Pure Programmer
Math Library
While the arithmetic and logic operators represent the basic operations that our [[CPU]] can perform, there are many other common math functions that com in handy. Since they are so common, programming languages usually have a math library that provides these functions. Logarithms, trigonometry and random number generation are just of few of the types of functions typically provided.
Math Constants
Math constants provide commonly used mathematical constanst to the highest precesion available. Some of the more useful math constants are summarized below.
| Constant | Description |
|---|---|
| E | Euler's constant [[ℯ]], base of the natural logarithm |
| PI | [[Pi]], Ratio of a circle's circumference to its diameter |
Math Functions
These most useful math functions are summarized below.
| Function | Description |
|---|---|
| abs(x) | [[Absolute value]] of x |
| acos(x) | [[Arc cosine]] of x, result is in the range [0,π] [[Radians]] |
| acosh(x) | [[Arc hyperbolic cosine]] of x |
| asin(x) | [[Arc sine]] of x, result is in the range [-π/2,π/2] [[Radians]] |
| asinh(x) | [[Arc hyperbolic sine]] of x |
| atan(x) | [[Arc tangent]] of x, result is in the range [-π/2,π/2] [[Radians]] |
| atan2(y,x) | Angle θ from the conversion of [[rectangular coordinates]] (x,y), result is in the range [-π/2,π/2] [[Radians]] |
| atanh(x) | [[Arc hyperbolic tangent]] of x |
| ceil(x) | Smallest integer value greater than x |
| cos(x) | [[Cosine]] of x (in [[Radians]]) |
| cosh(x) | [[Hyperbolic cosine]] of x |
| exp(x) | [[ℯ]] rasied to the power x, i.e. ℯx |
| floor(x) | Largest integer less than x |
| log(x) | [[Natural logarithm]] of x |
| log10(x) | [[Common logarithm]] of x |
| max(x,y) | Larger of x and y |
| min(x,y) | Smaller of x and y |
| pow(x,y) | x raised to the power y, i.e. xy |
| random() | [[Pseudorandom]] number on the interval [0,1) |
| round(x) | Nearest integer to x |
| sin(x) | [[Sine]] of x (in [[Radians]]) |
| sqrt(x) | [[Square root]] of x |
| tan(x) | [[Tangent]] of x (in [[Radians]]) |
| tanh(x) | [[Hyperbolic tangent]] of x |
Be sure to add the following import to the top of your program in order to use these math functions and constants.
import static java.lang.Math.*;
Math1.java
/******************************************************************************
* This program demonstrates the math library.
*
* Copyright © 2016 Richard Lesh. All rights reserved.
*****************************************************************************/
import org.pureprogrammer.Utils;
public class Math1 {
public static void main(String[] args) {
final double a = Math.PI / 6;
final double b = Math.PI / 4;
final double c = -a * 2;
final double d = -b * 2;
final double e = Math.E;
System.out.println(Utils.format("pi = {0:f}", Math.PI));
System.out.println(Utils.format("e = {0:f}", Math.E));
// abs, floor, ceil, round, trunc, min, max
System.out.println(Utils.format("abs({0:f}) = {1:f}", a, Math.abs(a)));
System.out.println(Utils.format("abs({0:f}) = {1:f}", c, Math.abs(c)));
System.out.println(Utils.format("floor({0:f}) = {1:f}", a, Math.floor(a)));
System.out.println(Utils.format("floor({0:f}) = {1:f}", c, Math.floor(c)));
System.out.println(Utils.format("ceil({0:f}) = {1:f}", a, Math.ceil(a)));
System.out.println(Utils.format("ceil({0:f}) = {1:f}", c, Math.ceil(c)));
System.out.println(Utils.format("round({0:f}) = {1:f}", a, Math.round(a)));
System.out.println(Utils.format("round({0:f}) = {1:f}", c, Math.round(c)));
System.out.println(Utils.format("trunc({0:f}) = {1:f}", a, a < 0.0 ? Math.ceil(a) : Math.floor(a)));
System.out.println(Utils.format("trunc({0:f}) = {1:f}", c, c < 0.0 ? Math.ceil(c) : Math.floor(c)));
System.out.println(Utils.format("min({0:f}, {1:f}) = {2:f}", a, c, Math.min(a, c)));
System.out.println(Utils.format("max({0:f}, {1:f}) = {2:f}", a, c, Math.max(a, c)));
// sin, cos, tan, atan, atan2, acos, asin
System.out.println(Utils.format("sin({0:f}) = {1:f}", a, Math.sin(a)));
System.out.println(Utils.format("sin({0:f}) = {1:f}", b, Math.sin(b)));
System.out.println(Utils.format("sin({0:f}) = {1:f}", c, Math.sin(c)));
System.out.println(Utils.format("sin({0:f}) = {1:f}", d, Math.sin(d)));
System.out.println(Utils.format("cos({0:f}) = {1:f}", a, Math.cos(a)));
System.out.println(Utils.format("cos({0:f}) = {1:f}", b, Math.cos(b)));
System.out.println(Utils.format("cos({0:f}) = {1:f}", c, Math.cos(c)));
System.out.println(Utils.format("cos({0:f}) = {1:f}", d, Math.cos(d)));
System.out.println(Utils.format("tan({0:f}) = {1:f}", a, Math.tan(a)));
System.out.println(Utils.format("tan({0:f}) = {1:f}", b, Math.tan(b)));
System.out.println(Utils.format("tan({0:f}) = {1:f}", c, Math.tan(c)));
System.out.println(Utils.format("asin({0:f}) = {1:f}", Math.sin(a), Math.asin(Math.sin(a))));
System.out.println(Utils.format("asin({0:f}) = {1:f}", Math.sin(b), Math.asin(Math.sin(b))));
System.out.println(Utils.format("asin({0:f}) = {1:f}", Math.sin(c), Math.asin(Math.sin(c))));
System.out.println(Utils.format("asin({0:f}) = {1:f}", Math.sin(d), Math.asin(Math.sin(d))));
System.out.println(Utils.format("acos({0:f}) = {1:f}", Math.cos(a), Math.acos(Math.cos(a))));
System.out.println(Utils.format("acos({0:f}) = {1:f}", Math.cos(b), Math.acos(Math.cos(b))));
System.out.println(Utils.format("acos({0:f}) = {1:f}", Math.cos(c), Math.acos(Math.cos(c))));
System.out.println(Utils.format("acos({0:f}) = {1:f}", Math.cos(d), Math.acos(Math.cos(d))));
System.out.println(Utils.format("atan({0:f}) = {1:f}", Math.tan(a), Math.atan(Math.tan(a))));
System.out.println(Utils.format("atan({0:f}) = {1:f}", Math.tan(b), Math.atan(Math.tan(b))));
System.out.println(Utils.format("atan({0:f}) = {1:f}", Math.tan(c), Math.atan(Math.tan(c))));
// 45 degrees
System.out.println(Utils.format("atan2({0:f}, {1:f}) = {2:f}", 1.0, 1.0, Math.atan2(1.0, 1.0)));
// 30 degrees
System.out.println(Utils.format("atan2({0:f}, {1:f}) = {2:f}", 1.0, Math.sqrt(3.0), Math.atan2(1.0, Math.sqrt(3.0))));
// sinh, cosh, tanh, atanh, acosh, asinh
System.out.println(Utils.format("sinh({0:f}) = {1:f}", a, Math.sinh(a)));
System.out.println(Utils.format("sinh({0:f}) = {1:f}", b, Math.sinh(b)));
System.out.println(Utils.format("sinh({0:f}) = {1:f}", c, Math.sinh(c)));
System.out.println(Utils.format("sinh({0:f}) = {1:f}", d, Math.sinh(d)));
System.out.println(Utils.format("cosh({0:f}) = {1:f}", a, Math.cosh(a)));
System.out.println(Utils.format("cosh({0:f}) = {1:f}", b, Math.cosh(b)));
System.out.println(Utils.format("cosh({0:f}) = {1:f}", c, Math.cosh(c)));
System.out.println(Utils.format("cosh({0:f}) = {1:f}", d, Math.cosh(d)));
System.out.println(Utils.format("tanh({0:f}) = {1:f}", a, Math.tanh(a)));
System.out.println(Utils.format("tanh({0:f}) = {1:f}", b, Math.tanh(b)));
System.out.println(Utils.format("tanh({0:f}) = {1:f}", c, Math.tanh(c)));
System.out.println(Utils.format("tanh({0:f}) = {1:f}", d, Math.tanh(d)));
System.out.println(Utils.format("asinh({0:f}) = {1:f}", Math.sinh(a), Math.log(Math.sinh(a) + Math.sqrt(Math.sinh(a) * Math.sinh(a) + 1.0))));
System.out.println(Utils.format("asinh({0:f}) = {1:f}", Math.sinh(b), Math.log(Math.sinh(b) + Math.sqrt(Math.sinh(b) * Math.sinh(b) + 1.0))));
System.out.println(Utils.format("asinh({0:f}) = {1:f}", Math.sinh(c), Math.log(Math.sinh(c) + Math.sqrt(Math.sinh(c) * Math.sinh(c) + 1.0))));
System.out.println(Utils.format("asinh({0:f}) = {1:f}", Math.sinh(d), Math.log(Math.sinh(d) + Math.sqrt(Math.sinh(d) * Math.sinh(d) + 1.0))));
System.out.println(Utils.format("acosh({0:f}) = {1:f}", Math.cosh(a), Math.log(Math.cosh(a) + Math.sqrt(Math.cosh(a) * Math.cosh(a) - 1.0))));
System.out.println(Utils.format("acosh({0:f}) = {1:f}", Math.cosh(b), Math.log(Math.cosh(b) + Math.sqrt(Math.cosh(b) * Math.cosh(b) - 1.0))));
System.out.println(Utils.format("acosh({0:f}) = {1:f}", Math.cosh(c), Math.log(Math.cosh(c) + Math.sqrt(Math.cosh(c) * Math.cosh(c) - 1.0))));
System.out.println(Utils.format("acosh({0:f}) = {1:f}", Math.cosh(d), Math.log(Math.cosh(d) + Math.sqrt(Math.cosh(d) * Math.cosh(d) - 1.0))));
System.out.println(Utils.format("atanh({0:f}) = {1:f}", Math.tanh(a), 0.5 * Math.log((Math.tanh(a) + 1.0)/(Math.tanh(a) - 1.0))));
System.out.println(Utils.format("atanh({0:f}) = {1:f}", Math.tanh(b), 0.5 * Math.log((Math.tanh(b) + 1.0)/(Math.tanh(b) - 1.0))));
System.out.println(Utils.format("atanh({0:f}) = {1:f}", Math.tanh(c), 0.5 * Math.log((Math.tanh(c) + 1.0)/(Math.tanh(c) - 1.0))));
System.out.println(Utils.format("atanh({0:f}) = {1:f}", Math.tanh(d), 0.5 * Math.log((Math.tanh(d) + 1.0)/(Math.tanh(d) - 1.0))));
// log, log10, exp, pow, sqrt
System.out.println(Utils.format("log({0:f}) = {1:f}", a, Math.log(a)));
System.out.println(Utils.format("log({0:f}) = {1:f}", b, Math.log(b)));
System.out.println(Utils.format("log({0:f}) = {1:f}", -c, Math.log(-c)));
System.out.println(Utils.format("log({0:f}) = {1:f}", -d, Math.log(-d)));
System.out.println(Utils.format("log({0:f}) = {1:f}", e, Math.log(e)));
System.out.println(Utils.format("log10({0:f}) = {1:f}", a, Math.log10(a)));
System.out.println(Utils.format("log10({0:f}) = {1:f}", b, Math.log10(b)));
System.out.println(Utils.format("log10({0:f}) = {1:f}", -c, Math.log10(-c)));
System.out.println(Utils.format("log10({0:f}) = {1:f}", -d, Math.log10(-d)));
System.out.println(Utils.format("log10({0:f}) = {1:f}", e, Math.log10(e)));
System.out.println(Utils.format("exp({0:f}) = {1:f}", 0.5, Math.exp(0.5)));
System.out.println(Utils.format("exp({0:f}) = {1:f}", 1.0, Math.exp(1.0)));
System.out.println(Utils.format("exp({0:f}) = {1:f}", 2.0, Math.exp(2.0)));
System.out.println(Utils.format("pow({0:f}, {1:f}) = {2:f}", 10.0, 0.5, Math.pow(10.0, 0.5)));
System.out.println(Utils.format("pow({0:f}, {1:f}) = {2:f}", 10.0, 1.0, Math.pow(10.0, 1.0)));
System.out.println(Utils.format("pow({0:f}, {1:f}) = {2:f}", 10.0, 2.0, Math.pow(10.0, 2.0)));
System.out.println(Utils.format("sqrt({0:f}) = {1:f}", 0.5, Math.sqrt(0.5)));
System.out.println(Utils.format("sqrt({0:f}) = {1:f}", 2.0, Math.sqrt(2.0)));
System.out.println(Utils.format("sqrt({0:f}) = {1:f}", 10.0, Math.sqrt(10.0)));
// random numbers
System.out.println(Utils.format("random() = {0:f}", Math.random()));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
}
}
Output
$ javac -Xlint Math1.java
$ java -ea Math1
pi = 3.141593
e = 2.718282
abs(0.523599) = 0.523599
abs(-1.047198) = 1.047198
floor(0.523599) = 0.000000
floor(-1.047198) = -2.000000
ceil(0.523599) = 1.000000
ceil(-1.047198) = -1.000000
round(0.523599) = 1.000000
round(-1.047198) = -1.000000
trunc(0.523599) = 0.000000
trunc(-1.047198) = -1.000000
min(0.523599, -1.047198) = -1.047198
max(0.523599, -1.047198) = 0.523599
sin(0.523599) = 0.500000
sin(0.785398) = 0.707107
sin(-1.047198) = -0.866025
sin(-1.570796) = -1.000000
cos(0.523599) = 0.866025
cos(0.785398) = 0.707107
cos(-1.047198) = 0.500000
cos(-1.570796) = 0.000000
tan(0.523599) = 0.577350
tan(0.785398) = 1.000000
tan(-1.047198) = -1.732051
asin(0.500000) = 0.523599
asin(0.707107) = 0.785398
asin(-0.866025) = -1.047198
asin(-1.000000) = -1.570796
acos(0.866025) = 0.523599
acos(0.707107) = 0.785398
acos(0.500000) = 1.047198
acos(0.000000) = 1.570796
atan(0.577350) = 0.523599
atan(1.000000) = 0.785398
atan(-1.732051) = -1.047198
atan2(1.000000, 1.000000) = 0.785398
atan2(1.000000, 1.732051) = 0.523599
sinh(0.523599) = 0.547853
sinh(0.785398) = 0.868671
sinh(-1.047198) = -1.249367
sinh(-1.570796) = -2.301299
cosh(0.523599) = 1.140238
cosh(0.785398) = 1.324609
cosh(-1.047198) = 1.600287
cosh(-1.570796) = 2.509178
tanh(0.523599) = 0.480473
tanh(0.785398) = 0.655794
tanh(-1.047198) = -0.780714
tanh(-1.570796) = -0.917152
asinh(0.547853) = 0.523599
asinh(0.868671) = 0.785398
asinh(-1.249367) = -1.047198
asinh(-2.301299) = -1.570796
acosh(1.140238) = 0.523599
acosh(1.324609) = 0.785398
acosh(1.600287) = 1.047198
acosh(2.509178) = 1.570796
atanh(0.480473) = nan
atanh(0.655794) = nan
atanh(-0.780714) = nan
atanh(-0.917152) = nan
log(0.523599) = -0.647030
log(0.785398) = -0.241564
log(1.047198) = 0.046118
log(1.570796) = 0.451583
log(2.718282) = 1.000000
log10(0.523599) = -0.281001
log10(0.785398) = -0.104910
log10(1.047198) = 0.020029
log10(1.570796) = 0.196120
log10(2.718282) = 0.434294
exp(0.500000) = 1.648721
exp(1.000000) = 2.718282
exp(2.000000) = 7.389056
pow(10.000000, 0.500000) = 3.162278
pow(10.000000, 1.000000) = 10.000000
pow(10.000000, 2.000000) = 100.000000
sqrt(0.500000) = 0.707107
sqrt(2.000000) = 1.414214
sqrt(10.000000) = 3.162278
random() = 0.560174
random() = 0.895638
random() = 0.828598
Math2.java
/******************************************************************************
* This program demonstrates the math integer functions.
*
* Copyright © 2020 Richard Lesh. All rights reserved.
*****************************************************************************/
import org.pureprogrammer.Utils;
public class Math2 {
public static void main(String[] args) {
final int a = 5;
final int b = 10;
final int c = -2;
// abs, floor, ceil, round, trunc, min, max
System.out.println(Utils.format("abs({0:d}) = {1:d}", a, Math.abs(a)));
System.out.println(Utils.format("abs({0:d}) = {1:d}", c, Math.abs(c)));
System.out.println(Utils.format("min({0:d}, {1:d}) = {2:d}", a, b, Math.min(a, b)));
System.out.println(Utils.format("max({0:d}, {1:d}) = {2:d}", a, b, Math.max(a, b)));
System.out.println(Utils.format("min({0:d}, {1:d}) = {2:d}", b, c, Math.min(b, c)));
System.out.println(Utils.format("max({0:d}, {1:d}) = {2:d}", b, c, Math.max(b, c)));
// random numbers
System.out.println(Utils.format("random({0:d}) = {1:d}", a, (int)(a * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", a, (int)(a * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", a, (int)(a * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", a, (int)(a * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", a, (int)(a * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", b, (int)(b * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", b, (int)(b * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", b, (int)(b * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", b, (int)(b * Math.random())));
System.out.println(Utils.format("random({0:d}) = {1:d}", b, (int)(b * Math.random())));
System.out.println(Utils.format("random(2) = {0:d}", (int)(2 * Math.random())));
System.out.println(Utils.format("random(2) = {0:d}", (int)(2 * Math.random())));
System.out.println(Utils.format("random(2) = {0:d}", (int)(2 * Math.random())));
System.out.println(Utils.format("random(2) = {0:d}", (int)(2 * Math.random())));
System.out.println(Utils.format("random(2) = {0:d}", (int)(2 * Math.random())));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
System.out.println(Utils.format("random() = {0:f}", Math.random()));
}
}
Output
$ javac -Xlint Math2.java
$ java -ea Math2
abs(5) = 5
abs(-2) = 2
min(5, 10) = 5
max(5, 10) = 10
min(10, -2) = -2
max(10, -2) = 10
random(5) = 2
random(5) = 0
random(5) = 1
random(5) = 3
random(5) = 4
random(10) = 6
random(10) = 1
random(10) = 6
random(10) = 9
random(10) = 7
random(2) = 0
random(2) = 1
random(2) = 1
random(2) = 0
random(2) = 1
random() = 0.057886
random() = 0.555563
random() = 0.513039
random() = 0.894659
random() = 0.069807
Random Numbers
Explain how to generate uniform random numbers, int and fp.
Questions
- {{Write an expression that yields √5.}}
- {{Write an expression that yields 3√5.}}
- {{Write an expression that yields the secant of π/4.}}
- {{Write an expression that yields log16 100.}}
Projects
More ★'s indicate higher difficulty level.
References
- [[Java Language Specification]], Java SE 17 Edition, Gosling, et. al., 2021.
- [[Java Tutorials]]
- [[Java at TutorialsPoint]]
- Download Java at [[Amazon Corretto]], [[Azul Zulu]], [[Eclipse Temurin]] or [[Oracle JDK]]