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This paper was written as an assignment for Ian Walton's Math G -Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

To explore other such papers go to theMath G Projects Page.

This paper was submitted by Jeanette Turkus for her final in Spring 2000 Math G at Mission College.

If you use material from this paper, please acknowledge it.

In the history of culture the discovery of zero will always stand out as one of the greatest achievments of the human race. -Tobias Danzig; Number: The Language of Science.

As I began my research on zero, the only number which can be divided by every other number, and the only number which can divide no other number, I was both fascinated and surprised to read that early in history, zero was feared, misunderstood and rejected! However, by the time I reached the end of my research I was even more surprised with my ironic conclusion .

To begin with, zero is the digit many of us do not commonly think of as using as a number and so I have always taken the familiar zero for granted as meaning nothing, none, zip! In the following quote by Alfred North Whitehead, he both confirms my thinking and points to my fascination of what I have been learning about zero. Whitehead said, "The point about zero is that we do not need to use it in operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought."

But how and why could there be such extreme negative reaction in early history over a little zero? Well, my thoughts about zero have dramatically expanded to now seeing zero as much more than nothing.

For instance, zero is actually often a starting point or place, as for example, in a race, the starting point is often called the zero part of the race. Also, score keeping in sports, both teams start with zero. Zero is also the starting point of any measure of weight, like a scale when weighing yourself, or measurement with a ruler, even thought most rulers do not show zero, it is taken for granted (again). Looking at zero this way makes sense to me remembering that zero is the first in the sequence of whole numbers. Yet, among the whole numbers, or also called real numbers, zero is unique because it is neither negative or positive. Furthermore, there is a difference between the set of zero and the other sets. All the numbers represent an infinite number of sets except zero, which represents only one -- the empty set.

Of course, besides all this, an important role zero plays is as a place holder in the column of the place value system: ones, tens, hundreds, thousands, etc. This important function, holding of a place in a column, brings me all the way back to the origin of zero.

Some say, early in the Christian era, in India, an unknown Hindu put down a symbol he invented to represent an empty column on his counting board called Sunja: meaning empty. Others say the zero was first discovered by the Babylonians in Iraq, others say the Mayan Indians, and in "Math G" it was mentioned some claim the Orient was where the zero was first discovered. In any case, one or possibly more than one of these ancient cultures did develop zero and many of these great old civilizations used some type of counting board that was called by different names. For instance, the Germans called it Rechenbank, and that is why we call money lenders banks. In some other parts of the world the counting boards were known an Abacus. These counting boards were basically a frame, divided into parallel columns, each column having a value of a power of ten, the number of times a particular power occurred in a total being represented by markers of some sort, usually various kinds of beads. The beads all stood for one unit and the value of the unit varied depending on the column. A bead in the first column would have a value of one. A bead in the second column had a value of ten; in the third column a value of one hundred and so on. (Modern Positional Rotation is the notation of counting boards made permanent.) Besides these counting boards, ancient cultures used many other devices to count. For example, body counting, by using body parts arranged in a certain order. Even the Romans counted with fingers, and finger counting was commonly used up through the middle ages. The Egyptians used pictograph symbols or pictures. Another early counting technique was the idea of tallying or matching the collection to be counted with a set of objects like shells or stones. Another counting style was counting by scratching notches in a bone or tying knots in cords. The Incas in Peru used string with knots and loops with varied colors. But the oldest is reported to be the Tally stick and fingers. It is thought the advancement from finger counting probably came from the expansion of the Roman Empire. The Barbarian tribes speaking different languages, needing to trade with merchants as Rome expanded. . Further the Arabs may have brought the zero from India when they came to trade with Italian and German merchants. This is where the Abacus (or counting Board) was useful. This new place holder, the zero, importantly cleared up what position a number was in.

Originally there was strong resistance to Zero. Sometimes the resistance took the form in ridicule like... "just as the donkey....wanted to be a lion....the Cifra (or Sifr called in Arabic) puts on airs and pretended to be a digit."

On the other hand, The Salem Monastery astrologers accepted the new numeral -- but had their own twist on it. One interpreted zero as, "the great sacred mystery; the zero neither increases or diminishes another number it is added to or from which it is subtracted. So does HE neither wax nor wane....Nay, to speak more correctly, HE creates all out of nothing...." In this way zero acquired some profound significance and began to represent something. Yet initially most were not comfortable with zero and therefore did not accept it. Although zero finally came to be adopted in the middle ages it was not because of realizing the place value notation, but because Indian numbers and zero were seen as abbreviations on the counting board. The place value notation was later gradually understood. However, it was the merchants use through trade, not the mathematicians, scholars or scientists, that advanced this new movement.

In fact, the Greek scholars and legendary thinkers of Greece, quite to the contrary, had a huge impact that held back the advancement of zero. This brings me to the fear, misunderstanding, and rejection discussion in connection with zero. For example, zero is said to have disrupted philosophy, science, math and religion. In fact the Greeks saw zero as dangerous! Because for them it was linked with the void and nothingness and disrupted their beliefs and that is why it was dangerous. Their ancient culture thought the world was chaotic and void before God added light and creatures. Further, adding to the rejection about zero was the way zero did not behave like other numbers. They believed that when two numbers are added together they get bigger, as one plus one equals two, but zero plus zero equals zero. Obviously, in subtraction the reverse is true. Zero did not get bigger contrary to their reasoning. The bottom line to all this was zero was in opposition with the Western philosophy of that time and therefore seen as dangerous and was rejected. Greeks believed there was no void. God had cured that, and there was no such thing as nothing.

Pythagoras, a renowned scholar and teacher of Philosophy thought, "all is numbers" and every number’s shape had a hidden meaning and the most beautiful shapes were sacred. For example, the mystical pentagram, the five pointed star, was a special one. When connecting the corners of that pentagram with lines it creates a small up side-down five pointed star which is exactly the same as the original star in its proportions. This star contains an even smaller pentagram which contains a tinier star inside and so on. (see figure 1) They thought the most important thing about this was hidden within the lines because they contained a number shape which was the ultimate symbol of the "Pythagorean View" of the universe: the Golden Ratio. In ancient Greece Pythagoras was remembered for inventing the musical scale (not the square of the Hypotenuse of a right triangle is equal to the sum of the squares of the other two sides)

Pythagoras, thought playing music was a mathematical act. Thus thereby concluding ratios govern music and other types of beauty. Further, the planets moving made a "harmony of spheres" and, simply put that is the "all is numbers thinking." Furthermore, the Golden Ratio came from dividing a line in a certain way: divide a shape so that the ratio of the small part to the large part is the same as the ratio of the large part to the whole, a rectangle. Even today objects with this ratio are still thought of as most pleasing in art and architecture.(see figure 2) The pentagram is full of the Golden Ratio and this was the symbol of the Pythagorean Brotherhood. Therefore the conclusion of the Pythagorean mind was ratios controlled the universe. Importantly, this supernatural link between ratios and the universe became a strong belief in western civilization and thought. However, the zero did not fit here at all. I f "all is numbers", then something with no height or width was a problem and did not fit in. There could be no rectangle with zero height or width. Therefore the Greeks chose to totally reject the zero, because zero and a ratio could not mix and no longer would proportion be a relationship between two objects.

Also, another concept the Greeks originally rejected was the irrational. They wanted the universe to stay governed by ratios and everything making sense by being related and proportioned and rational. Problem: a square. The diagonal of a square is irrational

( nowadays recognized as a square root of two). Then it was discovered the Golden rule, the symbol of beauty and rationality, was an irrational number. To keep things from ruining the Pythagorean belief it was kept secret. Of course eventually this secret would become known because irrationals occur in so many geometrical constructions. However, the Greeks tried to keep zero out of the picture for as long as they could.

Moreover, another great Greek mathematician and thinker was Archimedes who knew that a cone cut -up creates circles, ellipses, parabolas and hyperbolas, depending how cut. Further, he knew taking the parabola, it could take the light from the sun and focus to a point, the lights energy focused on a very small area. This lead Archimedes to study the parabola and with it infinity, by inscribing a triangle in the parabola. Then , in the gaps ,adding more triangles, repeating this, adding more triangles, in other words- an infinite series . Each getting smaller and smaller with the areas of the little triangles approaching zero. (see figure 3) This was revealing something different from Archimedes teachings that any number added to itself over and over can exceed any other number. Of course zero did not work-out with this thinking. Archimedes got close, with his triangles, to the concept of limits and calculus. Although he went that far, Archimedes still did not accept zero, which is a connection to the finite and the infinite and needed for calculus. Archimedes believed in the concept that the universe was contained within a huge sphere. (see figure 4)

In contrast to this Western avoidance of the void, in the East, in India and in parts of

Arabia, it was accepted and flourished. Beginning as a place-holder and from there growing. Many of the Eastern religions, such as Hinduism were steeped in duality. Yin and Yang and Shiva - both creator and destroyer. The God Shiva represented nothingness, a void. Because of this India was comfortable with the void, the infinite and accepting zero. They, in contrast to the Greeks, didn’t pursue squares in square numbers or areas of rectangles . On the other hand, they saw the interplay of numerals without the geometric importance. Because the Indians did not put so much emphasis on numbers geometric importance like the Greeks, Indians did not concern themselves about their operations making geometric sense. Negative numbers would not make sense to Greeks thinking in geometric terms. However, in India and China negative numbers made sense Just as 2-3 was now a number, so was 2 - 2. It was zero. No longer a place holder only, but a number. Now zero had a place on the number line, a number line that without zero, could not exist- so it was in the East where negative numbers first appeared.

Although, zero and infinity were present during the Italian Renaissance in art with perspective and the vanishing point. Here the vanishing point is a connection to zero and infinity. The point is an infinitesimal dot on the artwork that represents a spot infinitely far from the viewer and this zero of the art contains an infinity of space. Zero and infinity are linked to the vanishing point - just as multiplying by zero causes the number line to collapse- the vanishing point causes some of the universe to be on the dot of the vanishing point.

On that note of the vanishing point, fast forwarding to modern times and Albert Einstein. Einstein said if a group of people watch the same phenomenon such as a dog running to it’s owner, the law of physics are the same for everyone in the group. But the situation would be different if some of the people in the group were on a parallel moving train and some on the ground. The group would not agree on the speed of the dog. Yet, all would agree that the dog met up with its master, even if the details varied. Further, Einstein thought the speed of light in a vacuum was about three hundred million meters per second, a constant denoted by the letter C . For example, a flashlight sheds light at a speed of C whether the person holding the light is still or moving back and forth. This all leads to Einstein concluding the flow of time changes depending on the observers speed. Not only time changes but length and mass too. As objects speed up they get shorter and heavier. The speed of light is the ultimate speed that can not be reached. Seemingly nature avoiding zero, but it turns out -not so. Einstein extended the theory of relativity to include gravity and with it lead to describing zero and infinity and in that connection, led to the black hole.

The black hole very simply put, is dying stars getting smaller and smaller, then zero. The star sort of pushes itself into zero space. Part of the strangeness of the black holes are the way they curve space-time, taking no space, but having mass and with mass causing space-time to curve. The black hole is a point taking up zero space with no outer edge when space flattens out. The curve of space gets greater closer to a black hole and the curve goes to infinity because of the zero space. The black hole is a division by zero.

Moving forward in time we now see zero used all over the place. For instance, modern studies about absolute zero have concluded that absolute zero could never be reached, which has helped a new branch of physics called thermodynamics, which is the study of heat and energy behavior. In which there are laws that say it is impossible to create a machine that forever generates power without any source of energy.

Again, zero shows up now called the zero point energy- a zero in quantum mechanics means the entire universe and the vacuum, is filled with an infinite amount of energy. All this leads to what is called the phantom force of nothing. Quite frankly, for me this sounds a lot like a sequel to a Star Wars movie!

However, on a much more serious note, in his book, " Zero, the Biography of a Dangerous Idea" , the author Charles Sief claims "zero is so powerful because it unhinges the laws of physics...not only does zero hold the secret to our existence, it will also be responsible for the end of the universe." Sief further discusses the end of the universe may be by external expansion, cooling and heat death, or the cosmos expanding forever and ending in cold. This opinion of Sief’s makes me even more fascinated with zero and its history of rejection by the great Greek minds of the past. Now I find Sief’s forecast of zero being responsible for the end of the universe ironic because it puts new life into Zero’s early dangerous reputation.

In closing, I find the following quote quite poetic and appropriate;

"What is man in nature? Nothing in relation to the infinite,

everything in relation to nothing, a mean between nothing

and everything,"

-Blaise Pascal, Pensees.

REFERENCES

"O Is it something?" by Claudia Zaslasky

" From O to Infinity". by . Constance Reid

"Burton’s History of Mathematics". by D. M. Burton

"Zero is not Nothing." by M. & H. Sitomer

"Number Words & Number Symbols". by K. Menninger

"Zero, the Biography of a Dangerous Idea". by Charles Seif