Monday, September 28, 2015

The Value of Nothing: How Did We Get From One to Zero?

History of Mathematics

At the very beginning of my K-12, and maybe even my preschool education, zero has always been a number. The concept of zero - as a number that stands for something that isn't there - is as acceptable, and natural to me as the numbers that represent things that are there (1 thing, 2 things etc.). Mathematics wouldn't make sense without it. For counting, I could either start at nothing, 0, or 1. For arithmetic, 2-2=0. Two things exist. Take those two things away and those two things aren't there anymore. Nothing is left.

But the concept of zero as a number isn't natural at all, especially in Western Civilization (Europe) (0). The concept of zero started in the East, most notably in India, and it took until the Renaissance for Western Civilization to accept a representative number for nothing (0 + 1). So how did Western Civilization finally place a value for nothing? How did we get from one to zero? The time period and from what civilization zero was born depends on how one defines a number.

According to the MacTutor History of Mathematics Archive, the expression of zero throughout history can be separated into two categories (1). The zeroeth category (see what I did there) is zero as an indicator for empty space in a number system or as a positional notation in context with other numbers (1). The first category is zero as an abstract, independent number with value and properties, which is the way we think of zero today (1+ 2).

We will first discuss the first category: zero as an abstract number. Brahmagupta, an Indian mathematician living from 598 to 665 AD, is considered to be the first to think of zero this way (0 + 3). In his work titled Brahmasphutasiddhanta, he developed properties for negative and positive numbers, and for the number zero (1 + 4). The idea of zero as an actual number spread to China and the Middle East (1). Fibonacci in the 1200s tried to get Europe to replace Roman numerals with the 0-9 number system that we use today, but met opposition (0 + 1 + 7). In Christianity,  God was associated with infinity and the idea of nothingness, the opposite of infinity, was therefore associated with the devil (0). Furthermore, Fibonacci was campaigning for the new number system during the Crusades, and the new number system was associated with Islam, since they had already taken the same number system from the Hindus (6 + 7). Finally, in the 15th century , Europe converted to the 0-9 number system (1).

But before zero was seen as an abstract number, the concept of zero started as a positional value, which is the zeroeth category (1). In an interview on BBC's In Our Time Radio program, Robert Kaplan, Ian Stewart, and Lisa Jardine describe the importance of the number zero in history, and the significance of it's use in positional notation. The concept of zero started as a symbol to signify no-number about 5,000 years ago in the Sumerian number system, which used positional notation (0). Eventually, zero as a positional value became common in bookkeeping in different civilizations over time because of trade, except in Western Civilization (0). The ancient Greek mathematicians considered basic arithmetic that was used in trade not intellectually stimulating enough to think about (0). Furthermore, the ancient Greeks decided that the concept of nothing couldn't exist in the real world (5). So, the zero that meant no-number in trade didn't cross over to become an abstract mathematical concept for the ancient Greeks, nor the Romans, who based their number system off of the ancient Greeks (0).

So when in mathematical history should the advent of the number zero be marked? Was Sumerian Civilization the first when they used zero has a positional value? Or was it Indian Civilization the first when they treated zero as an abstract, independent number?

As I discussed in a previous post, I believe counting to be a mathematical concept, and therefore consider assigning numbers and values to the physical, tangible world as one of the first mathematical activities humans did. Historically speaking, humans did not start counting at zero, as indicated by the number systems of many ancient civilizations, since nothing isn't tangible and therefore has no value (1).

Yet, the concept of zero was used as a positional value in number systems, and arithmetic was used in trade, even though people of these civilizations didn't treat zero as a number that had value, and therefore didn't compute it in their calculations like we do today (0). They probably skipped right over the symbol, considering that the symbol for zero in position notation was often represented as an empty space (1). But they were inadvertently placing a mathematical property on that empty space when they did arithmetic, a property that Brahmagupta would attribute to the number zero: any number plus or minus zero is the same number (4). With that said, this action of skipping over the symbols for zero as a positional value is equivalent to us using the number 0 in 101+190 = 291.

Therefore, I believe the number zero was born when it was just a positional value at the time of Sumerian Civilization. Very humble beginnings indeed, especially when we look ahead in time and see all that was achieved during the Renaissance when zero was finally accepted as a number in Western Civilization, like Newton discovering calculus (0 + 2). Who knew nothing could amount to so much?

Sources: *No person or organization has endorsed my work, nor am I affiliated with anyone or organization. I am a college student.*

0. Podcast. Interviewees: Kaplan, Robert. Stewart, Ian. Jardine, Lisa. Interviewer: Bragg, Melvyn. "In Our Time: Zero." BBC Radio 4. Date of Access: September 28, 2015. http://www.bbc.co.uk/programmes/p004y254

1. O'Connor, John J. and Robertson, Edmund F. "A History of Zero." MacTutor History of Mathematics Archive. Date of Access: September 28, 2015. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html

2. "Who Invented the Zero?" Ask History. Date of Access: September 28, 2015. http://www.history.com/news/ask-history/who-invented-the-zero

3. Hayashi, Takao. "Brahmagupta: Indian Astronomer." Encyclopedia Britannica. Date of Access: September 28, 2015. http://www.britannica.com/biography/Brahmagupta

4. Mastin, Luke. "Indian Mathematics - Brahmagupta." The Story of Mathematics. Date of Access: September 28, 2015. http://www.storyofmathematics.com/indian_brahmagupta.html

5. "A Brief and Early History of Zero (ca. 2nd C BC - Onward)." The Ancient Standard. Date of Access: September 28, 2015. http://ancientstandard.com/2007/08/22/a-brief-and-early-history-of-zero-ca-2nd-c-bc-%E2%80%93-onward/

6. Mastin, Luke. "Islamic Mathematics - Al-Khwarizmi."  The Story of Mathematics. Date of Access: September 28, 2015. http://www.storyofmathematics.com/islamic_alkhwarizmi.html

7. Mastin, Luke. "Medieval Mathematics - Fibonacci."  The Story of Mathematics. Date of Access: September 28, 2015. http://www.storyofmathematics.com/medieval_fibonacci.html