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This chapter is from the book

This chapter is from the book

2.6 Arithmetic

Most programs perform arithmetic calculations. Figure 2.9 summarizes the C++ arithmetic operators. Note the use of various special symbols not used in algebra. The asterisk (*) indicates multiplication and the percent sign (%) is the modulus operator that will be discussed shortly. The arithmetic operators in Fig. 2.9 are all binary operators, i.e., operators that take two operands. For example, the expression number1 + number2 contains the binary operator + and the two operands number1 and number2.

Fig 2.9. Arithmetic operators.

C++ operation

C++ arithmetic operator

Algebraic expression

C++ expression

Addition

+

f + 7

f + 7

Subtraction

-

p – c

p - c

Multiplication

*

bm or b · m

b * m

Division

/

x / y or xbyy.jpg or x ÷ y

x / y

Modulus

%

r mod s

r % s

Integer division (i.e., where both the numerator and the denominator are integers) yields an integer quotient; for example, the expression 7 / 4 evaluates to 1 and the expression 17 / 5 evaluates to 3. Any fractional part in integer division is discarded (i.e., truncated)—no rounding occurs.

C++ provides the modulus operator, %, that yields the remainder after integer division. The modulus operator can be used only with integer operands. The expression x % y yields the remainder after x is divided by y. Thus, 7 % 4 yields 3 and 17 % 5 yields 2. In later chapters, we discuss many interesting applications of the modulus operator, such as determining whether one number is a multiple of another (a special case of this is determining whether a number is odd or even).

Arithmetic Expressions in Straight-Line Form

Arithmetic expressions in C++ must be entered into the computer in straight-line form. Thus, expressions such as "a divided by b" must be written as a / b, so that all constants, variables and operators appear in a straight line. The algebraic notation

048equ01.jpg

is generally not acceptable to compilers, although some special-purpose software packages do support more natural notation for complex mathematical expressions.

Parentheses for Grouping Subexpressions

Parentheses are used in C++ expressions in the same manner as in algebraic expressions. For example, to multiply a times the quantity b + c we write a * ( b + c ).

Rules of Operator Precedence

C++ applies the operators in arithmetic expressions in a precise order determined by these rules of operator precedence, which are generally the same as those in algebra:

  1. Operators in expressions contained within pairs of parentheses are evaluated first. Parentheses are said to be at the "highest level of precedence." In cases of nested, or embedded, parentheses, such as

    ( a * ( b + c ) )

    the operators in the innermost pair of parentheses are applied first.

  2. Multiplication, division and modulus operations are applied next. If an expression contains several multiplication, division and modulus operations, operators are applied from left to right. Multiplication, division and modulus are said to be on the same level of precedence.
  3. Addition and subtraction operations are applied last. If an expression contains several addition and subtraction operations, operators are applied from left to right. Addition and subtraction also have the same level of precedence.

The set of rules of operator precedence defines the order in which C++ applies operators. When we say that certain operators are applied from left to right, we are referring to the associativity of the operators. For example, the addition operators (+) in the expression

a + b + c

associate from left to right, so a + b is calculated first, then c is added to that sum to determine the whole expression's value. We'll see that some operators associate from right to left. Figure 2.10 summarizes these rules of operator precedence. We expand this table as we introduce additional C++ operators. A complete precedence chart is included in Appendix A.

Fig 2.10. Precedence of arithmetic operators.

Operator(s)

Operation(s)

Order of evaluation (precedence)

( )

Parentheses

Evaluated first. If the parentheses are nested, the expression in the innermost pair is evaluated first. [Caution: If you have an expression such as (a + b) * (c - d) in which two sets of parentheses are not nested, but appear "on the same level," the C++ Standard does not specify the order in which these parenthesized subexpressions will be evaluated.]

*, /, %

Multiplication, Division, Modulus

Evaluated second. If there are several, they're evaluated left to right.

+

-

Addition

Subtraction

Evaluated last. If there are several, they're evaluated left to right.

Sample Algebraic and C++ Expressions

Now consider several expressions in light of the rules of operator precedence. Each example lists an algebraic expression and its C++ equivalent. The following is an example of an arithmetic mean (average) of five terms:

049equ01.jpg

The parentheses are required because division has higher precedence than addition. The entire quantity ( a + b + c + d + e ) is to be divided by 5. If the parentheses are erroneously omitted, we obtain a + b + c + d + e / 5, which evaluates incorrectly as

050equ01.jpg

The following is an example of the equation of a straight line:

Algebra:

y = mx + b

C++:

y = m * x + b;

No parentheses are required. The multiplication is applied first because multiplication has a higher precedence than addition.

The following example contains modulus (%), multiplication, division, addition, subtraction and assignment operations:

050fig01.jpg

The circled numbers under the statement indicate the order in which C++ applies the operators. The multiplication, modulus and division are evaluated first in left-to-right order (i.e., they associate from left to right) because they have higher precedence than addition and subtraction. The addition and subtraction are applied next. These are also applied left to right. The assignment operator is applied last because its precedence is lower than that of any of the arithmetic operators.

Evaluation of a Second-Degree Polynomial

To develop a better understanding of the rules of operator precedence, consider the evaluation of a second-degree polynomial y = ax 2 + bx + c:

050fig02.jpg

The circled numbers under the statement indicate the order in which C++ applies the operators. There is no arithmetic operator for exponentiation in C++, so we've represented x 2 as x * x. We'll soon discuss the standard library function pow ("power") that performs exponentiation. Because of some subtle issues related to the data types required by pow, we defer a detailed explanation of pow until Chapter 6.

Suppose variables a, b, c and x in the preceding second-degree polynomial are initialized as follows: a = 2, b = 3, c = 7 and x = 5. Figure 2.11 illustrates the order in which the operators are applied and the final value of the expression.

Fig. 2.11

Fig. 2.11 Order in which a second-degree polynomial is evaluated.

As in algebra, it's acceptable to place unnecessary parentheses in an expression to make the expression clearer. These are called redundant parentheses. For example, the preceding assignment statement could be parenthesized as follows:

y = ( a * x * x ) + ( b * x ) + c;
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