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CSE202: Lecture 4The Ohio State University1 Mathematical Functions.

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1 CSE202: Lecture 4The Ohio State University1 Mathematical Functions

2 CSE202: Lecture 4The Ohio State University2 Mathematical Functions (1) The math library file cmath contains common and often used mathematical functions such as sine, cosine, log, square root, etc. To use, your program must have the following line in the header: #include As in math, functions take input and produce an output. For example, the square root function, sqrt(), needs a number input to product an output.

3 CSE202: Lecture 4The Ohio State University3 Mathematical Functions (2) These math functions can be used like a variable of the same data type: double a = 81; double b = sqrt(a); double c = sqrt(sqrt(a)) * b; Here, a is 81, b is 9, and c is 27 This statement would output 11: cout << sqrt(121.0) << endl;

4 CSE202: Lecture 4The Ohio State University4 Mathematical Functions (3) When data needs to be passed into the function, you place it inside the parentheses (). The passed data are called arguments or parameters. A function can have more than one input. For instance, the power function needs a base and an exponent to produce an output. –For example, cout << pow(3.0, 4.0); would output 81 which is 3 to the 4 th power.

5 CSE202: Lecture 4The Ohio State University5 Mathematical Functions (4) Arguments passed into any function can be a constant, variable, an expression that evaluates to the appropriate data type such as the following: double j = 2.0; cout << sqrt(j * 32.0); //outputs 8

6 CSE202: Lecture 4The Ohio State University6 Mathematical Functions (5) Functions can themselves be part of an expression. double x = 10; cout << 5 * pow( (x - 3), 2.0 ); //output?

7 CSE202: Lecture 4The Ohio State University7 Common Math Functions FunctionReturned ValueData Type of Returned Value abs(a) Absolute value of a Data type of a pow(b,e) Value of b raised to the e th power float or double Data type of b sqrt(a) Square root of a double sin(a) Sine of a in radians double cos(a) Cosine of a in radians double tan(a) Tangent of a in radians double log(a) Natural log of a double log10(a) Common log (base 10) of a double exp(a)e raised to the a th power double

8 CSE202: Lecture 4The Ohio State University8 Example: sqrt() #include using namespace std; int main() { double x(0.0), y(0.0); double dist(0.0); cout << "Enter x, y: "; cin >> x >> y; dist = sqrt(x*x + y*y); cout << "Distance of (" << x << "," << y << ") to (0,0) = " << dist << endl; return 0; }

9 CSE202: Lecture 4The Ohio State University9 > distance.exe Enter x, y: 3 4 Distance of (3,4) to (0,0) = 5 > distance.exe Enter x, y: 5 8 Distance of (5,8) to (0,0) = 9.43398 … cout << "Enter x, y: "; cin >> x >> y; dist = sqrt(x*x + y*y); cout << "Distance of (" << x << "," << y << ") to (0,0) = " << dist << endl; …

10 CSE202: Lecture 4The Ohio State University10 Example 2: sqrt() #include #include // math function library using namespace std; int main () { double height(0), time(0.0); cout << "Enter height (feet): "; cin >> height; time = sqrt(2.0 * height / 32.2); cout << "It will take " << time << " seconds to fall " << height << " feet. " << endl; return 0; }

11 CSE202: Lecture 4The Ohio State University11 > gravity.exe Enter height: 100 It will take 2.49222 seconds to fall 100 feet. > gravity.exe Enter height (feet): 500 It will take 5.57278 seconds to fall 500 feet. … cout << "Enter height (feet): "; cin >> height; time = sqrt(2.0 * height / 32.2); cout << "It will take " << time << " seconds to fall " << height << " feet. " << endl; …

12 CSE202: Lecture 4The Ohio State University12 Example: log() #include using namespace std; int main() { double rate(0.0), years(0.0); cout << "Enter annual interest rate (percentage): "; cin >> rate; years = log(2.0)/log(1+(rate/100.0)); cout << "Your money will double in " << years << " years." << endl; return 0; }

13 CSE202: Lecture 4The Ohio State University13 > interest.exe Enter annual interest rate (percentage): 7 Your money will double in 10.2448 years. > interest.exe Enter annual interest rate (percentage): 2 Your money will double in 35.0028 years. … cout << "Enter annual interest rate (percentage): "; cin >> rate; years = log(2.0)/log(1+(rate/100.0)); cout << "Your money will double in " << years << " years." << endl; …

14 Trigonometric Math Functions Trigonometric math functions (sin, cos, tan) require input in radians. CSE202: Lecture 4The Ohio State University14

15 CSE202: Lecture 4The Ohio State University15 Example: sin(), cos() #include using namespace std; int main() { double angle(0.0), x(0.0), y(0.0); cout << "Enter rotation angle (radians): "; cin >> angle; x = cos(angle); y = sin(angle); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl; return 0; }

16 CSE202: Lecture 4The Ohio State University16 > rotate_radians.exe Enter rotation angle (radians): 1.57 Point (1,0) rotates to point (0.000796327,1) > rotate_radians.exe Enter rotation angle (radians): 0.78 Point (1,0) rotates to point (0.710914,0.703279) … cout << "Enter rotation angle (radians): "; cin >> angle; x = cos(angle); y = sin(angle); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl; …

17 CSE202: Lecture 4The Ohio State University17 Common Math Constants In addition to functions, cmath also defines some commonly used math constants. These are used like variables, except that they cannot be assigned a new value (hence “constants”). ConstantValue M_PI π, 3.14159… M_E e, the base of natural logarithms M_LOG2E Base-2 logarithm of e M_LOG10E Base-10 logarithm of e M_LN2 Natural log of 2 M_LN10 Natural log of 10

18 CSE202: Lecture 4The Ohio State University18 degrees2radians.cpp #include #include // cmath contains definitions of math constants using namespace std; int main() { double degrees(0.0), radians(0.0); cout << "Enter angle in degrees: "; cin >> degrees; radians = (degrees * M_PI) / 180.0; cout << "Angle in radians = " << radians << endl; return 0; }

19 CSE202: Lecture 4The Ohio State University19 rotate_degrees.cpp #include #include // cmath contains definitions of math constants using namespace std; int main() { double degrees(0.0), radians(0.0), x(0.0), y(0.0); cout << "Enter rotation angle (degrees): "; cin >> degrees; radians = (degrees * M_PI)/180.0; x = cos(radians); y = sin(radians); cout << "Point (1,0) rotates to point (" << x << "," << y << ")" << endl; return 0; }

20 CSE202: Lecture 4The Ohio State University20 > rotate_degrees.exe Enter rotation angle (degrees): 90 Point (1,0) rotates to point (6.12323e-17,1) > rotate_degrees.exe Enter rotation angle (degrees): 45 Point (1,0) rotates to point (0.707107,0.707107) … cout << "Enter rotation angle (degrees): "; cin >> degrees; radians = (degrees * M_PI)/180.0; x = cos(radians); y = sin(radians); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl; …

21 CSE202: Lecture 4The Ohio State University21 log_2.cpp #include #include // cmath contains definitions of math constants using namespace std; int main() { double x(0.0), y(0.0); cout << "Enter number: "; cin >> x; y = log(x) / M_LN2; cout << "log_e(" << x << ") = " << log(x) << endl; cout << "log_2(" << x << ") = " << y << endl; return 0; }

22 Modulus Operator: % a mod b is the remainder after integer a is divided by integer b; In C++ a mod b is written as: a%b Examples: 25 % 3 = ? 137 % 10 = ? 2751 % 2 = ? CSE202: Lecture 4The Ohio State University22

23 Arguments to Math Functions Arguments to math functions should have type double, e.g.: double x(3.6), y(0.3), z(0.0); z = sin(1.2); z = sqrt(x); z = log(3.2 * x); z = pow(x / 0.5, 1.2 * y); CSE202: Lecture 4The Ohio State University23

24 CSE202: Lecture 4The Ohio State University24 logError.cpp... int main() { int value(0); cout << "Enter value: "; cin >> value; // log(value) generates a compiler error cout << "At 10% interest, it will take " << log(value)/log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; return 0; }

25 CSE202: Lecture 4The Ohio State University25 > g++ logError.cpp –o logError.exe Compiling logError.cpp into logError.exe. logError.cpp: In function `int main()': logError.cpp:16: call of overloaded `log(int&)' is ambiguous /usr/include/iso/math_iso.h:52: candidates are: double log(double) /usr/local/include/g++-v3/bits/std_cmath.h:333: long double std::log(long double) /usr/local/include/g++-v3/bits/std_cmath.h:323: float std::log(float) … 15. // log(value) generates a compiler error 16. cout << "At 10% interest, it will take " << log(value) / log(1.1) 17. << " years for a $1 investment to be worth $" << value << "." << endl; …

26 CSE202: Lecture 4The Ohio State University26 logExample.cpp... int main() { int value(0); double x(0.0); cout << "Enter value: "; cin >> value; x = value; // implicit conversion to double cout << "At 10% interest, it will take " << log(x)/log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; return 0; }

27 CSE202: Lecture 4The Ohio State University27 > logExample.cpp Enter value: 10 At 10% interest, it will take 24.1589 years for a $1 investment to be worth $10. > … double x(0.0);... x = value; // implicit conversion to double cout << "At 10% interest, it will take " << log(x)/log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; …

28 Mixed Mode Expressions CSE202: Lecture 4The Ohio State University28

29 CSE202: Lecture 4The Ohio State University29 Expression Types An expression that contains only integer operands is an integer expression. An expression that contains only floating point operands is a floating-point expression.

30 CSE202: Lecture 4The Ohio State University30 Mixing Expression Types A mixed-mode expression has both floating-point and integer data types. The rules governing the data type of the result are: 1.If both operands are integers, the result is an integer 2.If one operand is a floating-point number, then the result is a double

31 Mixed Mode Expressions int a(3); double x(3.5), y(5), z(0.0); z = 3.0 * 25; z = a * x; z = a * y; What about: z = x + (a/2); CSE202: Lecture 4The Ohio State University31

32 Exercises int a(3), b(2); double y(5), z(0.0); After each operation, what is z? z = (y + a) / b; z = (y * a) / b; z = y * (a / b); z = (y / b) * (a / b); z = (a + b) / (b*y); CSE202: Lecture 4The Ohio State University32

33 CSE202: Lecture 4The Ohio State University33 logError.cpp... int main() { int value(0); cout << "Enter value: "; cin >> value; // log(value) generates a compiler error cout << "At 10% interest, it will take " << log(value)/log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; return 0; }

34 CSE202: Lecture 4The Ohio State University34 logExample2.cpp... int main() { int value(0); cout << "Enter value: "; cin >> value; // Mixed mode operation "value*1.0" returns double. cout << "At 10% interest, it will take " << log(value*1.0) /log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; return 0; }

35 Operator Precedence CSE202: Lecture 4The Ohio State University35

36 Evaluate Evaluate: 1 + 3*6 – 4/2 = ??? CSE202: Lecture 4The Ohio State University36

37 Evaluate Evaluate: 20 – 16 / 2 + 2 * 3 = ??? (((20 – 16) /2) +2) *3 = 12. ((20 – 16) / (2 + 2)) * 3 = 3. 20 – ((16/2) + (2*3)) = -4. CSE202: Lecture 4The Ohio State University37

38 CSE202: Lecture 4The Ohio State University38 Operator Precedence and Associativity The minus sign can also be used as a unary operator that serves to negate the sign of a number PrecedenceAssociativity Unary - * / %Left to right + -Left to right

39 CSE202: Lecture 4The Ohio State University39 arithmetic3.cpp // Precedence of arithmetic operators #include using namespace std; int main() { cout << "-3+5*2 = " << -3+5*2 << endl << endl; // Is this? cout << "((-3)+5)*2 = " << ((-3)+5)*2 << endl; cout << "(-(3+5))*2 = " << (-(3+5))*2 << endl; cout << "(-3)+(5*2) = " << (-3)+(5*2) << endl; cout << "-(3+(5*2)) = " << -(3+(5*2)) << endl; return 0; }

40 Math in C++ Use #include for math functions; Common math functions: abs(a), pow(b,e), sqrt(a), sin(a), cos(a), tan(a), log(a), log10(a), exp(a); Common math constants: M_PI, M_E, M_LN2, M_LN10; CSE202: Lecture 4The Ohio State University40

41 Math in C++ Arguments to functions should always be double; Mixed mode operations: (3.0 + 5) or (3.0 * 5) have type double; Operator precedence: Multiplication and division before addition and subtraction. CSE202: Lecture 4The Ohio State University41

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