May 9, 20255 min readZero Is Foundational to Modern Mathematics. But It Was Rejected for CenturiesConceptual problems, ideology clashes and xenophobia prevented the concept of zero from catching on for a long time. Today all mathematics is based on itBy Manon Bischoff edited by Daisy Yuhas akinbostanci/Getty ImagesI’m a zero at mental arithmetic. It’s true—I struggle with this skill—but I want to focus on the phrase itself. In our language, we often equate zero with something negative. But zero is the only real number that is neither positive nor negative. It is neutral.Why the negative association? Humankind has long harbored strong feelings toward zero; it was even banned in some places at one point. Xenophobia and ideology held back this powerful concept. Yet today all of mathematics is based on this number.Defining “zilch,” “nil” or “0” is not easy. In fact, neuroscientists have studied how we conceive of nothing in varied ways. It should be no surprise, then, that cultures have approached zero in different ways over time.On supporting science journalismIf you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.But what is surprising is just how long people got on without this concept. Numbers have accompanied humankind throughout history. The oldest documents record them. Trade cannot be conducted without them, and they are needed to measure land or record a recipe for beer. Zero is somewhat unusual and not strictly necessary for all of these activities.As a result, it took several millennia for zero to be accepted as a number in its own right. People have repeatedly resisted it. Yet today we know that all other numbers—and all of modern mathematics—would truly be nothing without zero.A History of AbsenceZero may have been invented more than once, with different functions. For example, around 5,000 years ago the Babylonians had a concept of zero, but it wasn’t a number that stood for itself. Instead they—like us—used a place-value system to indicate numbers: if I write down three digits in a row, such as 145, then the first number corresponds to the place of hundreds, the second to tens and the last to ones (or units).The Babylonians used a similar approach except that their system was not based on 10 but on 60. In a place-value system, you need a zero to distinguish a number such as 105 from 15. The Babylonians usually made do with inserting a space, which is one of the oldest references to something like a zero.It’s also notable that many ancient societies got by without this concept. In ancient Greece, all kinds of advanced mathematical considerations were made (just think of Pythagoras’ theorem or the basic pillars of logic by Aristotle) without a zero per se. The abstract concept of nothingness was well known to the ancient Greeks, but they regarded it as part of logic, not mathematics. Zero is strange, after all. For example, no number can be divided by zero. The ancient Greeks disliked this property.The exact origin of zero as we use it today has been the subject of some debate, but we know that in the seventh century C.E., the brilliant Indian scholar Brahmagupta introduced zero as a number, along with negative numbers, which had not been used before.Previously, mathematical problems were usually illustrated using geometric objects. For example, you might want to know how two rectangular fields can be connected to form a square piece of land of equal size. Negative numbers are irrelevant for such tasks, as is zero.Brahmagupta was also interested in such abstract problems, however. To use these new numbers correctly, he first needed a functioning set of rules that clearly specified how to deal with these quantities. In his book Brāhmasphuṭasiddhantā, for example, he wrote that the sum of two positives is positive, the sum of two negatives negative, and the sum of a positive and a negative is their difference; if they are equal, it is zero. He also wrote that the sum of a negative and zero is negative, that of a positive and zero is positive, and the sum of two zeros is zero.In a similar style, Brahmagupta also described how to multiply and divide the new numbers. The rules he laid down around 1,400 years ago are the same we learn at school today—except for one. He defined zero by zero as zero, which is wrong from current mathematical perspectives.The Zero Gradually SpreadsBrahmagupta’s rules, together with the Indian decimal number system, quickly spread throughout the world. Arabic scholars took up the concepts and developed the Arabic number system, on which our modern numbers are based. From there, the zero and Arabic numerals arrived in Europe—albeit at the worst possible time. The Crusades took place between the 11th and 13th centuries, and with them came an immense rejection of all ideas and knowledge of Arab or Islamic origin.In Florence, Italy, this development culminated in the banning of the number zero in 1299. At that time, the economy in that city was flourishing, and merchants from all over the world came together to sell their goods. In a town famous for banking and trade, the zero posed a real problem: it was very easy to increase the size of a number on a piece of paper by simply adding a few zeros. A 10 quickly became a 100 or even a 1,000, whereas the Roman numeral system did not allow such manipulation. City leaders therefore decided to banish the zero and rely on the tried and tested Roman numerals.But calculation with Roman numerals is incredibly complicated and cumbersome. So gradually, over more than 100 years, Arabic numerals, including the zero, prevailed. In the 15th century the concepts finally became accepted by society at large.Much Ado about NothingAt the beginning of the 20th century, mathematician Ernst Zermelo created the set of rules on which modern mathematics is based. At that time, logicians were looking for the simplest possible rules from which everything in mathematics could be derived. Whether numbers, systems of equations, derivations or geometric objects, everything should spring from a few basic assumptions.Zermelo developed nine simple axioms, that is, unproven basic assumptions, on which everything is based in math. These are still used today. One of the axioms is: “There is an empty set.” This is something like the zero of set theory. That’s where it all starts—it’s the “Let there be light!” of mathematics. And, in fact, this is the only set that Zermelo constructed so explicitly. The other rules say, for example, that you can “combine two sets to form a third set” or “select an element from a set.”Everything else follows from the empty set, the “zero.” For example, the numbers are constructed from it. To do this, it helps to imagine a set as a bag into which you can pack objects. An empty set corresponds to an empty bag.When constructing the numbers, Zermelo started with zero. It corresponds to the empty set or empty bag. “One” is the quantity into which the previously defined zero is packed, so it’s a bag with an empty bag inside. Two is the quantity that contains the 1 and the 0, or a bag containing a bag that itself contains a bag. The 3 is then the quantity that contains the 2, the 1 and the 0—okay, I admit it gets confusing.Graphically, this can be represented a little better if ∅ symbolizes the empty set:0 = ∅ 1 = {0} = {∅} 2 = {0, 1} = {∅, {∅}} 3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}Zermelo has thus laid the foundations for integers. From here, all other numbers can be defined, including negative numbers, fractions, irrational numbers, and so on.Mathematical concepts other than numbers can also be obtained in this way. You can gradually work your way up in complexity until you end up with the most abstract structures of modern mathematics. It is fortunate for humanity that we eventually came to realize the power of zero as a starting point and to accept it.This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.