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It starts with n=1 and continues 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. Note 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. This implements the Euclidean Algorithm: int main() { int m, n, r; cout << "Enter two positive integers: "; cin >> m >> n; if (m < n) { int temp = m; m = n; n = temp; } // make m >= n cout << "The g.c.d. of " << m << " and " << n << " is "; while (n > 0) { r = m % n; m = n; n = r; } cout << m << endl; } Enter two positive integers: 532 112 The g.c.d. of 532 and 112 is 28
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5.1 INTRODUCTION Most useful programs are much larger than the programs that we have considered so far. To make large programs manageable, programmers modularize them into subprograms. These subprograms are called functions. They can be compiled and tested separately and reused in different programs. This modularization is characteristic of successful object-oriented software. 5.2 STANDARD C++ LIBRARY FUNCTIONS The Standard C++ Library is a collection of pre-defined functions and other program elements which are accessed through header files. We have used some of these already: the INT_MAX constant defined in <climits> (Example 2.3 on page 19), the sqrt() function defined in <cmath> (Example 2.15 on page 28), the rand() function defined in <cstdlib> (Example 4.26 on page 76), and the time() function defined in <ctime> (Example 4.28 on page 78). Our first example illustrates the use of one of these mathematical functions. EXAMPLE 5.1 The Square Root Function sqrt()
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The square root of a given positive number is the number whose square is the given number. The square root of 9 is 3 because the square of 3 is 9. We can think of the square root function as a black box. When you put in a 9, out comes a 3. When the number 2 is input, the number 1.41421 is output. This function has the same input-process-output nature that complete programs have. However, the processing step is hidden: we do not need to know what the function does to 2 to produce 1.41421. All we need to know is that the output 1.41421 does have the square root property: its square is the input 2. Here is a simple program that uses the predefined square root function: #include <cmath> // defines the sqrt() function #include <iostream> // defines the cout object using namespace std; int main() { // tests the sqrt() function: for (int x=0; x < 6; x++) cout << "\t" << x << "\t" << sqrt(x) << endl; } 0 0 1 1 2 1.41421 3 1.73205 4 2 5 2.23607 This program prints the square roots of the numbers 0 through 5. Each time the expression sqrt(x) is evaluated in the for loop, the sqrt() function is executed. Its actual code is hidden away within the Standard C++ Library. In using it, we may confidently assume that the expression sqrt(x) will be
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replaced by the actual square root of whatever value x has at that moment. Notice the directive #include <cmath> on the first line of the program. This is necessary for the compiler to find the definition of the sqrt() function. It tells the compiler that the function is declared in the <cmath> header file.
A function like sqrt() is executed by using its name as a variable in a statement, like this:
y = sqrt(x);
This is called invoking or calling the function. Thus in Example 5.1, the code sqrt(x) calls the sqrt() function. The expression x in the parentheses is called the argument or actual parameter of the function call, and we say that it is passed by value to the function. So when x is 3, the value 3 is passed to the sqrt() function by the call sqrt(x). This process is illustrated by main() this diagram. The variables x and sqrt() 3 y are declared in main() . The x 3 int value of x is passed to the sqrt() function which then returns the 1.73205 value 1.73205 back to main(). y 1.73205 double Note that the box representing the sqrt() function is shaded, indicating that its internal working mechanism is not visible. EXAMPLE 5.2 Testing a Trigonometry Identity
Here is another program that uses the <cmath> header. Its purpose is to verify empirically the identity sin2x = 2 sinx cosx. int main() { // tests the identity sin 2x = 2 sin x cos x: for (float x=0; x < 2; x += 0.2) cout << x << "\t\t" << sin(2*x) << "\t" << 2*sin(x)*cos(x) << endl; } 0 0 0 0.2 0.389418 0.389418 0.4 0.717356 0.717356 0.6 0.932039 0.932039 0.8 0.999574 0.999574 1 0.909297 0.909297 1.2 0.675463 0.675463 1.4 0.334988 0.334988 1.6 -0.0583744 -0.0583744 1.8 -0.442521 -0.442521 . The program prints x in the first column, sin 2x in the second column, and 2 sin x cos x in the third column. For each value of x tested, sin 2x = 2 sin x cos x. Of course, this does not prove the identity, but it does provide convincing empirical evidence of its truth. Note that x has type float instead of int. This allows the increment x += 0.2 to work correctly.
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