Section 3: On Binary Numbers

Modern electronic computers are capable of carrying out millions of mathematical calculations every second: adding, multiplying and juggling numbers at a speed hard for us to comprehend, let alone match. And they do this using what is to us a rather peculiar type of arithmetic, using what are called "binary numbers". Indeed, within the computer, these numbers are represented not as familiar digits, but as states of electrical charge, or pulses of high and low voltage.

How can numbers be represented this way?

Consider that the number system that is familiar to us from our earliest school days is based on the number ten for the very good reason that we have ten fingers - digits (including our thumbs!). Primitive people counted on their fingers, and so the number ten assumed a particular importance. But the fact that we have ten fingers is just an accident of evolution - we could just as easily have ended up with eight or twelve fingers and we would then no doubt use a number system with eight or twelve as the base.

What does it mean to say that our number system is based on the number ten? Well, it means that we use just ten distinct symbols - the digits "O", "1", "2", "3", "4", "5", "6", "7", "8", and "9" - to enable us to write down combined symbols which represent any number at all. We do this by using a system called "positional notation", which means that the value of a particular digit depends on its position. Thus, when we write down a number like "564", the digit "5" represents, in this particular case, five hundred, the "6" represents sixty, and the "4" just four units. You may remember doing simple arithmetic when you were very young, with the numbers arranged in columns, like this:

For every column to the left of the units column, we must multiply the digit in that column by another power of ten. A number like "564" represents a process or sum like this:

564 = 5 x 10 x 10 + 6 x 10 + 4

This system of positional notation is by far the easiest we have been able to devise for representing numbers. It was developed by the Arabs in about 700 - 800 AD, based on work by the Hindu mathematicians of India.

As we have discussed, our use of the base ten is quite arbitrary. Other bases are quite possible, and some of them turn out to be more useful for certain purposes than the base ten. The simplest base we can imagine for a system of positional notation is the number two. This is the binary system used by computers.

In the binary system, just two symbols are used. Usually, these are written down as "0" or "1" - but it is very easy to confuse these with decimal numbers. As soon as we see the combined symbol "10", we automatically think "ten", which is what it represents in the decimal system, but it can mean another number if the base is different. So to avoid that confusion, we will use special bold symbols for the time being to represent binary numbers.

So, in the binary system, we use two special symbols "0" for zero and "1" for one. Imagine, if you like, that these are symbols invented by a strange race of two fingered beings who can only count up to two on their hands.

Using just two symbols, it is perfectly possible to represent any number that could be written down with our familiar decimal system. In the binary system, positional notation works just as it does in the decimal system, except that the columns to the left of the units column represent increasing powers of two instead of ten. Let's consider we have a number which might have been written down by our imaginary beings, say "101". Remember these bold symbols represent something different than in the decimal system, so this number is not the same as "one hundred and one". To find out what number it really represents, we might write it down like this:

For every column to the left of the units column, we multiply the digit in that column by another power of two. Thus a number like "101" represents a sum like this:
101 = 1 x two x two + 0 x two + 1 unit
= four + zero + one
= five (5 decimal)

See the table at the end of this section for some more binary arithmetic using the two special symbols.

The point to recognise is that any number can be represented in binary notation just as it can in our familiar decimal notation, and arithmetic can be carried out perfectly well in such a binary system.

The great advantage of binary numbers is that they can be represented so easily by electrical means because there are only two symbols involved. In a computer, for example, a row of switches with only two positions each - "on" or "off" - can be used to represent binary numbers. Similarly, in transmitting information, a train of electrical pulses which are either "high" or "low" in voltage can also represent a binary number. See Diagram A. This is far easier than trying to represent the ten distinct symbols of decimal notation, and consequently it enormously simplifies the design of computer circuitry, and provides a simple method of digital transmission.