Zero

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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.

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 (0² = 0).

Exponent property: Any number other than zero raised to the power 0 equals 1. That is, b⁰ = 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.

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.

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.

Division by Zero; Limit.

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.

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