A numerical system is a set of symbols and rules used to represent numbers. All civilizations had to deal with expressing ever larger numerical quantities and they all faced the same issues. A look at the different ways humans across the planet and across the ages handled this challenge is fascinating. Among the many accounts of this long intellectual adventure, a couple are particularly readable and useful for our ends: Steven Strogatz’s section in his book “The Joy of X", excerpts of which are available online, and the article “Historical Counting Systems” in the book “Math in Society” by Lippman (Pierce College) which also provides clarifying examples and practice exercises. This fun video is also an amusing yet clear introduction to the origin and development of number systems. For a more in-depth look at the history of mathematics in various cultures, visit the MacTutor History of Mathematics created by the School of Mathematics and Statistics of St. Andrew’s University.

The three major areas of exploration with regard to number systems are: 1) their variety and diversity; 2) the bases they use and that can be used; and 3) the symbols used to represents digits.

Variety of Number Systems: if we think we may have taken for granted the (Hindu-Arabic) number system we are so familiar with, studying how other systems work allows us to understand and appreciate ours better while at the same time having fun with the many ways a numerical system can be built and numbers can be written. To this end, we can look at the influence of Ancient Egypt, read how the Babylonians managed without a zero, learn about the Roman or the Mayan systems, and even play online Maya math games.

Number Bases: In any numerical system, the basis or radix is the number of unique symbols used to represent numbers. For example the decimal system, the most commonly used system today, uses base 10, or 10 unique symbols (from 0 to 9). However, 10 is not the only possible base, as other systems used base 20 (Aztec and Mayan), base 60 (Babylonians) and so on. Using other bases can deepen our understanding of our own system and it is also a lot of fun: try it out here. In fact, base 2, 8 and 16 are used in all our electronic devices, as this is the way we communicate with computers and they communicate with each other. To better understand how these bases work, you can look at how to translate a number from hexadecimal to binary here or play the binary game. The National Council of Teachers’ of Mathematics has even published a set of problems based on a cartoon strip to engage middle school students in studying different numerical bases. Finally, some people would argue that 12 would be preferable to 10 as base for a number system and explain their reasoning here.

Glyphs: Various symbol sets can be used to express numerical quantities. Each culture (from the Egyptian to the Ancient Chinese) had their own set of symbols that also evolved with time. So it is for the glyphs that we are now using in the decimal system – the history of their origin and their transmission is not a simple one… but that’s a story for another time.

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