# Zero

# Zero

The symbol zero, in mathematics, is the representation of the absence of any magnitude or quantity. Zero is often equated with nothing, but that is not a good analogy. Zero, sometimes called naught, can be the absence of a quality, but it can also be a starting point, such as 0° on a temperature scale. In a mathematical system, zero is the additive identity. It is a number that can be added to any given number to yield a sum equal to the given number. Symbolically, it is a number 0, such that a + 0 = a for any number a.

In the Hindi-Arabic numeration system, zero is used as a placeholder as well as a number. The number 205 is distinguished from 25 by having a 0 in the tens place. This can be interpreted as no tens, but the early use of 0 in this way was more to show that 2 was in the hundreds place than to show no tens.

Zero is used in some ways that take it beyond ordinary addition and multiplication. One use is as an exponent. In an exponential function such as y = 10^{x} the exponent is not limited to the counting numbers. One of its possible values is 0. If 10^{0} is to obey the rule for exponents—10^{m} = 10^{m} + 0 = 10^{m}× 10^{0}—10^{0} must equal 1. This is true not only for 10^{0} but for any a^{0}, where a is any positive number. That is, a^{0} =1.Another curious use of zero is the expression 0!. Ordinarily n! is the product 1× 2× 3×...× n, of all the integers from 1 to n. In a formula such as n!/r!(n - r)!, which represents the number of different combinations of things that can be chosen from n things r at a time, 0! can occur. If 0! is assigned the value 1, the formula works. This happens in other instances as well.

The symbol for zero does not appear before about 800 AD, when it appears in connection with Hindu-Arabic base-10 numerals. In these numerals it functions as a placeholder. The Mayans also used a zero in writing their base-20 numerals. It was a symbol that looked something like an eye, and it acted as a place holder.

The reason that the symbol appeared so late in history is that the number systems used by the Greeks, Romans, Chinese, Egyptians, and others did not need it. For example, one can write the Roman numeral for 1056 as MLVI. No zero as a placeholder is needed. The Babylonians did have a place-value system with their base-60 numerals, and a symbol for zero would have eliminated some of the ambiguity that shows up in their clay tablets, but was probably overlooked because, within each place, the numbers from 1 to 59 were represented with wedge-shaped tallies. In a tally system all that is required to represent zero is the absence of a tally. Sometimes Babylonians did use a dot or a space as a placeholder, but failed to see that this could be a number of its own.

The word zero appears to be a much metamorphosed translation of the Hindu word sunya, meaning void or empty.

Zero also has the property a× 0 = 0 for any number a. This property is a consequence of zero’s additive property.

In ordinary arithmetic, the statement ab = 0 implies that a, b, or both are equal to 0; that is, the only way for a product to equal zero is for one or more of its factors to equal zero. This property is used when one solves equations such as (x - 2)(x + 3) = 0 by setting each factor equal to zero.

The multiplicative property of zero is also used in the argument for not allowing zero to be used as a divisor or a denominator. The law that defines a/b is (a/b) b = a. If one substitutes 0 for b, the result is (a/0)0 = a, which forces a to be 0. But, even when a is 0, the law allows 0/0 to be any number, which is intolerable.

Zero sometimes appears in disguise. In even-andodd arithmetic, one has ‘even plus odd equals odd,’ ‘odd times odd equals odd,’ and so on. The various combinations can be listed in the tables

Is there a zero? Is there an element 0 such that 0 + a = a for either of the possible values of a? The top line in the addition table says that there is. Even is such an element. Does 0× a = 0 for both values of a? The top line of the multiplication table says that it does. Does ab = 0 imply that one or both of the factors is 0? Only if both factors are odd, is the product odd; so, yes, it does.

Thus, this miniature arithmetic has a zero, and it is even.

Another arithmetic is clock arithmetic. In this arithmetic, 3 is three hours past 12; 3 + 7 is 10 hours past 12; and 3 + 12 is 15 hours past 12. But on a clock, every 12 hours the hands return to their original position; so 15 hours past 12 is the same as three hours past 12. For any a, a + 12 = a. [In number-theory symbolism, this would be written a + 12 integral a (mod 12).] So, in clock arithmetic, 12 behaves like 0 in ordinary arithmetic.

It also multiplies like 0. Twelve 3-hour periods equal 36 hours, which the hands show as 12. Twelve periods of a hours each leave the hands at 12 for any a (a is limited to whole numbers in clock arithmetic), so 12× a = 12.

Thus, in clock arithmetic 12 does not look like zero, but it behaves like zero. It could be called 0, and on a digital 24-hour clock, where the number 24 behaves like 0, 24 is called 0. The next number after 23:59:59 is 0:0:0.

In this arithmetic, unlike ordinary arithmetic, the law ‘ab = 12 if and only if a, b, or both equal 12^{’} does not hold.’ The ‘if’ part does, but not the ‘only if.’ Six times 2 is 12, but neither 6 nor 2 is 12. Three times 8 is 12 (the hands go around twice, passing 12 once and ending at 12), but neither 3 nor 8 is 12. Thus, in clock arithmetic there can be two numbers, neither of them zero, whose product is zero. Such numbers are called divisors of zero. This happens because people use 12-hour (or 24-hour) clocks. If humans used, for instance, 11-hour clocks, it would not.

*See also* Numeration systems.

## Resources

### BOOKS

Burton, David M. *The History of Mathematics:* An Introduction. New York: McGraw-Hill, 2007.

Clawson, Calvin C. *The Mathematical Traveler: Exploring the Grand History of Numbers Cambridge,* MA: Perseus Publishing, 2003.

Gelfond, A.O. *Transcendental and Algebraic Numbers.* New York: Dover Publications, 2003.

Kaplan, Robert. *The Nothing That Is: A Natural History of Zero.* Oxford, UK, and New York: Oxford University Press, 2000.

Seife, Charles. Zero: The Biography of a Dangerous Idea. New York: Viking, 2000.

Stopple, Jeffrey. A Primer of Analytic Number Theory: From Pythagoras to Riemann. Cambridge, UK: Cambridge University Press, 2003.

J. Paul Moulton

# 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

xequalsx. That is,x+ 0 =x.Zero is called the additive identity.Multiplication property: Multiplying any number

bby 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 ×xmust be equal to 12. No value ofxwill 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: McGraw-Hill, 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: Addison-Wesley, 2001.

# Zero

# Zero

Zero is often equated with "nothing," but that is not a good analogy. Zero can be the absence of a quality, but it can also be a starting point, such as 0° on a **temperature** scale. In a mathematical system, zero is the *additive identity*. It is a number which can be added to any given number to yield a sum equal to the given number. Symbolically, it is a number 0, such that a + 0 = a for any number a.

In the Hindi-Arabic numeration system, zero is used as a placeholder as well as a number. The number 205 is distinguished from 25 by having a 0 in the tens place. This can be interpreted as no tens, but the early use of 0 in this way was more to show that 2 was in the hundreds place than to show no tens.

Zero is used in some ways that take it beyond ordinary **addition** and **multiplication** . One use is as an **exponent** . In an exponential function such as y = 10x the exponent is not limited to the counting numbers. One of its possible values is 0. If 100 is to obey the rule for exponents—10m = 10m + 0 = 10m × 100—100 must equal 1. This is true not only for 100 but for any a0, where a is any **positive number** . That is, a0 = 1.

Another curious use of zero is the expression 0!. Ordinarily n! is the product 1 × 2 × 3 ×... × n, of all the **integers** from 1 to n. In a formula such as n!/r!(n - r)!, which represents the number of different combinations of things which can be chosen from n things r at a time, 0! can occur. If 0! is assigned the value 1, the formula works. This happens in other instances as well.

The symbol for zero does not appear before about a.d. 800, when it appears in connection with Hindu-Arabic base-10 numerals. In these numerals it functions as a place holder. The Mayans also used a zero in writing their base-20 numerals. It was a symbol which looked something like an **eye** , and it acted as a place holder.

The reason that the symbol appeared so late in history is that the number systems used by the Greeks, Romans, Chinese, Egyptians, and others did not need it. For example, one can write the Roman numeral for 1056 as MLVI. No zero as a place holder is needed. The Babylonians did have a place-value system with their base-60 numerals, and a symbol for zero would have eliminated some of the ambiguity that shows up in their clay tablets, but was probably overlooked because, within each place, the numbers from 1 to 59 were represented with wedge-shaped tallies. In a tally system all that is required to represent zero is the absence of a tally. Sometimes Babylonians did use a dot or a space as a placeholder, but failed to see that this could be a number of its own.

The word zero appears to be a much metamorphosed translation of the Hindu word "sunya," meaning void or empty.

Zero also has the property a × 0 = 0 for any number a. This property is a consequence of zero's additive property.

In ordinary **arithmetic** the statement ab = 0 implies that a, b, or both are equal to 0; that is, the only way for a product to equal zero is for one or more of its factors to equal zero. This property is used when one solves equations such as (x - 2)(x + 3) = 0 by setting each **factor** equal to zero.

The multiplicative property of zero is also used in the argument for not allowing zero to be used as a divisor or a denominator. The law which defines a/b is (a/b)b = a. If one substitutes 0 for b, the result is (a/0)0 = a, which forces a to be 0. But even when a is 0, the law allows 0/0 to be any number, which is intolerable.

Zero sometimes appears in disguise. In even-andodd arithmetic we have "even plus odd equals odd," "odd times odd equals odd," and so on. The various combinations can be listed in the tables

+ | even | odd | x | even | odd | ||||

even | even | odd | even | even | even | ||||

odd | odd | even | odd | even | odd |

Is there a zero? Is there an element 0 such that 0 + a = a for either of the possible values of a? The top line in the addition table says that there is. "Even" is such an element. Does 0 × a = 0 for both values of a? The top line of the multiplication table says that it does. Does ab = 0 imply that one or both of the factors is 0? Only if both factors are odd, is the product odd; so, yes, it does.

Thus this miniature arithmetic has a zero, and it is "even."

Another arithmetic is clock arithmetic. In this arithmetic 3 is three hours past 12; 3 + 7 is 10 hours past 12; and 3 + 12 is 15 hours past 12. But on a clock, every 12 hours the hands return to their original position; so 15 hours past 12 is the same as three hours past 12. For any a, a + 12 = a. [In number-theory symbolism this would be written a + 12 intergral a (mod 12).] So in clock arithmetic, 12 behaves like 0 in ordinary arithmetic.

It also multiplies like 0. Twelve 3-hour periods equal 36 hours, which the hands show as 12. Twelve periods of a hours each leave the hands at 12 for any a (a is limited to whole numbers in clock arithmetic), so 12 × a = 12.

Thus, in clock arithmetic 12 does n0t look like zero, but it behaves like zero. It could be called 0, and on a digital 24-hour clock, where the number 24 behaves like 0, 24 is called 0. The next number after 23:59:59 is 0:0:0.

In this arithmetic, unlike ordinary arithmetic, the law "ab = 12 if and only if a, b, or both equal 12" does not hold. The "if" part does, but not the "only if." Six times 2 is 12, but neither 6 nor 2 is 12. Three times 8 is 12 (the hands go around twice, passing 12 once and ending at 12), but neither 3 nor 8 is 12. Thus in clock arithmetic there can be two numbers, neither of them zero, whose product is zero. Such numbers are called divisors of zero. This happens because we use 12-hour (or 24-hour) clocks. If we used 11-hour clocks, it would not.

See also Numeration systems.

## Resources

### books

Clawson, Calvin C. *The Mathematical Traveler: Exploring the Grand History of Numbers.* Cambridge, MA: Perseus Publishing, 2003.

Gelfond, A.O. *Transcendental and Algebraic Numbers.* Dover Publications, 2003.

Gullberg, Jan, and Peter Hilton. *Mathematics: From the Birth of Numbers.* W.W. Norton & Company, 1997.

Stopple, Jeffrey. *A Primer of Analytic Number Theory: From Pythagoras to Riemann.* Cambridge: Cambridge University Press, 2003.

J. Paul Moulton

# 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*zero out phase out or reduce to zero:

**zeroed in on**the clues he gave away about.*the bill would zero out capital gains taxes.*

# 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 Hindu-Arabic 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 Hindu-Arabic 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 Hindu-Arabic numeration system—including the zero—was widely accepted and adopted throughout Europe.

[*See also* **Numeration systems** ]

# 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.

# 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.

# Zero

# Zero 1984

WWII story about the building of the zero fighter that was used to devastating effect in the invasion of Pearl Harbor. **128m/C VHS** . Yuzo Kayama, Tetsuro Tamba; ** D:** Toshio Masuda.

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