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CSE202: Lecture 4 The Ohio State University 2
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 <cmath>
• 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.
CSE202: Lecture 4 The Ohio State University 3
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;
CSE202: Lecture 4 The Ohio State University 4
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 4th power.
CSE202: Lecture 4 The Ohio State University 5
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
CSE202: Lecture 4 The Ohio State University 6
Mathematical Functions (5)
• Functions can themselves be part of an expression.
double x = 10;
cout << 5 * pow( (x - 3), 2.0 ); //output?
CSE202: Lecture 4 The Ohio State University 7
Common Math FunctionsFunction Returned Value Data Type of
Returned Value
abs(a) Absolute value of a Data type of a
pow(b,e) Value of b raised to the eth 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 ath power double
CSE202: Lecture 4 The Ohio State University 8
Example: sqrt()#include <iostream>#include <cmath>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;}
CSE202: Lecture 4 The Ohio State University 9
> distance.exeEnter x, y: 3 4Distance of (3,4) to (0,0) = 5
> distance.exeEnter x, y: 5 8Distance 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;
…
CSE202: Lecture 4 The Ohio State University 10
Example 2: sqrt()#include <iostream>#include <cmath> // math function libraryusing 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;}
CSE202: Lecture 4 The Ohio State University 11
> gravity.exeEnter height: 100It will take 2.49222 seconds to fall 100 feet.
> gravity.exeEnter height (feet): 500It 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;
…
CSE202: Lecture 4 The Ohio State University 12
Example: log()#include <iostream>#include <cmath>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;}
CSE202: Lecture 4 The Ohio State University 13
> interest.exeEnter annual interest rate (percentage): 7Your money will double in 10.2448 years.
> interest.exeEnter annual interest rate (percentage): 2Your 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;…
Trigonometric Math Functions
• Trigonometric math functions (sin, cos, tan) require input in radians.
CSE202: Lecture 4 The Ohio State University 14
CSE202: Lecture 4 The Ohio State University 15
Example: sin(), cos()#include <iostream>#include <cmath>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;}
CSE202: Lecture 4 The Ohio State University 16
> rotate_radians.exeEnter rotation angle (radians): 1.57Point (1,0) rotates to point (0.000796327,1)
> rotate_radians.exeEnter rotation angle (radians): 0.78Point (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;…
CSE202: Lecture 4 The Ohio State University 17
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”).
Constant ValueM_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
CSE202: Lecture 4 The Ohio State University 18
degrees2radians.cpp#include <iostream>#include <cmath> // cmath contains definitions of math constantsusing 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;}
CSE202: Lecture 4 The Ohio State University 19
rotate_degrees.cpp#include <iostream>#include <cmath> // cmath contains definitions of math constantsusing 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;}
CSE202: Lecture 4 The Ohio State University 20
> rotate_degrees.exeEnter rotation angle (degrees): 90Point (1,0) rotates to point (6.12323e-17,1)
> rotate_degrees.exeEnter rotation angle (degrees): 45Point (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;…
CSE202: Lecture 4 The Ohio State University 21
log_2.cpp#include <iostream>#include <cmath> // 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;}
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 4 The Ohio State University 22
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 4 The Ohio State University 23
CSE202: Lecture 4 The Ohio State University 24
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;}
CSE202: Lecture 4 The Ohio State University 25
> g++ logError.cpp –o logError.exeCompiling 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;
…
CSE202: Lecture 4 The Ohio State University 26
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;}
CSE202: Lecture 4 The Ohio State University 27
> 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;
…
CSE202: Lecture 4 The Ohio State University 29
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.
CSE202: Lecture 4 The Ohio State University 30
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
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 4 The Ohio State University 31
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 4 The Ohio State University 32
CSE202: Lecture 4 The Ohio State University 33
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;}
CSE202: Lecture 4 The Ohio State University 34
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;}
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 4 The Ohio State University 37
CSE202: Lecture 4 The Ohio State University 38
Operator Precedence and Associativity
• The minus sign can also be used as a unary operator that serves to negate the sign of a number
Precedence Associativity
Unary -
* / % Left to right
+ - Left to right
CSE202: Lecture 4 The Ohio State University 39
arithmetic3.cpp// Precedence of arithmetic operators
#include <iostream>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;}
Math in C++• Use #include<cmath> 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 4 The Ohio State University 40