Number Systems

This explanation is quite complex. The rest of the website is easier!

Here is an analysis of how the different systems on this website work. See below for a table summarising the essential differences, but first, a description of the different types.

Description: Number systems fall into categories. Unary and positional are described below. Roman numbers are described as modified unary, as they have the strange concept of subtraction, so IV meaning one less than 5. Several systems are based on abacuses. The Chinese standard system is really a system of words rather than numerals, very similar to our own words for numbers. The Greek system uses the Greek alphabet to represent numbers. This is quite common - the Hebrews and Arabs did the same. As a system it is different from both unary and positional systems.

Base: All number systems have a base, where the numbers below are treated differently to the numbers above. Several system have more than one base. For example, Roman numbers use multiple ones for 1,2,3, but a different symbol for 5, and again for 10. (There are other complications, but that is the underlying principal). Some systems not only have a double base, but have a different way of treating the different bases. For example, Babylonian numbers have a unary system base 10 until they get to 60. Then they have a positional system for multiples and powers of sixty.

Unary: The oldest number systems were unary systems. One is represented by a single symbol, two is represented by two of these symbols, and so on. After a bit they get bored with this, and introduce a new symbol, usually at 5 or 10, and then have combinations of these new symbols with the old ones. It is a feature of unary systems that they tend to run out of symbols, and have to keep inventing new ones. This means that they are limited and can't express any number, however big. Some systems get over this problem by adopting a double unary/positional system, such as the Mayans. Tallies is the simplest of unary systems. I have called it base 5, but it is even simpler than that. There certainly is a symbol for 5, but this is repeated indefinitely. There is no extra symbol for bigger numbers. So you can express any number, but it does get rather long.

Positional: Abacuses are naturally positional. The number of beads in one column represents a different number to the same number of beads in another. Some number systems do this as well. They have a limited number of symbols, and express bigger numbers by where these symbols appear in the number. Its advantage is that you can represent any number, no matter how big. Arithmetic is also easier. Word systems, like our own words for numbers, are not really positional. They use the words from one to nine throughout the number, but also have words for the powers of ten to fix how big the number is.

Zero: Zero has two functions in a number system. You can use it to express nothing at all. All cultures had to have a way to say this, even if they used words for it. (Nonetheless it is an important mathematical concept, and having a symbol for it is vital.) However, you need a zero in a positional system, and you get problems if you don't have it. You need a way to say 'no value in this particular column'. In an abacus, there are no beads, and that doesn't matter. But in a written number system without a zero, you must leave a space, and that can be left out by mistake or misread. Positional systems gradually acquired zeroes, but seemed to use them intermittently. It was the constant use of the zero in the Arabic system, together with the use of multiple symbols rather than a double unary/positional system, which made it such a good system. I have said that zero is not used in 'Numbers as words'. This sounds silly - there is a word - it's zero! But we don't use zero in the middle of a number. I also have said that there is no zero on an abacus. An empty column represents zero. But a space represents zero in some positional systems without zero. For this type of zero, which is known as a place holder, you need a specific symbol, not the absence of one.

Order essential: One of the advantages of pure unary systems is that it doesn't matter what order the digits are in. They did tend to have a conventional order, but it wasn't necessary. Positional systems require order, as the position of the digits gives their value. 123 means something different than 321. However, certain non-positional systems also have to be in the right order. The Romans had a modified unary system, and IX means something different from XI. When we use numbers as words, two hundred and three means something different from three hundred and two.

Still in use: All these systems have been used for counting or calculation in the past, even if some of them were so complicated that they needed abacuses for the serious stuff! Now Arabic numbers are used throughout the world for calculation. Computers use binary and programmers sometimes use hexadecimal as short-hand for binary. Tallies are used for counting. Eastern abacuses are used as a machine for calculation. The other systems have a more formal use.

Easy/hard: This is my own opinion! It depends whether you have met the system before or whether you know the symbols already. I have described Greek and Chinese as medium. They are simple systems but use a lot of symbols which would confuse you if you haven't met them before. Generally speaking, I say that a system is harder if it uses a base other than 10, or a double base, particularly if they operate in different ways.

System	Description	Base	Unary	Positional	Zero	Order essential	Still in use
Egyptian	unary	10	yes	no	no	no	no
Babylonian	double base, unary/positional	10/60	yes	yes	intermittently	yes	no
Chinese standard	words	10	no	no	intermittently	yes	yes
Chinese rod	abacus	5/10/100	yes	yes	no	yes	yes
Greek	letters	10	no	no	no	no	yes
Roman	modified unary	5/10	yes	no	no	yes	yes
Mayan	double base, unary/positional	5/20	yes	yes	yes	yes	no
Arabic	positional	10	no	yes	yes	yes	yes
Binary	positional	2	no	yes	yes	yes	yes
Hexadecimal	positional	16	no	yes	yes	yes	yes
Chisenbop	abacus	5/10	yes	yes	no	yes	yes
Simple abacus	abacus	10	yes	yes	no	yes	no
Eastern abacus	abacus	5/10	yes	yes	no	yes	yes
Words	words	10	no	no	no	yes	yes
Tallies	simple unary	5	yes	no	no	no	yes
System	Base	Unary	Positional	Zero	Order essential	Still in use	Easy/hard