Another innovation we take for granted yet has a surprisingly short history is the number zero. My research on this particular subject is principally from three sources: The Book of Nothing by John D. Barrow, The Nothing that Is: A Natural History of Zero by Robert Kaplan and Zero: The biography of a dangerous idea by Charles Seife.
The idea of zero isn’t as obvious as we might think. It does several important jobs in our number system. It plays an important role as a placeholder in our numerals. The number 106 could easily be mistaken for 16 of we didn’t explicitly show that there is no value in the tens column. No value doesn’t mean unimportant, as we discover when we add a zero to the end of a number.
Zero is also what you get when you take a number a way from itself. Or, to put it another way, it is the number that lies half-way between one and negative one. This makes mathematics much more straightforward. Consider, for example, our dating system, which was invented by monks who didn’t utilise the concept. The year Jesus was believed to have been born was chosen as the year 1, and all other years were calculated from then. Jesus’ tenth birthday was, therefore, on the year 11. More confusingly, the year 1 AD was immediately preceded by the year 1 BC. This meant there were only nine years between 5 BC and 5 AD (cf. Kaplan, 1999, p. 103). Any mathematics utilising negative numbers needs a zero to make the arithmetic work without introducing extra steps.
The zero as we know it didn’t arrive in the West until the end of the tenth century CE when it was introduced by the French mathematician Gerbert of Aurillac (who had a distinguished career, culminating in his election as Pope Sylvester II in 999 — Barrow, 2009, pp. 47–48). Gerbert came across the idea in Spain, which was heavily influenced by the Arab culture. The Arabs themselves had discovered zero in India, where it had been a part of the number system for centuries (Cooke, pp. 244–245). The question of why Europe failed to come up with anything like a zero is puzzling at first. It’s not like the Europeans weren’t able to do mathematics–after all, they could boast Euclid, Archimedes and Pythagoras as their leading lights. That heavy Greek influence appears to have also been part of the problem. Philosophy and mathematics were inseparable for the ancient Greeks, and the relations between numbers had philosophical implications. The Greeks had come across zero, a version of which existed in the more ancient Babylonian mathematics. However, it did strange things. If you add it to a number, that number doesn’t get bigger, even though addition is expressly about making numbers bigger. Worse, multiplying by zero obliterates numbers. Dividing by zero is impossible. It was the mathematical representation of the Void, something that Greek philosophy (and its Western successors) simply could not tolerate (Seife, 2000, pp.19–23).
Indian philosophy didn’t have this fear of Nothingness. In fact, it delighted in it. Nothingness is a natural state from which Being can proceed, and to which Being can return (Barrow, 2009, p. 42). Nothingness was often represented by the Indians as a simple dot or circle and (according to one theory, at least. Barrow, 2009, p. 38ff.) this circle came to be used to represent the nothing that made mathematics so much easier.
Whilst the Greeks resiled from what zero meant for mathematics, the Indians systematised it. The astronomer Brahmagupta defined the rules needed to use zero and negative numbers arithmetically, even going so far as to define infinity as the result of dividing any number by zero (Barrow, 2009, p. 38). Far from threatening the fabric of the Greek Cosmos, this trick allowed Nothing to become an all-encompassing Everything.
Once the idea of zero got to the West, it started to spread, although it did run into trouble on occasion. In Florence, for example, zero-based mathematics was banned in 1299 due to fears of fraud. It was felt that it would be very easy for an unscrupulous banker to surreptitiously enter a zero on the end of a record of debt, rendering the total ten times it’s actual value. The Roman system in use at the time had ways of minimising the possibility of fraud (Barrow, 2009, p. 48). There was also suspicion as Europeans considered the new numerals as ‘Saracen magic’ (Kaplan, 1999, p. 102). However, the Indian system slowly became more common, until it was the de facto standard for trade in Europe in the Sixteenth century (Barrow, 2009, p. 48).
The advantages of this system were considerable, especially when mathematics became more than simply a way to record financial transactions. Moreover, zero came to be understood as a number in its own right. Mathematics dealing with negative numbers became trivially easy, which had important implications for commercial activities like double-entry book-keeping (Kaplan, 1999, p. 110). When zero is introduced into exponent arithmetic, exponents become exponentially more useful. Being able to explicitly name zero as a remainder allows us to do modulus arithmetic. Modulus arithmetic is at the heart of Fermat’s Little Theorem, which gives us the ability to find prime numbers of arbitrary length and thus opens up the possibility of modern computer cryptography (Kaplan, 1999, pp. 120ff).
It astounds me that such a powerful idea, and one which is now such an integral part of our culture, has only been in universal use for the last few centuries. The idea of zero seems so obvious. And yet, it wasn’t always so. It also means that it won’t necessarily be understood as easily as I might think today. Whilst it is a powerful idea that is taught to children from a very young age, there is nothing inherent about the idea of zero or the mathematics it enables that means I can just assume that everyone finds it obvious.
There are implications for my practice as a teacher. For one, I must be careful to recognise when students have missed out on fundamental elements of their mathematical education. For example, Di Siemon has identified several fundamental skills learners need to develop in order to successfully ‘do’ maths. For instance, students need to learn to ‘trust the count.’ This means recognising how the numbers we use for counting correspond to things being counted. We can only say we have counted something correctly if the each number in our counting matches exactly one item in the set of objects being counted. When we run out of objects, the last number we listed is the number of objects in the set (Siemon, 2019). This is such an obvious thing that we assume everyone knows, it’s easy to miss when it is going wrong. The fact that a child can list numbers in numerical order while pointing to objects doesn’t mean they are counting them. I have to be careful of my assumptions of what they actually know.
I find it rather fascinating that the number system we use wasn’t devised by any of the great mathematicians of the West. Euclid and Pythagoras’ geometry is still taught, both in primary schools and universities. We are still heavily influenced by the thought of Aristotle and Augustine. Yet the numbers we use come from the East, first from the Indians and then the Arabs. This suggests to me that there is no room for cultural hubris, least of all in mathematics. The history of our numbers and the little round nothing that gives them so much power makes for fascinating reading, but also tells us that a good idea is a good idea, no matter where it comes from. The Europeans were probably right to be suspicious of ideas coming from the land of the Saracens, but this particular idea transcends politics and war. They literally had Nothing to fear.
Barrow, J. D. (2009). The book of nothing: Vacuums, voids, and the latest ideas about the origins of the universe. Vintage.
Kaplan, R. (1999). The nothing that is: A natural history of zero. Oxford University Press.
Seife, C. (2000). Zero: The biography of a dangerous idea. Penguin.
Siemon, D. (2019). Common Misunderstandings – Level 1 Trusting the Count. State Government of Victoria. https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/lvl1trust.aspx