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Each function computed by calc, such as exp() or sqrt(), requires a set of parentheses to delimit the function’s argument. 6. The function ln() and the function log() are identical, both being the “natural logarithm” function ln. Thus ln(exp(1)) is 1. To obtain other loga­ rithms, use the identity logb x = ln x ln b Thus to obtain log10 97 use the entry ln(97)/ln(10). 7. We are ready now to evaluate the other constant mentioned at the beginning of this chapter, tan 31 . The three trigonometric functions sin(), cos(), and tan() all use radian angle measure, so degrees must be specified in terms radians, so one degree is /180 radians.

6 are not mutually exclusive. It is quite possible that at a certain precision we decide that a is within 10−k of b, but at a higher precision we decide that one of the other possibilities occurs. 7 For any real number a and any two unequal real numbers b1 and b2 , choose either b1 or b2 as a number unequal to a. Because we know that b1 and b2 are distinct, by computation to sufficiently high precision the intervals b1 and b2 will not intersect, and a certain rational width will separate them. 4 Some solvable and nonsolvable decimal place problems 29 that the a width < , then a can intersect at most one of the b1 and b2 intervals, and so at least one of the b1 and b2 numbers is found to be unequal to a.

Then it is always true that a = b or a = b, because two intervals either intersect or they do not intersect. 7 Computing standard functions in range arithmetic 17 In this textbook, because range arithmetic is the computer arithemetic used, the phrase “if a < b, then” should be interpreted as meaning “if at the current precision a < b, then”. Similarly, the phrase “if a = b, then” should be interpreted as “if at the current precision a = b, then”. 7 Computing standard functions in range arithmetic After the four arithmetic operations + − × and ÷ have been coded for range arithmetic, one can compose simple programs to use this arithmetic.