Zero
Zero
The idea of nothingness and emptiness has inspired and puzzled mathematicians, physicists, and even philosophers. What does empty space mean? If the space is empty, does it have any physical meaning or purpose?
From the mathematical point of view, the concept of zero has eluded humans for a very long time. In his book, The Nothing That Is, author Robert Kaplan writes, "Zero's path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West." But our own familiarity with zero makes it difficult to imagine a time when the concept of zero did not exist. When the last pancake is devoured and the plate is empty, there are zero pancakes left. This simple example illustrates the connection between counting and zero.
Counting is a universal human activity. Many ancient cultures, such as the Sumerians, Indians, Chinese, Egyptians, Romans, and Greeks, developed different symbols and rules for counting. But the concept of zero did not appear in number systems for a long time; and even then, the Roman number system had no symbol for zero. Sometime between the sixth and third centuries b.c.e., zero made its appearance in the Sumerian number system as a slanted double wedge.
To appreciate the significance of zero in counting, compare the decimal and Roman number system. In the decimal system, all numbers are composed of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. After counting to nine, the digits are repeated in different sequences so that any number can be written with just ten digits. Also, the position of the number indicates the value of the number. For example, in 407, 4 stands for four hundreds, 0 stands for no tens, and 7 stands for seven.
The Roman number system consists of the following few basic symbols: I for 1, V for 5, and X for 10. Here are some examples of numbers written with Roman numerals.
IV = 4 XV = 15
VIII = 8 XX = 20
XIII = 13 XXX = 30
Without a symbol for zero, it becomes very awkward to write large numbers. For 50, instead of writing five Xs, the Roman system has a new symbol, L.
Performing a simple addition, such as 33 + 22, in both number systems further shows the efficiency of the decimal system. In the decimal number system, the two numbers are aligned right on top of each other and the corresponding digits are added.
In the Roman number system, the same problem is expressed as XXXIII + XXII, and the answer is expressed as LV. Placing the two Roman numbers on top of each other does not give the digits LV, and therefore when adding, it is easier to find the sum with the decimal system.
Properties of Zero
All real numbers, except 0, are either positive (x > 0) or negative (x < 0). But 0 is neither positive nor negative. Zero has many unique and curious properties, listed below.
Additive Identity: Adding 0 to any number x equals x. That is, x + 0 = x. Zero is called the additive identity.
Multiplication property: Multiplying any number b by 0 gives 0. That is, b × 0 = 0. Therefore, the square of 0 is equal to zero (0^{2} = 0).
Exponent property: Any number other than zero raised to the power 0 equals 1. That is, b ^{0} = 1.
Division property: A number cannot be divided by 0. Consider the problem 12/0 = x. This means that 0 × x must be equal to 12. No value of x will make 0 × x = 12. Therefore, division by 0 is undefined.
Undefined Division
Because division by 0 is undefined, many functions in which the denominator becomes 0 are not defined at certain points in their domain sets. For instance, is not defined at x = 0; is not defined at x = 1; is not defined at either x = 1 or x = −1.
Even though the function is not defined at 0, it is possible to see the behavior of the function around 0. Points can be chosen close to 0; for instance, x equal to 0.001, 0.0001, and 0.00001. The function values at these points are f (0.001) 1/0.001 1,000; f (0.0001) = 10,000; and f (0.00001) = 100,000.
As x becomes smaller and approaches 0, the function values become larger. In fact, the function grows without bound; that is, the function values has no upper ceiling, or limit, at x = 0. In mathematics, this behavior is described by saying that as x approaches 0, the function approaches infinity.
Approaching Zero
Consider a sequence of numbers which in decimal notation is expressed as 1, 0.5, 0.33, 0.25, 0.2, 0.16, 0.14, and so on. Each number in the sequence is called a term. As n becomes larger, becomes increasingly smaller. When n = 10,000 is 0.0001.
The sequence approaches 0, but its terms never equals 0. However, the terms of the sequence can be as close to 0 as wanted. For instance, it is possible for the terms of the sequence to get close enough to 0 so that the difference between the two is less than a billionth, or 10^{−6}. If one takes , then the sequence terms will be smaller than 10^{−6}.
see also Division by Zero; Limit.
Rafiq Ladhani
Bibliography
Dugopolski, Mark. Elementary Algebra, 3rd ed. Boston: McGrawHill, 2000.
Dunham, William. The Mathematical Universe. John Wiley & Sons Inc., 1994.
Kaplan, Robert. The Nothing That Is. New York: Oxford University Press, 1999.
Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: AddisonWesley, 2001.
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Zero
Zero
The most common meaning of the term zero is the absence of any magnitude or quantity. For example, a person might say that he or she has zero children, meaning that he or she has no children. In this respect, zero is a number, like 2, +9, −45, or 0.392. It can be used in mathematical operations in nearly all of the same ways that nonzero numbers can be used. For example, 4 + 0 = 4 is a legitimate mathematical operation. One mathematical operation from which zero is omitted is division. One can divide 0 by any number (in which case the answer is always zero), but one cannot divide any number by zero. That is, the mathematical operation 4 ÷ 0 has no meaning.
Zeroes also have other functions. For example, a zero may indicate the beginning of some counting system. A temperature of zero degrees kelvin (0 K), for example, is the starting point for the absolute temperature scale.
Zero also is used as a placeholder in the HinduArabic numeration system. The zero in the number 405 means that the number contains no tens. An expanded definition of the number is that 405 = 4 hundreds (4 × 100) plus 0 tens (0 × 10) plus 5 ones (5 × 1).
History
The history of the zero in numeration systems is a fascinating one. The symbol for zero (0) was not used by early Greek, Roman, Chinese, Egyptian, and other civilizations because they did not need it. In the Roman numeration system, for example, the number 405 is represented by CDIV.
The symbol for zero is believed to have first been used in the fourth century b.c. by an unknown Indian mathematician. When he wanted to record a more permanent answer on his beaded counting board, he used a simple dot. This dot was called a sunya and indicated columns in which there were no beads. While the sunya was not a true zero symbol, its use in place value notation was very important.
The actual 0 symbol for zero first appeared in about a.d. 800 when it was adopted as part of the HinduArabic numeration system. The symbol was originally a dot, or sifr, as it was called in Arabic. Over time, the dot gradually evolved to a small circle and then to the familiar oval we recognize today.
The zero symbol reached Europe around the twelfth century. However, Europeans did not adopt the symbol eagerly. In fact, many were reluctant to abandon their familiar Roman numerals, and hostile battles took place between supporters of the two systems. Such battles sometimes took the form of bloody physical encounters. It was not until three centuries later, therefore, that the HinduArabic numeration system—including the zero—was widely accepted and adopted throughout Europe.
[See also Numeration systems ]
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zero
zero, that number which, when added to any number, leaves the latter unchanged; its symbol is 0. The introduction of zero into the decimal system was the most significant achievement in the development of a number system in which calculation with large numbers was feasible. Without it, modern astronomy, physics, and chemistry would have been unthinkable as we know them. The lack of such a symbol was one of the serious drawbacks of Greek mathematics. Its existence in the West is probably due to the Arabs, who, having obtained it from the Hindus, passed it on to European mathematicians in the latter part of the Middle Ages. The Maya of Central America and probably the Babylonians also invented zero. With the extension of the number system to negative as well as positive numbers, zero became the name for that position on the scale of integers between 1 and +1. It is used in this sense in speaking of zero degrees on the Fahrenheit and Celsius temperature scales;
"absolute zero"
is a term used by physicists and chemists to indicate the theoretically lowest possible temperature—a use reminiscent of zero as a symbol for nothing. Unlike other numbers, zero has certain special properties in connection with the four fundamental operations. By definition zero added to or subtracted from any number leaves the number unchanged. Any number multiplied by zero gives zero. Zero multiplied by or divided by any number (other than zero) is still zero. But division by zero is undefined; i.e., there is no number that is the value of a number divided by zero.
See C. Seife, Zero (2000).
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zero
ze·ro / ˈzi(ə)rō/ • cardinal number (pl. ros) no quantity or number; the figure 0: figures from zero to nine. ∎ a point on a scale or instrument from which a positive or negative quantity is reckoned. ∎ the temperature corresponding to 0° on the Celsius scale (32° Fahrenheit), marking the freezing point of water: the temperature was below zero. ∎ the temperature corresponding to 0° on the Fahrenheit scale (approx. minus 18° Celsius), considered a very cold temperature, esp. for outdoor activities. See also subzero. ∎ [usu. as adj.] Linguistics the absence of an actual word or morpheme to realize a syntactic or morphological phenomenon: the zero plural in “three sheep.” ∎ the lowest possible amount or level; nothing at all: I rated my chances as zero. ∎ inf. a worthless or contemptibly undistinguished person: her husband is an absolute zero. • v. (roes, roed) [tr.] 1. adjust (an instrument) to zero: zero the counter when the tape has rewound. 2. set the sights of (a gun) for firing. PHRASAL VERBS: zero in take aim with a gun or missile: jet fighters zeroed in on the rebel positions. ∎ focus one's attention: they zeroed in on the clues he gave away about.zero out phase out or reduce to zero: the bill would zero out capital gains taxes.
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zero
zero no quantity or number; nought; the figure 0. The word is recorded from the early 17th century, and comes via French or Italian from Old Spanish and ultimately from Arabic ṣifr ‘cypher’.
zero hour the time at which a planned operation, typically a military one, is set to begin.
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zero
zero cipher, 0 XVII; point marked 0 on a scale, temperature denoted by this XVIII; nought, nothing XIX. — F. zéro or its source It. zero — OSp. zero (mod. cero) — Arab. ṣifr CIPHER.
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zero
zero
•arrow, barrow, farrow, harrow, Jarrow, marrow, narrow, sparrow, taro, tarot, Varro, yarrow
•gabbro • Avogadro • Afro • aggro
•macro • cilantro • Castro
•wheelbarrow
•Faro, Kilimanjaro, Pissarro, Pizarro, Tupamaro
•Pedro • allegro • hedgerow • velcro
•escrow
•metro, retro
•electro • Jethro
•bolero, caballero, dinero, Faeroe, pharaoh, ranchero, sombrero, torero
•scarecrow • Ebro
•Montenegro, Negro
•repro • in vitro • Pyrrho • synchro
•windrow • impro • intro • bistro
•Babygro • McEnroe
•biro, Cairo, giro, gyro, tyro
•fibro • micro • maestro
•borrow, Corot, morrow, sorrow, tomorrow
•cockcrow • cointreau
•Moro, Sapporo, Thoreau
•Mindoro • Yamoussoukro
•Woodrow
•burro, burrow, furrow
•upthrow
•De Niro, hero, Nero, Pierrot, Pinero, Rio de Janeiro, subzero, zero
•bureau, chiaroscuro, Douro, enduro, euro, Ishiguro, Oruro, Truro
•Politburo • guacharo • Diderot
•vigoro • Prospero • Cicero • in utero
•Devereux • Jivaro • overthrow
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