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Addition, indicated by a + sign, is a mathematical method of combining numbers; that is, of increasing one amount by another amount. The result of adding two numbers (such as a and b) is called their sum. For example, if a = 5 and b = 6, then their sum (c) is: c = a + b = 5 + 6 = 11. Addition can also include the process of adding together any of a number of objects; such as adding applies and cherries.

## Adding natural numbers

Consider the natural, or counting, numbers 1, 2, 3, 4. . . . Each natural number can be defined in terms of sets. The number 1 is the name of the collection containing every conceivable set with one element, such as the set containing 0 or the set containing the Washington Monument. The number 2 is the name of the collection containing every conceivable set with two elements, and so on. The sum of two natural numbers is determined by counting the number of elements in the union of two sets chosen to represent them. For example, let the set {A, B, C} represent 3 and the set {W, X, Y, Z} represent 4. Then 3 + 4 is determined by counting the elements in {A, B, C, W, X, Y, Z}, which is the union of {A, B, C} and {W, X, Y, Z}. The result is seven, and we write 3 + 4 = 7. In this way, the operation of addition is carried out by counting.

## The addition algorithm

Addition of natural numbers is independent of the numerals used to represent the numbers being added. However, some forms of notation make addition of large numbers easier than other forms. In particular, the Hindu-Arabic positional notation (in general use today) facilitates addition of large numbers, while the use of Roman numerals, for instance, is quite cumbersome. In the Hindu-Arabic positional notation, numerals are arranged in columns, each column corresponding to numbers that are ten times larger than those in the column to the immediate right.

For example, 724 consists of 4 ones, 2 tens, and 7 hundreds. The addition algorithm amounts to counting by ones in the right hand column, counting by tens in the next column left, counting by hundreds in the next column left, and so on. When the sum of two numbers in any column exceeds nine, the amount over 10 is retained and the rest transferred, or carried, to the next column left. Suppose it is desired to add 724 and 897. Adding each column gives 11 ones, 11 tens, and 15 hundreds. However, 11 ones is equal to 1 ten and 1 one so one has 1 one, 12 tens and 15 hundreds. Checking the tens column one finds 12 tens equals 2 tens and 1 hundred, so one actually has 1 one, 2 tens and 16 hundreds. Finally, 16 hundreds is 6 hundreds and 1 thousand, so the end result is 1 thousand, 6 hundreds, 2 tens, and 1 one, or 1, 621.

## Adding common fractions

Historically, the number system expanded as it became apparent that certain problems of interest had no solution in the then-current system. Fractions were included to deal with the problem of dividing a whole thing into a number of parts. Common fractions are numbers expressed as a ratio, such as 2/3, 7/9, and 3/2. When both parts of the fraction are integers, the result is a rational number. Each rational number may be thought of as representing a number of pieces; the numerator (top number) tells how many pieces the fraction represents; the denominator (bottom number) tells how many pieces the whole was divided into. Suppose a cake is divided into two pieces, after which one-half is further divided into six pieces and the other half into three pieces, making a total of nine pieces. If a person takes one piece from each half, what part of the whole cake has been taken? This amounts to a simple counting problem if both halves are cut into the same number of pieces, because then there are a total of six or 12 equal pieces, of which one takes two. One gets either 2/6, or 2/12, of the cake. The essence of adding rational numbers, then, is to turn the problem into one of counting equal size pieces. This is done by rewriting one or both of the fractions to be added so that each has the same denominator (called a common denominator). In this way, each fraction represents a number of equal size pieces. A general formula for the sum of two fractions is a/b + c/d = (ad + bc)/bd.

## Adding decimal fractions

Together, the rational and irrational numbers constitute the set of real numbers. Addition of real numbers is facilitated by extending the positional notation used for integers to decimal fractions. Place a period (called a decimal point) to the right of the ones column, and let each column to its right contain numbers that are successively smaller by a factor of ten. Thus, columns to the right of the decimal point represent numbers less than one, in particular, tenths, hundredths, thousandths, and so on. Addition of real numbers, then, continues to be defined in terms of counting and carrying, in the manner described above.

## Adding signed numbers

Real numbers can be positive, negative, or zero. Addition of two negative numbers always results in a negative number and is carried out in the same fashion that positive numbers are added, after which a negative sign is placed in front of the result, such as -4 + (-21) = -25. Adding a positive and a negative number is the equivalent of subtraction, and, while it also proceeds by counting, the sum does not correspond to counting the members in the union of two sets, but to counting the members not in the intersection of two sets.

## Addition in algebra

In algebra, which is a generalization of arithmetic, addition is also carried out by counting. For example, to sum the expressions 5x and 6x one notices that 5x means five xs and 6x means six xs, making a total of 11 xs. Thus 5x + 6x = (5 + 6)x = 11x, which is usually established on the basis of the distributive law, an important property that the real numbers obey. In general, only like variables or powers can be added algebraically. In adding two polynomial expressions, only similar terms are combined; thus, (3x2 + 2x + 7y + z) + (x3 + 3x + 4z + 2yz) = (x3 + 3x2 + 5x + 7y + 5z + 2yz).

## Resources

### BOOKS

Berinde, Vasile. Exploring, Investigating, and Discovering in Mathematics. Boston, MA: Birkhauser, 2004.

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

Cooke, Roger. The History of Mathematics: A Brief Course. Hoboken, NJ: Wiley-Interscience, 2005.

Grahm, Alan. Teach Yourself Basic Mathematics. Chicago, IL: McGraw-Hill Contemporary, 2001. Lorenz, Falko. Algebra. New York: Springer, 2006.

Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

Trivieri, Lawrence. Basic Mathematics. New York: HarperCollins Publishers, 2006.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2003.

views updated

Addition, indicated by a + sign, is a method of combining numbers. The result of adding two numbers is called their sum.

## Adding natural numbers

Consider the natural, or counting, numbers 1, 2, 3, 4,... Each natural number can be defined in terms of sets. The number 1 is the name of the collection containing every conceivable set with one element, such as the set containing 0 or the set containing the Washington Monument. The number 2 is the name of the collection containing every conceivable set with two elements, and so on. The sum of two natural numbers is determined by counting the number of elements in the union of two sets chosen to represent them. For example, let the set {A, B, C} represent 3 and the set {W, X, Y, Z} represent 4. Then 3 + 4 is determined by counting the elements in {A, B, C, W, X, Y, Z}, which is the union of {A, B, C} and {W, X, Y, Z}. The result is seven, and we write 3 + 4 = 7. In this way, the operation of addition is carried out by counting.

## The addition algorithm

Addition of natural numbers is independent of the numerals used to represent the numbers being added. However, some forms of notation make addition of large numbers easier than other forms. In particular, the Hindu-Arabic positional notation (in general use today) facilitates addition of large numbers, while the use of Roman numerals, for instance, is quite cumbersome. In the Hindu-Arabic positional notation, numerals are arranged in columns, each column corresponding to numbers that are 10 times larger than those in the column to the immediate right. For example, 724 consists of 4 ones, 2 tens, and 7 hundreds. The addition algorithm amounts to counting by ones in the right hand column, counting by tens in the next column left, counting by hundreds in the next column left and so on. When the sum of two numbers in any column exceeds nine, the amount over 10 is retained and the rest transferred or "carried" to the next column left. Suppose it is desired to add 724 and 897. Adding each column gives 11 ones, 11 tens, and 15 hundreds. But 11 ones is equal to 1 ten and 1 one so we have 1 one, 12 tens and 15 hundreds. Checking the tens column we find 12 tens equals 2 tens and 1 hundred, so we actually have 1 one, 2 tens and 16 hundreds. Finally, 16 hundreds is 6 hundreds and 1 thousand, so the end result is 1 thousand, 6 hundreds, 2 tens, and 1 one, or 1,621.

## Adding common fractions

Historically, the number system expanded as it became apparent that certain problems of interest had no solution in the then-current system. Fractions were included to deal with the problem of dividing a whole thing into a number of parts. Common fractions are numbers expressed as a ratio , such as 2/3, 7/9, and 3/2. When both parts of the fraction are integers , the result is a rational number . Each rational number may be thought of as representing a number of pieces; the numerator (top number) tells how many pieces the fraction represents; the denominator (bottom number) tells us how many pieces the whole was divided into. Suppose a cake is divided into two pieces, after which one half is further divided into six pieces and the other half into three pieces, making a total of nine pieces. If you take one piece from each half, what part of the whole cake do you get? This amounts to a simple counting problem if both halves are cut into the same number of pieces, because then there are a total of six or 12 equal pieces, of which you take two. You get either 2/6 or 2/12 of the cake. The essence of adding rational numbers, then, is to turn the problem into one of counting equal size pieces. This is done by rewriting one or both of the fractions to be added so that each has the same denominator (called a common denominator). In this way, each fraction represents a number of equal size pieces. A general formula for the sum of two fractions is a/b + c/d = (ad + bc)/bd.

## Adding decimal fractions

Together, the rational and irrational numbers constitute the set of real numbers . Addition of real numbers is facilitated by extending the positional notation used for integers to decimal fractions. Place a period (called a decimal point) to the right of the ones column, and let each column to its right contain numbers that are successively smaller by a factor of ten. Thus, columns to the right of the decimal point represent numbers less than one, in particular, "tenths," "hundredths," "thousandths," and so on. Addition of real numbers, then, continues to be defined in terms of counting and carrying, in the manner described above.

## Adding signed numbers

Real numbers can be positive, negative , or zero . Addition of two negative numbers always results in a negative number and is carried out in the same fashion that positive numbers are added, after which a negative sign is placed in front of the result, such as -4 + (-21) = -25. Adding a positive and a negative number is the equivalent of subtraction , and, while it also proceeds by counting, the sum does not correspond to counting the members in the union of two sets, but to counting the members not in the intersection of two sets.

## Addition in algebra

In algebra , which is a generalization of arithmetic , addition is also carried out by counting. For example, to sum the expressions 5x and 6x we notice that 5x means we have five xs and 6x means we have six xs, making a total of 11 xs. Thus 5x + 6x = (5 + 6)x = 11x, which is usually established on the basis of the distributive law, an important property that the real numbers obey. In general, only like variables or powers can be added algebraically. In adding two polynomial expressions, only similar terms are combined; thus, (3x2 + 2x +7y + z) + (x3 + 3x + 4z + 2yz) = (x3 + 3x2 + 5x + 7y + 5z + 2yz).

## Resources

### books

Eves, Howard Whitley. Foundations and Fundamental Concepts of Mathematics. NewYork: Dover, 1997.

Grahm, Alan. Teach Yourself Basic Mathematics. Chicago,IL: McGraw-Hill Contemporary, 2001.

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

Paulos, John Allen. Beyond Numeracy, Ruminations of a Numbers Man. New York: Alfred A. Knopf, 1991.

Tobey, John, and Jeffrey Slater. Beginning Algebra. 4th ed. NY: Prentice Hall, 1997.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.