# 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 = 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 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

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