The Binary Number System — Part 1: Decimal Notation

I’ve stated in a previous post that the digital revolution is all about measuring signals and representing them in a binary format that can then be processed with digital gadgets.

A big part of this processing is the ability to represent anything in a binary format. My definition of binary is a representation which only involves two states (or combinations of these two states).

This definition is rather terse. I’ll attempt to illustrate it in the next two posts. However, before we dive into binary, let’s review something that is almost so intrinsic to our thinking about numbers that we take it for granted:the decimal system. Our first step into understanding the decimal system will be to contrast it with the roman numeral system.

Roman Numerals

Most of us are familiar with the roman numeral system. If one were to count using Roman numerals, it would look like this:

I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
XIV
XV
XVI

Basically, I, V, X, etc are unit values. Learning Roman Numerals is akin to learning how to count change: there’s no regular pattern. There’s a penny (I), a nickel (V), and a dime (X). You have to break down any number into these units. In fact, it’s even worse than counting change: the numeral IX means “a dime minus a penny”.

The problem with Roman numerals is that they are very irregular. There’s no pattern to build off of. (Actually, there’s a pattern enough to program a computer to do it, but it’s not as simple as it could be.) Representing 9 (IX) is very different than representing 9000 (MMMMMMMMM).

Arabic Numerals

When I say the decimal system, I don’t mean the system of using meters (metres), millimeters (millimetres), etc. Instead, I mean our system of counting. When we need to represent a number, we all pull out the decimal system without realizing.

In reality, all numbers are really ideas that have a representation. For example, there’s the notion of nine apples. I can represent that by writing 9 apples or IX apples. It is exactly the same as writing cat or gato or chat.

Fortunately/Unfortunately, unlike the cat/gato/chat, which has different representations, almost all of the globe represents the number nine the same way: the Arabic numeral 9.

The Arabic Numerals are the numerals 0,1,2,3,4,5,6,7,8,9. Using this small alphabet of numerals, we can practically represent any number in the world.

Let’s contrast our method of counting by creating a table between Roman and Arabic numerals:

Unfortunately, the number ten has no Arabic Numeral representation; the numerals stop at 9. So, what do we do? Well, we add another digit: we create another space for numbers to the left of our original numbers and then start over at 0. I’ll show this extra digit in red.

Here’s what we did: We added another set of digits to the left of our previous set and started counting over from 0. It’s tempting to see the notation “14” and call it fourteen. However, when discussing number system, it’s better to call it “one-four”. The reason is that “14” is our best way of representing fourteen, but there are other ways. This post (and the next) attempts to detach numbers from the many ways of representing them.

The reason that we ended up with the decimal number system is that we have ten fingers: we can actually count from one to ten. Notice that there are ten Arabic numerals: 0,1,2,3,4,5,6,7,8,9–the same as the number of fingers we have, even though they don’t exactly correspond to how we count on fingers (one, two, three, four, five, six, seven, eight, nine, ten).

Generally, humans tend to count with one rather than zero. However, in numbering systems it’s better to start with 0. The reason is that when we reach 9, we add another digit (1) and reset the second digit to 0–getting 10. (Alternatively, we can just remember that after 9, we wrap back around to 0.)