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.

1. O'Connor, John J. and Robertson, Edmund F. "A History of Zero." MacTutor History of Mathematics Archive. Date of Access: September 28, 2015. 

2. "Who Invented the Zero?" Ask History. Date of Access: September 28, 2015.

3. Hayashi, Takao. "Brahmagupta: Indian Astronomer." Encyclopedia Britannica. Date of Access: September 28, 2015.

4. Mastin, Luke. "Indian Mathematics - Brahmagupta." The Story of Mathematics. Date of Access: September 28, 2015.

5. "A Brief and Early History of Zero (ca. 2nd C BC - Onward)." The Ancient Standard. Date of Access: September 28, 2015.

6. Mastin, Luke. "Islamic Mathematics - Al-Khwarizmi."  The Story of Mathematics. Date of Access: September 28, 2015.

7. Mastin, Luke. "Medieval Mathematics - Fibonacci."  The Story of Mathematics. Date of Access: September 28, 2015.

Sunday, September 13, 2015

Riddle Me This, Diophantus

Blog post for Doing Math

Diophantus was a great, ancient greek mathematician, who played a very important role in the subject of algebra ( To honor this man, my MTH 495 professor gave us a riddle - a riddle of algebra - to solve that would give Diophantus' age when he died. Here it is, as given on the class handout, below:

" 'Here lies Diophantus,' the wonder behold. Through art algebraic the stone tells how old: 'God gave him his boyhood one-sixth of his life, one-twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; in five years, there came a bouncing new son. Alas, the dear child of master and sage after attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' "

The language seems clear but riddles are tricks of the mind. What seems straightforward, isn't, and the challenge is finding where the riddle deliberately leads the mind astray, so it would make an erroneous assumption. In other words, how has the riddle tricked us into thinking one interpretation, when it really meant something else?

To demonstrate how challenging the riddle was for me, I went through 7 attempts or 7 different interpretations of the riddle before I got the correct answer. For the first attempt, I tried solving the riddle without using algebra, and that meant picking an age for Diophantus, and doing the series of computations laid out in the riddle to see if it was correct. Basically, guess-n-check. I picked 50 because people rarely lived for very long back then.

On this first attempt, I thought the challenge, or the trick in the riddle, was in the line "And then one-seventh ere marriage begun." The word ere means before, according to ( So, I had to subtract 1/7 from his boyhood and youth before continuing on with addition. So, the sequence of events laid out in the riddle is not in chronological order. Everything else seemed straightforward. I interpreted "God gave him one-sixth of his life," as 1/6 of his age. "One-twelfth more as youth," I interpreted as his youth was 1/12 more than his boyhood, or 1/6 of his age. "One-seventh ere marriage begun," I took as 1/7 of youth subtracted from his boyhood and youth. My interpretation for the rest: then 5 years after his marriage, a son was born, who grew to half the age that Diophantus eventually reached, 4 years after his son's death. As mentioned before, I picked 50 for Diophantus' age. So, I did the following computations:

Since the ages don't match, my guess was wrong. But the closeness between the two numbers gives a false impression that 50 was close. However, the fractions would divide any number in such a way that would make the two numbers seem close. At this point, I decided that I could keep on blindly guessing or try to solve the riddle in the most efficient way possible, with algebra.

When I continued with my other attempts, I realized the challenge was no longer with the chronological wording of the riddle. The trick lay with the wording of the fractions. In my middle school teaching course, the class went over fractions, and whenever someone said "Take 1/2," our professor would always say "1/2 of what?" The fraction needs to be applied to another number, but sometimes the riddle doesn't tell explicitly what the fraction is being applied to. For example, nowhere in the line "One-twelfth more as youth" does it say what 1/12 is being taken from. Is it 1/12 more of his entire life that is being added to his boyhood, or is it  1/12 of his boyhood, or is it just simply 1/12 of his life was his youth, and not in addition to his boyhood?   

I interpreted the riddle differently, with different algebraic equations, 6 more times. On the last attempt, I got it right. I will show the different combinations or ways I interpreted the riddle with the variations for each section of Diophantus' lifespan below. The straightforward interpretations are on top and the multiple interpretations are on the bottom:

Finally on the last, overall 7th attempt, I got the answer right, at least according to Wikipedia ( The interpretation necessary was youth equaling his boyhood plus 1/12 of his entire life, and his marriage was 1/7 of his boyhood before the end of his youth. My solution is below:

I enjoyed this riddle very much. It reminded me how the process of mathematics should be like. Doing math is about looking at problems or situations with different interpretations to see how it plays out and what's discovered in the process. After all, learning requires making mistakes. Figuring out what went wrong is necessary to find out what will make it right.

Figures were created using Microsoft Word and Geogebra.

Wednesday, September 2, 2015

Mathematics: It Started With Numbers

I’ve made a harrowing discovery. There is a gaping hole in my understanding and knowledge of mathematics. For most, my statement doesn’t seem concerning. Mathematics isn’t a popular subject. But for a future math teacher, like myself, it’s a confession of a big, terrible shortcoming.

My revelation came when my class was asked a very simple question, “What is mathematics?,” and I couldn’t think of a satisfactory answer. The confidence I had in the very subject I was preparing to teach was gone. How can someone teach a subject, when they can’t even define it in words?

Before that question, I believed I knew what mathematics was. I’ve been doing it for years - adding, subtracting, multiplication, matrices, algebra, geometry, and proofs! But those words are either what people do in mathematics or a specific area within mathematics. Not one of them encapsulates what mathematics is, and listing everything in mathematics does not describe what it is either. So, what do all those words have in common to help define mathematics in a clear, precise way?

At first, I thought numbers. Recalling my elementary years, learning math was learning adding, subtracting, and other operations. But before I could do or be taught math, I needed to know numbers. To learn numbers, I counted physical objects before and during kindergarten.

Others in the class shared my view, stating that math is about describing the physical world with numerical values. Some expanded on this concept and said putting numerical values in an order is math. Others mentioned measurement, explaining how trade spurred the use of mathematics, while others mentioned astronomy and how mathematics was used to find the length of a year and day. Finding patterns was also mentioned, and a particular person said, “mathematics is the science of patterns.”

“But is that enough?,” our professor asked and continued to ask with each new claim of what math is. Is having a pattern enough for something to be labeled math? How about counting? Are people doing math when they count? How about labeling a quantity with a specific value, like 4 or 9? Does knowing you have 9 cows make it math? Or could any conceptual form of measurement make it mathematical? A student used an example of a stick as an example of the first rudimentary means of measurement - walk 4 stick lengths and then turn left. How about time? Does knowing the position of the sun in the sky or time of the year make it mathematical? Does assigning that position with a numerical value like 1:00 pm make it mathematical? Just because it has a number or value or quantity, does it make it math?

Since the class period, I’ve been thinking a lot about these questions the professor posed to the class, and I’ve come to the tentative conclusion that yes, just because something has a number or could be described or substituted with a number, it is math. To do the simplest math, like adding, numbers needed to exist or at the very least some form of measurement had to be in place to describe and order our world. To figure out time, the concept of a start and end needed to be established, which could then be described more specifically with numbers. Then operations could have been conceptualized from there. Imagine two people looking at a sundial. One says to the other, “Meet me in two sun-lengths time from now.” The other person sees the shadow on the sundial, and then imagines the position of the shadow in two sun-lengths time. That person has just done addition. Mathematics started with numbers.

Lastly, I have another confession to make. My knowledge of the history of mathematics is very and utterly limited to pop culture. Another embarrassing fact about the math teacher to-be. When asked to think of five top moments or milestones in mathematics, I could barely think of five at all. The first one I thought of was Albert Einstein’s E=mc2. My knowledge of that equation is very lacking. Do not ask me to explain it or tell you what it is for. My embarrassment is bad enough.

The second milestone was the discovery of the number Zero, since it took civilizations a long time to create a number to describe nothing. A student in my group had also mentioned this when we were discussing the “What is mathematics?” question, and I remembered hearing that fact in high school when I learned about how certain civilizations influenced others when they traded or were conquered in war. Again, don’t ask me which civilization came up with the concept of Zero first.

The third milestone was Sir Isaac Newton discovering or creating Calculus. The fourth was Alan Turing’s use of mathematical theory (I’m assuming from the movie) to create a computer during World War II - thank you very much, The Imitation Game! The fifth…didn’t the first woman receive the Nobel Prize in mathematics for something last year? I remember reading an article about it. The math portion was completely over my head.  

Luckily, someone acknowledging they have a problem is the first step towards a solution, and my remedy to fix my lack of knowledge is to read Clifford Pickover’s The Math Book. According to, the book gives brief explanations to many historical moments in mathematics. Just the book I need.