Addition and subtraction

Summary: Addition and subtraction are binary mathematical operations, each the inverse of the other, and are among the oldest mathematical concepts.

Addition can be thought of as a process of accumulation. For example, if a flock of 3 sheep is joined with a flock of 4 sheep, the combined flock will have 7 sheep. Thus, 7 is the sum that results from the addition of the numbers 3 and 4. This can be written as 3+4=7 where the sign "+" is read "plus" and the sign "=" is read "equals." Both 3 and 4 are called addends. Addition is commutative; that is, the order of the addends is irrelevant to how the sum is formed. Subtraction finds the remainder after a quantity is diminished by a certain amount. If from a flock containing 5 sheep, 3 sheep are removed, then 2 sheep remain. In this example, 5 is the minuend, 3 is the subtrahend, and 2 is the remainder or difference. This can be written as 5-3=2 where "-"is read "minus." Subtraction is not commutative and therefore the ordering of the minuend and subtrahend affects the result: 5-3=2 , but 3-5=-2.

The concept of addition can be extended to have meaning for fractions, negative numbers, real numbers, measurements, and other mathematical entities. The algorithms used for computing the sum or difference, some of which have been taught for millennia, ultimately depend on the representation used for the numbers. For example, the approach used for adding Roman numerals is different from that used to add Hindu-Arabic numbers. Computers perform subtraction using the same circuits they use for addition.

History and Development of Addition and Subtraction

Human beings’ ability to add and subtract small whole numbers is probably innate. Some of the earliest descriptions of techniques for handling large numbers come from ancient China during the Warring States period (475–221 b.c.e.), when arithmetic operations were performed by manipulating rods on a flat surface that was partitioned by vertical and horizontal lines. The numbers were represented by a positional base-10 system. Some scholars believe that this system—after moving westward through India and the Islamic Empire—became the modern system of representing numbers.

The Greeks in the fifth century b.c.e., in addition to using a complex ciphered system for representing numbers, used a system that is very similar to Roman numerals. It is possible that the Greeks performed arithmetic operations by manipulating small stones on a large, flat surface partitioned by lines. A similar stone tablet was found on the island of Salamis in the 1800s and is believed to date from the fourth century b.c.e. The word "calculate" was derived from the Latin word for "little stone."

The Romans had arithmetic devices similar in appearance to the typical Chinese abacus. It is difficult to use modern paper-and-pencil techniques for adding and subtracting Roman numerals (with I as one, II as two, V as five, X as ten, L as fifty, C as one hundred, D as five hundred, M as one thousand)—but it worked well in its time, since it was devised for use with an abacus.

During the Middle Ages, counting boards were used to perform arithmetic. A counting board consisted of a series of actual or virtual horizontal lines that were labeled from the bottom by I, X, C, M, and so on. The system borrowed the symbols used for core numbers from the Roman system. The spaces between the lines were labeled starting from the bottom by V, L, and D. A number like MMDCCXXXVIIII (2739) would be represented by placing the appropriate number of counters on each line. The line labeled M would have 2 counters (for 2000, or two thousands). The space just below, labeled D, would have 1 counter (500, or one five-hundreds); the line labeled C, 2 counters (200, or two hundreds); the space labeled L, 0 counters; the line labeled X, 3 counters (for 30, or three tens); the line labeled V, 1 counter (5); and the line labeled I, 4 counters (4, or four ones). The total of all these numbers is 2739. Note that accountants used VIIII (denoting five plus four) to represent 9, whereas stonemasons used "IX"(denoting 10 less 1). To compute the sum MMDCCXXXVIIII + MCLXI, a person would simply transcribe the numbers to the counting board and then combine the counters following rules of carrying to ensure that no more than 4 counters were on any line and 1 counter on any space. This representation was then easily transcribed back into Roman numerals.

Many early books on arithmetic claim that this method of performing arithmetic was especially preferred by women, who at times had the responsibility for keeping the books for small family businesses. Hindu-Arabic numerals and paper-and-pencil methods for performing arithmetic began to appear in Europe in the twelfth century and replaced Roman numerals and the counting board by the nineteenth century.

Two Methods for Subtracting by Hand

Two popular methods for handling "borrowing" that are taught today are shown below. The method shown in the figure below on the left is popular in Italy, England, and the United States, while the one on the right is popular in Spain, France, and parts of Latin America. The example is to compute 3047-1964. Starting with the method on the left, first begin with the rightmost column and subtract 4 from 7. Write the result, 3, below the 4. Moving one column to the left, try to subtract 6 from 4, which cannot be done without using negative numbers. The method is thus to attempt to "borrow" 1 from 0, which is the digit to the left of the 4. Again, this cannot be done without using negative numbers. Therefore, the method is to borrow 1 from 3, which is the digit to the left of the 0 resulting in crossing out the 3 and replacing it with a 2. Then the zero becomes a 10, and it in turn can be replaced by a 9 so the borrowed 1 can be placed in front of the 4 to make it 14. Now, one can subtract 6 from 14 to get 8, which is written below the 6. Moving left to the next column, one can subtract 9 from 9 to get a 0, which is written below the 9. Finally, 1 is subtracted from 2 to get a 1, which is written below.

To solve the problem using subtraction with carry, use the example on the right. The carrying numbers (the small 1s) affect the numbers on a diagonal, as shown in the example. The number 1 adds 10 to the integer in the top row and adds 1 to the integer in the bottom row. Starting from the rightmost column, 4 is subtracted from 7, resulting in 3, which is written below. Then, try to subtract 6 from 4, which cannot be done, so insert a small 1 to the left of the space between the 4 and the 6. This is interpreted to mean that the 4 has become 14. Subtract 6 from 14 and record the answer, 8, below. Move left to the next column containing 0 and 9. The small 1, written above and to the right of the 9, is added to the 9 to get 10. Attempt to subtract the 10 from the 0 above, which cannot be done. Instead, write a small 1 just to the left of the space between the 0 and 9, and interpret this to mean that the 0 has become a 10. Now, 10 minus 10 is 0, which is written below. Move left to the next column. The small 1, written above and to the right of the 1, is added to the 1 giving 2, which is subtracted from 3 resulting in 1, which is written below.

Adding and Subtracting on a Computer

At the most basic level, whole numbers are represented in a computer in base-two by a sequence of the binary states "Hi" and "Lo" interpreted as "1" and "0." The circuits that perform addition are implemented by sequences of logical gates. Typically a "1" in the leftmost bit indicates that the number is negative, with the remaining bits indicating the magnitude of the number. Subtraction can be performed by the same circuits that perform addition. Two popular approaches are designated as "one’s complement" and "two’s complement." "One’s complement" can best be explained by performing subtraction in base-10 using "nine’s complement." Assume a computation of 3047-1964. To find the "nine’s complement" of 1964, subtract each digit from 9 to obtain 8035. This is added to 3047 resulting in 11,082. The leftmost 1 is viewed as a "carry" and brought around and added to the rightmost digit in an operation called "end-around carry" to obtain the final result: 1083.

Generalizing Addition and Subtraction

The sum of two fractions a/b and c/d is defined to be

The sum of irrational numbers (numbers that cannot be represented as fractions of whole numbers) can be approximated only by adding their approximating rationals. The exact sum of two irrational lengths, a and b, can be found exactly using geometry by first extending the segment representing a sufficiently on one end so that the length b can be marked off from that end with a compass.

Addition can be generalized to other mathematical objects, such as complex numbers and matrices. One of these objects, typically called the additive identity and denoted by "0," has the property such that if "a" is any object then the sum of 0 and a is a. The additive reciprocal of an object a is denoted by -a and is defined to the object so that the sum of a+(-a) is 0. The difference a - b is defined to be a+(-b).

Bibliography

Flegg, G. Numbers: Their History and Meaning. New York: Schocken Books, 1983.

Karpinski, L. C. The History of Arithmetic. New York: Russell & Russell, 1965.

Pullan, J. M. The History of the Abacus. New York: F. A. Praeger, 1969.

Rafiquzzaman, M. Fundamentals of Digital Logic and Microcomputer Design. Hoboken, NJ: Wiley, 2005.

Yong, L. L., and A. T. Se. Fleeting Footsteps. Singapore: Word Scientific Publishers, 2004.