If there’s one realization that’s guaranteed to send a mind reeling, it’s getting a glimpse at just how different human beings from one culture to another are at thinking. Even in modern day, the contrasts between two societies can be striking and staggering. Envision an American visiting Japan for the first time with no clue of just how different Japanese culture is from American culture. Coming from a culture that prizes individual accomplishment and direct communication, the tendency for Japanese culture to assign worth based on relationships and to consider addressing a subject bluntly as ‘clumsy’ would be shocking, if not dazzling (Western Washington University).
Archeologists deal with the difference of thought on a regular basis. In studying now-extinct civilizations, much of their deciphering of the artifacts they discover cannot be done by relying on modern ideas and understandings. Even in the subject of math, perhaps the most definite rules of the universe understood by mankind, ideas and understanding of mathematical concepts and uses have drastically evolved.
When I began this class, I was excited for the topics that I expected would be covered. I’ve still enjoyed the classes thus far, but I can say with absolute certainty that I was not prepared for the culture shock that would come with dealing with mathematical concepts that, in our modern society, are so basic and fundamental.
Mesopotamia. Image: Giusi Barbiani, via Flickr.
Zero. Zero is perhaps the most important digit in our system of numerals. It’s a place holder; it’s a starting point; it’s the middle of a number line that goes on for infinity in each direction. But go back to the inception of mathematics beyond 2+2=4, and you will find zero is as nonexistent as that which it represents. And boy, what a difference NOT having a zero makes. Perhaps the first function of zero that one misses when working without it is its job as a placeholder. How does one write 10, or 100, or 1000 without a zero? The Mesopotamians used a base 60 system, which meant instead of 10, 100, etc. their digits went up 60, 3600, 21600, and beyond. But still, the problem becomes: How does one write those without a zero?
Like this: 1. That’s it. 1. When the Mesopotamians landed on a power of 60, they wrote it as 1, because just as if you took the zeroes out from behind 10, 100, and 1000, all that’s left is a 1. This both creates a problem, but at the same time it provides a fascinating workaround. Since any power of 60 can be written as 1, the numbers prior to them can treated as fractions. For instance–and it’d be useful to think of a clock for this–if you wanted to write ‘half’ using Mesopotamian numbers, you would not write 1/2, but rather 30. Think minutes; thirty minutes is half of an hour, which is 60 minutes. 1/4 would be 15. 1/8 is a little more complex, as it comes out as 7;30 (that’s 7 sixties and 30 ones), but it’s still exactly like 7 minutes and thirty seconds is one eighth of an hour.
I could be a millionaire! If this was Mesopotamian. Image: David Guo, via Flickr.
Which brings the subject to reciprocals; reciprocals are fractions that, when multipled to a number, produce a 1. Again, because the Mesopotamians didn’t have a zero, their representations of 60, 3600, 216000, etc. all appear as 1. Because of this, reciprocals in the Mesopotamian numerals are sets of numbers that multiply not just to one, but any power of 60. Some examples would be 4 and 15, which multiply to 60, or 16 and 225, which produce 3600. Because these powers of 60 appear as 1, these sets count as reciprocals. It’s truly staggering what not having a zero does to math.
But when you consider the applications these ancient civilizations, such as the Mesopotamians, used math for, it does not make much sense to have a zero. For their purposes, a representation for zero would be irrelevant. Using a base 60 system, they could count to far higher with their digits before needing place holders. And when you’re counting cattle or grains or simple transactions at the market, zero is the last number you want to see on your accounting clay. This was a society that dealt entirely in positive numbers and practical, tangible concepts. We can look back at the Mesopotamian number system now and think, “Look at how hard it was for them to do even basic operations like completing the square,” but in a time when each man could only farm as much land as they and their family could do themselves and the technology for giant architectures was not common, completing the square was about as advanced in mathematics as any one person ever needed to attempt.
Like writing, mathematics is a largely intangible concept, and thus got off to slow start purely for practical purposes. Archaeological evidence indicates it would have simply started in counting animals and crops for the purposes of trade, or perhaps for counting people in some sort of rudimentary census. It wasn’t until humanity’s capability for the written language had advanced enough to express and record complex ideas that math began to see use for architecture and infrastructure. For the Mesopotamians, perhaps one of the most important uses of mathematics was in irrigation. Mathematical standards enabled uniform construction of materials, which was essential for carrying water the long distances necessary to hydrate the numerous farms of Mesopotamia. Advanced accounting and inventory ensured that construction had all of the materials a project would require without being oversupplied, as well as pay and supply the necessary manpower to work the construction project (Melville, Robson).
The mathematical system of the Mesopotamians can be quite a culture shock for American students. I myself was lost on the concept for the first week. It took a lot of practice for me to understand and comprehend the ‘reciprocals’ required due to the lack of a Mesopotaian zero. But it’s truly fascinating, regardless of its difficulty. It’s amazing to think this is one of the first advanced number systems to exist in human history. Its differences are shocking; for the unprepared mind, they can leave one feeling numb and lost. But once one manages to cross the bridge from the present to the past, the concepts ready to be rediscovered are truly staggering.
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