This is my geometry term paper about the number zero.
When asked to count to ten, a person would usually sound like this: one, two, three, four... et cetera; the chances are very unlikely that a person would start with zero, which is not considered a counting number. Zero is usually seen as just another number, and that a story of zero would simply have the length and value of the number itself. Despite what one would think, this one numeral was invented, or discovered, as some would consider it, and has an extensive history of its several names and the common "0" symbol. The well-disputed properties of it are unlike any other number. This is mainly because the concept of nothing as a number is hard for most humans to be able to understand. After all, can nothing be given a name and accepted as something?
History of the Modern Number Zero
The Egyptians used different hieroglyphs about 3500 B.C. to represent numbers using a decimal system. There were glyphs to represent 1, 10, 10^2, 10^3, 10^4, 10^5, and 10^6. These glyphs were written in descending order additively to show different numbers. If a category of numbers was missing, as in the number 207, this was easily visible by the glyphs that were used for the 2 and 7. The 2 would be shown with the symbol for 10^2, and the 7 with glyphs for 1. This was a very hard way to write numbers, as they would become very long, and the amount of numbers that could be written were limited to nine million. Though, at the time, Egyptians had no use for numbers as large as that. (Gullberg, p.34) The Egyptians did use a form of zero for the reference point during construction guidelines and as the answer to a number subtracted from itself. (Origin of a Formal Fallacy...)
The Sumerians, from around 3200 B.C., used a decimal system for everyday counting and a sexagesimal system, base 60, for astronomical calculations. Both did not include a number for zero. There were symbols for numbers 1 - 9, and 10 - 90 by tens, 60^2, 10x60^2, and 60^3. A group of symbols would signify multiplication. A subtraction symbol was sometimes used to make it simpler to show long numbers. This system used many different symbols for numbers, and had a limit of numbers that could be named. (Gullberg, p.35)
When the Babylonians came to power around 2000 B.C., their sexagesimal system became the most commonly used. This was the first counting system to use place value. Because there was no zero, differentiating 6001 from 61 or from 6100 was very confusing to read, and often a blank space was left. Around 4 B.C., a symbol came into use to show a void that looked like a triangle with a long tail. This symbol acted as a placeholder, like the modern zero, but it was not considered a number. (Gullberg, pp.56 - 57)
In Greek mathematics, as in Roman, there were words to show the absence of all numbers (nothingness). The Greeks and Romans used a decimal counting system too, and used the 24 letters with special notation to show numbers. The first ten letters were the first ten numbers, the eleventh letter was the number 10, the twelfth letter was 20, and so on. With one myriad totaling 10,000, larger numbers were sometimes shown as myriads of myriads. (Kaplan, pp.17 - 19, p.31)
Today the sort of counting system we use is the base ten system, or the decimal system. Zero is used as a starting number and as a placeholder. As a placeholder it serves after our number nine is used, the numbers start over again at one, in the "tens" position, with a zero after it to show that there are no "ones." Without a digit there, if a space was used instead, fast calculations would become very difficult and mistakes would be commonplace. It is used in this manner to show the "absence of countable or measurable magnitude whose precise nature is determined by the context;" or in other words, zero is an adjective that is used to show none of the noun is there. (Black, p.770)
History of the Name and Symbol for Zero
The Hindus are most credited to the invention of the symbol 0 and the true usage of positional notation. This is because they have well documented use of it just like a real number. In 876 A.D., the number 270 was written as 27° on a stone tablet that was for an order of flowers for a temple of Vishnu. The symbol was likely to have been used long before this, but the real question is why the empty circle symbol was used. Some consider this a Greek discovery, because on a drawing of a counting table an O was used where the 0 would be. This was most likely because the symbol looked like the first letter of the Boetian alphabet, the "Obol," which also was a coin that was considered to be worth almost nothing. (Kaplan, pp.23, 31)
There is no real evidence of where exactly this symbol could have come from first, but there are many different ideas of where it might have originated. In Sanskrit, the ° symbol was used to show a word or letter being omitted, like an apostrophe. The counting boards that were used by both the Greeks and the Indians were dusted with sand, most likely to catch errors in calculations, and the depressions left by markers in the sand resembled an empty circle. Also, the pebble counters that were used on these boards looked like dots, so the absence of one could have been an empty dot. (Kaplan, pp.23, 24, 43, 48, 50)
The names for zero has many different possible origins, most derived from Hindu words like sunya, meaning empty, and kha, once used in a book for the word "place" in place value (empty value). The Arab merchants that often used Indian math used the Indian sunya but it evolved to sifr and as-sifr. By the time this name had gotten to Venice, it had evolved into "zero." (Kaplan, pp.43, 44, 93)
Zero in Algebra
More important than the name for zero or its origin are the properties that sets zero apart all other numbers. Zero is often considered the identity of numbers because of the Law of Addition. Similar are its properties with multiplication. Dividing by zero is cause for questionably the most common math question. What may be even worse, is zero in exponential value.
The Law of Addition states that any real number added to zero is itself. Any real number subtracted from zero is the opposite of itself. ("Numerals," Microsoft Encarta Encyclopedia 2000) Because the original number will repeat itself in this way, the laws of addition are very similar to the Identity Property, also called the Reflexive Property of Equality. The Reflexive Property of Equality states that for any real number a, a = a. Using the Transitive Property of Equality, for any real number a, a + 0 = a + 0. This is why zero is often considered the identity of numbers.
By the Laws of Multiplication, any real number multiplied by zero equals zero. ("Numerals," Microsoft Encarta Encyclopedia 2000) Zero is the only real number in which everything multiplied by it equals the same thing. Multiplication is seen as taking a number and putting it into a certain number of groups, for example: if there were three bags with four apples in each, how many apples are in all the bags added together? (12). If there were five bags, with no apples in each one, how many apples are in all the bags added together? If there were no bags, and there were five apples sitting where the bags would be, how many apples are in the bags? The answer to both of these questions is no apples, or zero. So anything multiplied by zero ends up with nothing in those groups, or with an answer of zero.
Division, like multiplication, is also best described by groups. If there were 18 bananas, and you put them into 3 boxes, how many bananas would be in each box? There would be six. If there were no bananas and you put nothing into 3 boxes, how many bananas would be in each box? The answer would be no bananas, so this shows how zero divided by anything equals zero. What if there were 18 bananas; how many bananas would be in each box if there were no boxes? If the boxes were there, could we tell how many bananas would be in them if we don't know the total number of boxes? It is most commonly considered "undefined," because we don't know enough information to say how to divide the bananas up. A better way to look at this problem is by using an example from division's cousin, multiplication. 10/2=5 because 5x2=10, 9/3=3 because 3x3=9, but 4/0=?? Nothing times zero can equal four, because everything times zero equals zero. (Dr. Math FAQ...)
If this is true, then isn't 0/0 undefined? But also, any number divided by itself is one. For example, if there were nine ducks and you put them into nine boxes, that's one duck in each box. But if there were no ducks and you didn't put them into any boxes, then there would be nothing that you didn't put into anything. Isn't that just zero? Because there are too many questions about this function also, it too is "undefined."
Zero has caused many fears and confusion, especially during the Middle Ages because it was thought of as almost satanic. Zero is associated with darkness and nothingness and pretty much evil in