hundred - sixty-four is (equal to or) greater than zero (1 * sixty-four), remains thirty-six

thirty-six - sixty-two is (equal to or) greater than zero (1 * thirty-two), remains four

... (check for every other doubling from one)

Now you know how binary works, in a similar matter any other numbering system may be devised, some of the "more popular" are octal (eight-valued) and hexadecimal (sixteen-valued). Basically they are both ways to represent binary in a more readable way; each three binary digits may be grouped in one octal digit, each four binary digits may be grouped in one hexadecimal digit.

Now back to why we use binary. The relay is not used a low anymore, other, more compact, alternatives are currently in use.

The transistor is a small (about a match-head) device where the current flow from one point to another can be controlled by a third point, by changing the voltage on that third point. This is somewhat similar to the relay principle and is called semi-conduction; a piece of material either conducts or not depending on certain conditions (semi is a fancy name for half, so it conducts "half" the time).

Combining multiple transistors in one device (getting rid of the bulky housing of each transistor) saves a lot of space. The separate transistors are so to say integrated in a single device, hence these devices are named integrated circuits or ICs.

A computer contains a lot of these ICs, some of them housing over a million transistors. Yet they are still based on transistors, which are based on relays (or radiotubes, but let's not talk about these now), which can only be switch on or off. That's why we use binary.