Unlike most numbers, zero was not easily accepted in its beginning. In fact, when the concept of zero was beginning to emerge, there appears to have been many people that were afraid of its acceptance (Seife, 2000). Due to such fears, these people refused to use it or try to understand it for what it could be. Overtime though, we see that the development of this concept has become one of the greatest discoveries/inventions in the progression of mathematics. It is not known when exactly the concept began, but its progression has changed the way we look at the world and mathematics forever.
It appears that since the beginning of time, people all over the world have understood zero as the concept of nothing. However, it would be many millennia before zero received its name and its status as a number. Numbers began with the necessity for counting things. You could not see or count nothing and so it was obvious that it could not be a number. Thus, there was no such thing as zero. When the idea for zero was considered, it was met with much fear from the Greeks, Pythagoreans and western societies who would not accept it. The Pythagoreans especially believed that numbers were a way to get closer to God. Since they considered God to be infinite, then zero as the opposite of infinity must also be the opposite of God or the devil. Thus, since zero was of the devil, it was forbidden to be used (Seife, 2000).
The idea that lead to the acceptance of zero began in Samaria. It was the Samaritans that first created a positional counting system that they used to keep track of their trade (Braggs, 2004). Around 300 B.C. this system of mathematics was passed to the Babylonians (Szalay, 2017). Although they used different bases for their systems, the idea of representing numbers based on the position they were on the tablet or other writing device was the same. After a period of using this method, their writings with blank spaces to represent nothing in the columns started to become confusing. They would look back on the work that they had done and not be able to tell numbers apart from one another. Because of this, they produced a symbol that looked like two slanted wedges indicating that there was nothing in the specific columns where we would put a zero now days.
The Babylonians weren’t the only ones that used a symbol that we now recognize as zero as a sort of placeholder representing position of numbers. Nearly 350 A.D. the Mayan people discovered the usefulness of zero as a placeholder when creating their calendars (Szalay, 2017). The Mayan calendar is one of the only calendars that began with zero. Each month of the Mayan calendar also started with the number zero and was partitioned into 20 days (Nieder, 2016). However, they, like the Babylonians, were unaware of the potential that this seemingly uninteresting number would hold when used like any other number.
The first people to recognize zero and use zero as a number were the people of India (Bragg, 2004). A man by the name of Brahmagupta established the properties of addition, subtraction, multiplication, and his idea of division by zero around 650 A.D. (Wallin, 2002). Although he claims to not have invented zero, his work is the first recognized as treating zero as more than a placeholder and more like a number (Szalay, 2017). This information slowly made its way to Europe and was still not immediately accepted. However, about 1202 A.D., it was found by the Italian mathematician Fibonacci who was the first person known to have used it to make calculations without using an abacus or any other device besides a writing utensil and a form of paper.
The ability to make calculations without the use of an abacus took off from there. Accountants no longer had to use the abacus to check their balances, and the foundations of algebra were laid. The unique properties of zero as a number made it possible for Newton and Leibniz in the 1600’s to develop calculus which we now use with differential equations to help us understand the world in a way that no one ever dreamed would be possible. Although zero seems like an insignificant idea to us now, the advancements that have come because of it should not be taken for granted.
The development of zero throughout time came in a similar way to how children develop an understanding of zero today. In an article written by Andreas Nieder, we see four distinct stages that the concept of zero must pass through to be fully comprehended (2016). Unlike any other number, zero is the only one that must be conceptualized by the properties that it holds and not simply as a number or way of counting (Chuck, 2012).
In the first stage Nieder proposes that zero is considered as the absence of a stimulus (2016). Now what does he mean by that? If you were to step into a dark room, would you be able to tell if anything was in the room? What about if you were to turn on a light? Now turn off the light. Is the stuff you saw in the room still there? Take a moment to think about how these questions could relate to zero being referred to as “the absence of a stimulus”. Got it? Just as this is the first stage proposed for the development of understanding of zero in children, it is also the first stage that we see in history. At first ancient people, especially the Pythagoreans were focused on the things around them. The things that existed were the only things that mattered. The idea of a void was present, but the recognition of it as something was unthinkable. They could not see it and thus it could not be anything.
The next stage of understanding zero is recognizing zero as “nothing” verses “something” (Nieder, 2016). This should not to be confused with zero as a quantity. Unlike the first stage, in the second stage, we acknowledge that there is nothing there, but we aren’t entirely sure what that means or how to express it. Think about an experiment done in Nieder’s article that illustrates this stage. Children were given the task of determining, using sticky notes, how many cookies were on a set of different plates with varying amounts of cookies. When the children got to the plate with no cookies on it, they left the sticky note blank. When they were asked to interpret the blank sticky note, they gave the response that it meant there were no cookies on the plate (2016). How does this relate to representing zero?
Just as the children in the experiment placed blank sticky notes to represent that there was nothing there, we see the Babylonians and other cultures of this time frame do something very similar. As things to be counted increased, especially with trade, a positional system for interpreting numbers was created to help with resenting higher orders of numbers. Although calculators such as the Babylonian abacus and the Chinese rod method used the same positional system with objects in columns or rows, it was desirable to have a written record of the calculations. However, writing these calculations on paper proved to be very difficult. As more people kept records on tablets and other such materials, they would leave blank spaces to help differentiate between numbers. It was these blank spaces for which we see the second stage, proposed by Nieder, develop over time.
Although the understanding of zero as “nothing” verses “something” was useful for a time, eventually, with the similarity in symbols for numbers, it became too confusing to keep track. Thus, the Babylonians needed a new way to represent the empty column. What they came up with demonstrated the next stage of thinking about zero which is to recognize zero as a quantity (Nieder, 2016). In the Babylonians case, they placed two diagonal wedges in the column to represent that there was nothing there. This meant that for that specific column the quantity was zero. How do we still see this develop today? Think about if you taught a young child using apples that taking two apples away from two apples that I have means that I will have zero apples left. Would the child be able to do this same basic math with bananas instead?
If the children you are testing have reached the point where they understand zero as a quantity, they will be able tell you that the answer is zero bananas. They understand that zero is the smallest number in the number line and so if I have a number of any item and take all of those items away, then there are zero items left. However, if you were to ask them to add zero apples to a number of items, they would not fully understand how to do that. This stage is the recognition that there is such a thing as an empty set (Nieder, 2016).
The last stage that Nieder suggests is that of zero as a number (2016). What does it mean for zero to be a number? This is the stage in which we finally recognize and appreciate the properties of zero. This stage is shown throughout history beginning with Brahmagupta from India who was the first to write down rules for addition, subtraction, multiplication, and division of zero (Wallin, 2002). Although our understanding of division by zero has changed over time, it is clear to see Nieder’s last stage at work. The abstract thinking that is required to develop this concept of zero as a number is clearly developed throughout time as well as within each individual born today.
Podcast: The Concept of Zero
Podcast Script:
Should the Number Zero be Banned? [Gasp]
Crazy question, right?
We have lived with zero for so long now that life without it seems inconceivable.
However, if we take a look into history, we would find that ancient people refused to acknowledge zero. Even the idea of zero was foreign, frightening, and, in the case of the Pythagoreans, blasphemous.
But why? What were these people so afraid of? How could zero or the concept of nothing produce such fruit of fear and turmoil?
The answers to these questions lie in the nature and properties of zero. When we first look at zero, it seems like a simple idea. Even children understand that having a quantity of zero bananas, simply means that I have no bananas [child laughing]. However, this is just one of 4 stages outlined by Andreas Nieder in his article “Representing Something Out of Nothing: The Dawning of Zero” that is used for representing and understanding zero. According to his article, zero can also be understood as the absence of a stimulus, a representation for nothing verses having something, and finally as a number itself.
These different stages of understanding zero have caused great trouble and confusion for many logicians. Those that try to teach children to understand why adding zero to a number results in the same number or in other words zero does nothing to the number, then have great difficulty explaining why multiplying that same number by zero, the number that once did nothing, now changes the number entirely-to zero [huh?]. Further if we multiply of any number by zero, we see that we completely destroy the number line without any hope for reversing it. It is as though someone dropped an atomic bomb [background: explosion] on the existence of numbers with no hope of renewal. Any number times zero is zero, and when we try to divide the resulting number (zero) by zero there is no way of knowing what the original number was and thus it could be any number we choose. It is because of this fact that even trying to divide by zero one time creates a logical void where anything can be proven.
I think Charles Seife put this idea best in his book “Zero: The Biography of a Dangerous Idea” when he said, “Within zero there is the power to shatter [background: glass breaking] the framework of logic”. And that is exactly what the Greeks, Babylonians, and Pythagoreans were so afraid of.
It is obvious that life without zero is doable, I mean these ancient civilizations lived their lives and studied mathematics for nearly 2 millennia before zero was finally accepted. So then if we were to ban zero, what would we be missing out on? [clock: tick, tick, tick, tick]
Well, for one thing, our calculations would be much more difficult to make and decipher. As mathematics developed over time, writing down calculations allowed mathematicians to keep track of and formulate new ideas. However, although their logic was sound, when they looked back at their writings, they had to go to great lengths to determine what numbers were being displayed and what calculations had been made. This was because they used the same symbol to represent different numbers such as representing both the number 61 and 3601 with the same symbol. When zero was used as a placeholder however, this instantly made it easier to distinguish one number from another and thus rapidly advanced calculations.
From zero’s discovery, came many more discoveries and advancements in mathematics, as well as the existence of technology [light bulb]. Without the number zero, we would have no additive inverse for addition and thus calculations for algebra and higher math would not be possible. We would not be able to use mathematics to determine the best way to maximize profits while minimizing costs [cha ching], find inverse functions [page flipping sound], or use negative numbers to represent how cold the temperature is outside [shivering]. None of these would have meaning or value if zero were not recognized.
Although zero does have peculiar properties which can potentially destroy the logical foundation of mathematics, as we’ve seen, it also has incredible power to advance mathematics. It is no wonder that Andreas Nieder and many other mathematicians have claimed that even though zero stands for nothing, it is considered one of the greatest achievements of mankind.
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