Friday, December 29, 2006

ThE ZERO EFFEcT




Project Zero's
The mission is to understand and enhance learning, thinking, and creativity in the arts, as well as humanistic and scientific disciplines, at the individual and institutional levels.
The introduction of zero into the decimal system in the 13th century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. Without the notion of zero, the descriptive and prescriptive modeling processes in commerce, astronomy, physics, chemistry, and industry would have been unthinkable. The lack of such a symbol is one of the serious drawbacks in the Roman numeral system. In addition, the Roman numeral system is difficult to use in any arithmetic operations, such as multiplication. The purpose of this site is to raise students and teachers awareness of issues in working with zero and other numbers. Imprecise mathematical thinking is by no means unknown; however, we need to think more clearly if we are to keep out of confusions.

Zero in Four Dimensions:
Cultural, Historical, Mathematical, and Psychological Perspectives

The notion of zero was introduced to Europe in the Middle Ages by Leonardo Fibonacci who translated from Arabic the work of the Persian (from Usbekestan province) scholar Abu Ja'far Muhammad ibn (al)-Khwarizmi (the word "algorithm," Medieval Latin 'algorismus', is a contamination of his name and the Greek word arithmos, meaning "number,: has come to represent any iterative, step-by-step procedure) who in turn documented (in Arabic, in the 7th century) the original work of the Hindu mathematician Ma-hávíral as a superior mathematical construction compared with the then prevalent Roman numerals which do not contain the concept of zero. When these scholarly treatises were being translated by European accountants, they translated 1, 2, 3, ....; upon reaching zero, they pronounced, "empty", Nothing! The scribe asked what to write and was instructed to draw an empty hole, thus introducing the present notation for zero. Hindu and early Muslim mathematicians were using a heavy dot to mark zero's place in calculations. Perhaps we would not be tempted to divide by zero if we also express the zero as a dot rather that the 0 character.


You might ask then how did the Romans do calculations with their numerals notations? Romans typically relied on the Chinese abacus, their version of our modern calculator, visit, e.g., Ancient Civilizations Web site. By using pebbles as counters, there was no need to use Roman numerals. People known as "calculatores" (after "calcule", Latin for "pebbles"), did the math used to tally totals in addition, subtraction, division and multiplication. For us using Roman numerals system to perform arithmetic operations such as division, or multiplication are very difficult if not impossible. In modern days they are used for decorative purposes only.
Zero as a concept, was derived, perhaps from the concept of a void. The concept of void existed in Hindu philosophy and the Buddhist concept of Nirvana, that is: attaining salvation by merging into the void of eternity. Ma-hávíral (born, around 850 BC) was a Hindu mathematician, unfortunately, not much is known about him. As pointed out by George Wilhelm Friedrich Hegel, "India, such a vast country, has no documented history." In the West, the concept of void and nothingness appeared first in the works of Arthur Schopenhauer during the 19th century, although zero as a number has been adapted much earlier.


The Arabic writing mathematicians not only developed decimal notation, they also gave irrational numbers, such as square root of 2, equal rights in the realm of Number. And they developed the language, though not yet the notation, of algebra. One of the influential persons in both areas was Omar Khayyam, known in the west more as a poet. I consider that an important point; too many people still believe that mathematicians have to be dry and uninteresting.
Initially, there was some resistance to accepting this significant modification to the time-honored Roman numerical notation, in particular from the privileged job-secured Roman numerical calculation experts: The Tax Gatherers:


Among the trite objections to leaving Roman numerals for the new notation was the difficulty in distinguishing between the numeral 1 and 7. The solution, still employed in Europe, was to use a cross-hatch to distinguish the numeral 7.
The introduction of the new system indisputably marked the democratization of mathematical computation by its simplicity and lack of mystery. Up to then the "abacus" was the champion. Abacus was a favorite tool for a few and praised by Socrates. The Greek's emphasis on geometry (i.e., measuring the land for agricultural purposes, the earth, thus the world geography) so kept them from perfecting number notation system. They simply had no use for zero.
Greeks were not too much interested in arithmetic, believing in inherited nobility of a few, the Greeks had the adage "that arithmetic should be taught in democracies, for it teaches relations of equality, but that geometry alone should be reserved for oligarchies, as it demonstrates the proportions within inequality."

To ask, is the sum of the parts greater, lesser or equal to the sum of the whole? One thought would be to eliminate zero! As in the "Reverse Polish Notation" which eliminates the need of parentheses.

Sacrilegious as it may sound on first impression, the notation of zero is at heart nothing more than a directional separator as in the case of a thermometer. It is, in actuality, "not there." For example, in order to express the number 206, a symbol is needed to show that there are no tens. The digit 0 serves this purpose. Zero became a part of the Natural Numbers System in the last century when Giuseppe Peano puts it in his first of five axioms for his number theory.
One may think of an analogy. Zero is similar to the "color" black, which is not a color at all. It is the absence of color, while the Sun Light contains all the colors.

Zero is the only digit which cannot stand alone. It is a lonely number, lonelier than one. It requires some sort of companionship to give meaning to its life. It can go on the left. On the right. Or both ways! Or in the middle as part of a threesome. Witness "01", "10", or "102". Even "1000". A relationship with other numbers gives it meaning (i.e. it is a dependent number). By itself it is nothing!

When we write 10, we mean 1 ten and 0 ones. In some number systems, it would be redundant to mention the 0 ones, because zero means there are no objects there. Place value uses relative positions. So an understanding of the role of 0 as marking that a particular ‘place' is empty is essential, as is its role of maintaining the ‘place' of the other digits. The usage of zero here is more of a qualitative than quantitative. Therefore, it is called an operational zero.

In many languages you come across expressions which refer to "red numbers" and "black numbers" to denote negative and positive ones. For example, in the Ancient China the two colors were used in the arithmetic meaning, but in the opposite way on their counting rods. They were associated with Yin and Yang, the principal forces of the Tao cosmology. The use of colors, elsewhere was simply a convention by accountants: red ink to indicate losses, black ink for profits.

Natural numbers are positive integer numbers. One horse, two trees, etc. However, the arrival of zero caused the inevitable rise of the even more nefarious numbers: The negative numbers.
What about negative numbers? The negative sign is an extension of the number system used to indicate directionality. Zero must be distinguished from nothing. Zero belongs to the integer set of numbers. Zero is neither positive nor negative but psychologically it is negative. The concept of zero represents "something" that is "not there," while zero as a number represents the lowest of all non-negative numbers. For example, if a person has no account in a bank, his/her account is nothing (not there). If he/she has an account, he/she may have an account-balance of zero.

If one defines evenness or oddness on the integers (either positive or all), then zero seems to be taken to be even; and if one only defines evenness and oddness on the natural numbers, then zero seems to be neither. This dilemma is caused by the fact that the concepts of even and oddness predated zero and the negative integers. The problem posed by this question is that zero is not to be really a number not that it is even or odd.

Most modern textbooks apply concepts such as "even" only to "natural numbers," in connection with primes and factoring. By "natural numbers" they mean positive integers, not including zero. Those who work in foundations of mathematics, though, consider zero a natural number, and for them the integers are whole numbers. From that point of view, the question whether zero is even just does not arise, except by extension.

Judging from the treatment accorded to the concept of zero, we do practice a variety of avoidance mechanisms rather than confront the imagery associated with this seemingly difficult concept.
In reciting one's telephone number, social security number, postal zip code or post office box, room number, street number or any of a variety of other numeric nominals, we carefully avoid pronouncing the digit "zero" and instead substitute "oh." One may say "it is caused by our desire to communicate quickly, if we can say the same thing in one syllable, why not?" What about number seven, should we find a substitute for this too?

In some parts of the world, the phrasing "naught" and "aught" are used but it is quite uncommon to hear "zero." All the other digits are correctly enunciated with this one curious exception. However, in the US Army there is an additional curious habit of saying "duece" instead of "two". For example, the M102 105mm Artillery Cannon is called a "One oh Duece" (notice the "oh" therein).

Today zero has a meaning not just of a number, but as the bottom, or failure. He made no baskets, or, he made zero baskets -- meaning he failed to score. Or he gave zero assistance.
If you are familiar with numerology, you notice that there is no zero to work with in the numbers that correlate with the alphabet, strange? Not at all. The absence of zero may suggest that the Pythagorean who first developed the duality between numbers and letters were not aware of the zero notion. The notion of zero is much younger.
In tennis scores, zero is called "love," because zero looks like an egg, the French called it "l'oeuf," which is French for "egg." You may have also noticed the weird numbering in the tennis scores which goes back to medieval numerology, in which 60 was considered a "complete" number (much like 100 is considered a nice round number today). Back in medieval times, tennis's four points were 15, 30, 45 (later abbreviated to 40), and 60, or game.
On the telephone keypad, zero has the honor of representing the operator. There is no zero in most games, such as plying cards (after all who wants to win zero!). Zero is placed at the end of the keypad on the computer and at the bottom of the keypad on the telephone. Is zero the beginning or the end? Notice that on a calculator's keypad the numbers starts with the largest numbers on the top and work their way down to zero. What about the o and 0 being right next to each other on the PC keyboard? Numbers are located three places. First it is located on the keyboard keys with the range 1, 2,...,0; this is the same order that phone keypad. Second, on the right of the keyboard is a calculator-like pad where zero is the last listed number. Finally, there is a list of functions key, however there is no F0 because that could translate into no function and what would be the point of having a key "without" function. There will always be questions about the true meaning and function of zero. Is it the end or the beginning? What does ground zero mean? Some use it as starting point; the military uses it as an ending point.
The resistance against zero can be noted even at the architectural level in buildings where the ground-level is rarely denoted as the zeroth-level as it should be. However, for mathematicians it comes easily to label the floors of a building to include zero, for example, the Department of Mathematics' building at the University of Zagreb in Croatia has floors numbered as -1, 0, 1, 2, and 3. In fact, this is not a particularity of one building but a common practice in modern buildings in large cities such as Buenos Aires. In most European countries the floors are always numbered starting from 0. We do have a special word to say 'ground floor' in a conversation, not using 0, but the elevators will always offer you a "0 button" for the ground floor


Is the presence of nothing (reflecting non-existence) different from the absence of something (reflecting non-availability) or the absence of anything (reflecting non-existence)? Zero is a symbol for "not there" which is different from "nothing" "Not there" reflects that the number or item(s) exists but they are not just available. "Nothing" reflects nonexistence.

There is also "the Zero Factor about the US Presidents" known as the Zero Factor and Tecumseh's Curse which is the curse of Indian chief Tecumseh which has Killed every U.S. President before the end of their term in office, if they were elected in a year that ended with 0. The first victim of the curse was William Henry Harrison, whose troops killed the Indian chief in 1813 (the zero factor has one exception, i.e., Ronald Reagan who was elected in 1980).

Zero not only has the quality of being nothing, it is also a noun, verb, adverb, and an adjective as in "zero possibility". "We zeroed in on the cause," means we had isolated all the possibilities, and have discovered the one remaining. In this use as a verb, zero equals one. However, "The result was a big, fat, zero," uses the noun to express the idea of results of "nothing". Here, zero has the quality of not being there. Zero as an action appears in the Conservative Laws of physics.

The term "zeroing in on (whatever)" might have originated also with the military. The "zero" in this term might refer to the distance from the last bomb dropped or the last shell fired to some target. The aim is always to try in reducing this distance to zero.
On a roulette wheel, there is the number Zero which is neither Red nor Black. Zero is the GREEN number, for all the cash the house rakes in when it comes up. It is considered neither Even nor Odd.

While zero is a concept and a number, Infinity is not a number; it is the name for a concept. Infinity cannot be considered as a number since since it does not follow numbers' properties. For example, (infinity + 2) is not more than infinity. Since infinite is the opposite of finite, therefore whoever uses "infinite" must first give an indication for what is finite. For example, in the use of statistical tables, such as t-table, almost all textbooks denote symbol of infinity (¥) for the parameter of any t-distribution with values greater than 120. I share Cantor's view that "....in principle only finite numbers ought to be admitted as actual."

It may be considered frivolous hyperbole to suggest that the demise of the Roman Empire was due to the absence of zero in its number system, but one can only ponder the fate of our civilization given the difficulty our culture seems to have with the presence of zero in our number system.

The notion of zero brings another wearying and yet intriguing questions: Is our current century the 20th century or the 21st century? According to the Holy Scriptures (see, Matthew chapter 2), King Herod was alive when Jesus was born, and Herod died in 4 BC. Does that mean the millennium actually started in 1996?

As you may know, our calendar was improperly set up by a monk in about 525 AD. Since the zero's concept was not available yet, he began the calendar with year 1. Anno domini means "in the year of our lord"; but by starting at 1, the calendar does not correctly reflect the verbal statement. As year one begins, Christ is just born. As year two begins Christ is one year old. The second Century begins with 101 and the second millennium begins with 2002. Still there is a confusion when referring to any particular year in any century. For example, the American Independent achieved in 18th century, but we refer to it as 1776, "as if" it occurred in duting 17th century.

Ordinal numbers, which the Gregorian calendar uses, indicate sequence. Thus "A.D. 1" (or the first year A.D.) refers to the year that begins at the zero point and ends one year later. Think of a carpenter's ruler, if you will; the first inch is the interval between the edge and the one-inch mark. Thus, e.g., the millennium ended with the passing of the two-thousandth year, not with its inception. Cardinal numbers, which astronomers use in their calculations, indicate quantity.
Zero is a cardinal number and indicates a value; it does not name an interval. Thus "zero" indicates the division between B.C. and A.D., not the interval of the first year before or after this point. Continuing with our example, put two rulers end to end: although there is a zero point, there is no "zero'th" inch.
As it stands now, we refer to years with ordinal numbers and to ages with cardinal numbers. Thus a child less than a year old is usually said to be so many weeks or months old, rather than "zero years old." If we changed over to this system for our calendar (referring to the age of our era, rather than to the order of the year), then there would be "zero years" for both A.D. and B.C.! That is to say, the last twelve months before the birth of Christ and the first twelve months after the birth of Christ would be the years 0 B.C. and A.D. 0 respectively. For more on this, you may like also to visit the Web site Zero.

The main confusion is between the notions of "time window length" and a "point in time". There is an interval between 0 and 1. Considering whether the millennium starts in 2000 or 2001, depends on whether you look at a number as a points on time or a time interval. Years are intervals; numbers are points. Therefore, it is always a mistake to treat years as points. For example, consider the old arithmetic question: John was born in 1985 and Jane in 1986. How much older is John than Jane? The answer, of course, can be anywhere from a few seconds to two years, depending on when in those intervals the two people were born.

This is quite revealing of the cultural predilections of the time when the calendar was reorganized, first under the Julian scheme undertaken under the auspices of the Roman Emperor, Julius Caesar, after whom the month of July was named, and subsequently under the Gregorian calendar currently in use, which was devised during the reign of Pope Gregory. What is quietly yet magnificently revealed by this now-curious omission is the absence of the notion of zero in the numbering systems then in use. When the notion of zero was subsequently introduced in the west in the Middle Ages, it could hardly have been regarded as feasible to rewrite the entire calendar, if the debate occurred in the first place. Clearly then, our ideas about numbers permeate our culture.
How zero relates to time of day? For instance, what time would be at 12:30 if not for zero.

Continuous data come in the forms of Interval or Ratio measurements. The zero point in an Interval scale is arbitrary. The different scales for measuring temperature all have a zero, yet each has a different value! For example, on a Celsius thermometer, zero is set at the temperature at which pure water freezes at the sea level altitude. While zero degrees Fahrenheit is 32 ° degrees below freezing, and finally absolute zero is the theoretical point at which molecular movement ceases. Therefore, since the absolute temperature can be created in the laboratory, it is only a concept. So, here one must accept that the meaning of zero is relative to its context. Now the question is: does 80 ° degrees Fahrenheit temperature implies it's twice as hot as when it's 40 ° degrees? The answer is a No.

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