ITERATION AND FLOATING TYPES

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[CHAP. 3

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a. The 23 bits hold the 2nd through 24th bit of the mantissa. The first bit must be a 1, so it is not stored. Thus 24 bits are represented. These 24 bits can hold 224 numbers. And 224 = 16777,216, which has 7 digits with full range, so 7 complete digits can be represented. But the last digit is in doubt because of rounding. Thus, the 32-bit f loa t type yields 6 significant digits of precision. b. The 8 bits that the 32-bit float type uses for its exponent can hold 2* = 256 different numbers. Two of these are reserved for indicating underflow and overflow, leaving 254 numbers for exponents. So an exponent can range from -126 to +127, yielding a magnitude range of 2-126 = 1.175494 x 10m3* to 2127 = 1.70141 x 1038.

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Write a program that converts inches to centimeters. For example, if the user enters 16.9 for a length in inches, the output would be 42 .9 2 6 cm. (One inch equals 2.54 centimeters.)

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We use two variables of type f 1 oat: main0 float inches, cm; tout << "Enter length in inches: "; tin >> inches; cm = 2.54*inches; tout << inches << ' inches = ' C-C cm C=C ' centimeters.\n"; )

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Simply read the input into inches, 3.15

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convert it to centimeters in cm, and output it.

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Write a program to find the integer square root of a given number. That is the largest integer whose square is less than or equal to the given number. We use an exhaustive algorithm here: find &l the positive integers whose square is less than or equal to the given number; then the largest of those is the integer square root: main0 -t float x; tout << "Enter a positive number: "; tin >> x; for (int n = 1; n*n <= x; n++) ; // the null statement tout << "The integer square root of ' << x C-C ' is ' C-C n-l << endl;

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We start with n = 1 and continue to increment n until n*n > x. When the for loop terminates, n is the smallest integer whose square is greater than x, so n- 1 is the integer square root of x. Notice the use of the null statement in the for loop. Everything that needs to be done in the loop is done within the control parts of the loop. But the semicolon is still necessary at the end of the loop.

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CHAP. 31

ITERATION AND FLOATING TYPES

Write and run a program that directly implements the quotient operator / and the remainder operator % for the division of positive integers.

The algorithm used here, applied to the fraction n/d, repeatedly subtracts the d from the n until n is less than d. At that point, the value of n will be the remainder, and the number q of iterations required to reach it will be the quotient: main0 { int n, d, q, r; tout << "Enter numerator: 'I; tin >> n; tout << "Enter denominator: 'I; tin >> d; for (q = 0, r = n; r > d; q++) r -= tout << n << I1 / I -c-c d x-c 'I = I' -c-c tout << n << I1 % I1 -c-c d << I' = I' -c-c tout << "(I' -c-c q -c-c 'I) ('I -cc d << 'I) c-c n -x endl;

d; q c-c endl; r -cc endl; + (II -cc r -c-c

II) = I

This r u n iterated 4 times: 30 - 7 = 23,23 - 7 = 16, 16 - 7 = 9, a n d 9 - 7 = 2. So the quotient is 4, a n d the remainder is 2. Note that the following relationship must always be true for integer division: (quotient) (denominator) + (remainder) = numerator