Zero, Zilch, Nadda
Did you know it took the invention of the Calculus to nail down and finalize the concept of 0? That’s right it was just about the end of the 17th century (1687 is when Newton’s Principia Mathematicawas published). Newton and Leibnitz, two independent thinkers each tackled and used infinite limits to solve problems. Separately they solved the last standing concern about the meaning of 0. It hasn’t even been350 years since mankind knew once and for all what 0 was. This is a fascinating tale I think worthy of an “antrho-pyschologist”. These folks find out how we arrived at and conceptualized modern knowledge. I invented this field of study because we have enough history to piece together processes we now take for granted.
This doesn’t mean to imply that we didn’t have inkling about 0. It is just that we needed to formalize and standardize a number of other problems before we could have a clear understanding. It took us nearly our entire time as a species to understand what we understand as a simple idea today. Oh sure we understood the “absence” of something. If I told a neighbor that I didn’t have any apples they would know that I had 0 apples to lend them.
The Beginning of Understanding
Young children learn the Natural numbers. They know how to count 1,2,3,4 and so on. It isn’t until a bit later that they can even process and add 0 to this set of numbers to get the Whole Number set. This is exactly how our species came to understand these two sets of numbers as well.
Many of you may be scratching your heads wondering if I got this wrong. 0 is such a simple idea. Everyone knows what this number is and how to use it. The thing is it isn’t natural for us to know this concept. The easiest way to show this is to use Roman Numerals to illustrate the point. Even though this number system is base 10 it is made up of symbols that represent certain numbers. Roman numerals are still not far removed from any other ancient hieroglyphic system. Please click here to see my hub on Roman numbers.
Take the number CXXIII. This would be interpreted as 1 - 100 and 2 – 10s and 3 – single items. Think about looking at these as bills in your wallet. In our modern symbolism we would write this number as 123. The Romans didn’t have any need for the number 0. All values could be expressed without this symbol. For example MXI in modern symbolism would be 1011. The hieroglyphs were not placement valued. A modern symbol placement example would be to consider that 2 is different from 20 which is different from 200 because of where the 2 is living.
The Babylonians too had a hieroglyph system. Their number system was base 60. For quite a long time one would know that the number 215 should really be understood to mean 2105 because of the accompanying test description with the number. They were the first to recognize the need for a place holder in the event hieroglyph arrangement did not adequately express value. I don’t have any reference to support why this idea of a separate placement value happened. I imagine this became a problem because lawyers first came into existence. They probably began to win cases about misunderstanding between the text and written number. It’s just a guess. I may just be tainted with modern cynicism.
The Babylonians did understand there was a problem with their system. They begin to express placement value. Surprisingly they did not know to put 0’s on the ends of numbers. To know that 215 meant 2150 one had to have a textual message. The most common notation was to put a couple of hatch marks below a symbol to indicate the correct value. Using our modern keyboard and a single quotation mark means that 21”5 would be known as 2105 today. Some writing examples used a single hatch mark and there are examples where three hatch marks was the notation used. By using one of these three notations scribes initiated the first attempt at the number 0 as a placement symbol/value.
The First Real Steps to Formalize a New Number
It is considerably later before the next step is taken with our understanding of 0. About the middle of the 7th century the renowned Indiana Mathematician and Astronomer Brahmagupta published the next bit of understanding we have about 0. One of his “profound” observations is that if one has 10 apples and someone takes away 10 apples you now have 0 apples. Today we would write this problem as 10 – 10 = 0. He found other properties of 0 such as if 0 is added to any number whether it is positive or negative then the number is unchanged. He also demonstrated the product of any number whether it is positive or negative is equal to 0. He is the first to describe these properties in a published treatise. For this reason he is often referred to as the Father of 0.
Still, it was another two hundred years where these concepts first traveled to China and then back to the Middle East before the next important step enters into our understanding. It was the Father of Algebra Mohammed ibn-Musa al-Khowarizmi who introduced the concept solving some Mathematical problems leading to modern day symbolic Algebra. It was al-Khowarizmi who called 0 as “sifr”. He showed that the solution of some algorithms could be the value of 0 or “sifr”. The symbol used for 0 was an oval not much different than our symbol.
Our understanding of 0 continued traveling through the world west and north. It now began moving around Europe. It found its next benefactor in the Mathematical genius of Fibonacci. He expanded on Brahmagupta and al-Khowarizmi’s work. Through him Italian and German merchants and accountants came to appreciate 0 as a symbol placement value. Our modern numerical system quickly galvanized about this time. It was this practical group of professionals that profited from this way to balance assets against liabilities in Fibonacci’s “Abacus” book. It was these businessmen who introduced the cipher from “sifr”.
The Final Piece of the Puzzle
Finally the Calculus taught us in the 18th century what it meant to divide by 0. This was the last bit of trouble we had to resolve. We needed to have an understanding of instantaneous speed for example. The Calculus taught us to look at the ratio of distance differences to diminishing time intervals (which is how one figures average speed). When the diminishing time goes to 0 we have the instantaneous speed. This concept was truly profound. It is no less astonishing today. No wonder there was so much controversy over whom to credit so extraordinary a concept. I am grateful to both Leibnitz and Newton since each understood the concept in unique ways which helped me to appreciate the Calculus all the more.
Now do you see the importance of introducing some history and economics into your curriculum? Mathematics may be an intellectual pursuit today. It was simply a tool in the past. It helped us to understand processes. It helped us to regulate everything from accounting to gamboling. It helped us to trade and barter more fairly not only with ones neighbor but between governments. With time we found something pure and immune to opinion or political influence hiding in the very real and practical world. Please keep our old thoughts alive. Knowledge happened as natural processes for reasons too innate even for current students to be detached from yet.
Perhaps we should look at the independent development of 0 in the new world. The Maya’s too had discovered much about 0. This will hopefully be another blog another time. Some ideas and concepts are so important that where mankind was “growing up” is irrelevant. Some ideas were important to an emerging society independent of the location and social interaction. This may provide further insight into who and what and how we think as a species. To find and understand a concept independently is most unusual. How does this define us as a thinking creature? It really is a small, small world out there.