1. Chapter 4 Mathematical Functions, Characters, and Strings

2. Motivations
Suppose you need to estimate the area enclosed by four cities, given the GPS locations (latitude and longitude) of these cities, as shown in the following diagram. How would you write a program to solve this problem? You will be able to write such a program after completing this chapter.

3. Objectives
To solve mathematics problems by using the methods in the Math class (§4.2).
To represent characters using the char type (§4.3).
To encode characters using ASCII and Unicode (§4.3.1).
To represent special characters using the escape sequences (§4.4.2).
To cast a numeric value to a character and cast a character to an integer (§4.3.3).
To compare and test characters using the static methods in the Character class (§4.3.4).
To introduce objects and instance methods (§4.4).
To represent strings using the String objects (§4.4).
To return the string length using the length() method (§4.4.1).
To return a character in the string using the charAt(i) method (§4.4.2).
To use the + operator to concatenate strings (§4.4.3).
To read strings from the console (§4.4.4).
To read a character from the console (§4.4.5).
To compare strings using the equals method and the compareTo methods (§4.4.6).
To obtain substrings (§4.4.7).
To find a character or a substring in a string using the indexOf method (§4.4.8).
To program using characters and strings (GuessBirthday) (§4.5.1).
To convert a hexadecimal character to a decimal value (HexDigit2Dec) (§4.5.2).
To revise the lottery program using strings (LotteryUsingStrings) (§4.5.3).
To format output using the System.out.printf method (§4.6).

4. Mathematical Functions
Java provides many useful methods in the Math class for performing common mathematical functions.

5. The Math Class Class constants: Class methods: PI E
Trigonometric Methods
Exponent Methods
Service Methods:
Includes rounding min, max, abs, and random Methods

6. Trigonometric Methods Examples
Notes:
The parameter for sin, cos, and tan is an angle in radians.
The return value for asin, acos, and atan is a degree in radians in the range between -pi/2 and pi/2.
One degree is equal to pi/180 in radians, 90 degrees is equal to pi/2 in radians, and 30 degrees is equal to pi/6 in radians.
sin(double a)
cos(double a)
tan(double a)
acos(double a)
asin(double a)
atan(double a)
toDegree(double a)
toRadians(double a)

7. Trigonometric Methods Examples
Math.toDegrees(Math.PI / 2)
returns 90.0
Math.toRadians(30)
returns (same as π/6)
Math.sin(0)
returns 0.0
Math.sin(Math.toRadians(270))
returns -1.0
Math.sin(Math.PI / 6)
returns 0.5
Math.sin(Math.PI / 2)
returns 1.0

8. Trigonometric Methods
Examples:
Math.asin(0.5)
returns (same as π/6)
Math.acos(0.5)
returns (same as π/3)
Math.atan(1.0)
returns (same as π/4)

9. Exponent Methods Examples: exp(double a)
Math.exp(1) returns 2.71
Math.log(2.71) returns 1.0
Math.log10(10) returns 1.0
Math.pow(2, 3) returns 8.0
Math.pow(3, 2) returns 9.0
Math.pow(3.5, 2.5) returns
Math.sqrt(4) returns 2.0
Math.sqrt(10.5) returns 3.24
exp(double a)
Returns e raised to the power of a.
log(double a)
Returns the natural logarithm of a.
log10(double a)
Returns the 10-based logarithm of a.
pow(double a, double b)
Returns a raised to the power of b.
sqrt(double a)
Returns the square root of a.

10. Rounding Methods double ceil(double x)
x rounded up to its nearest integer. This integer is returned as a double value.
double floor(double x)
x is rounded down to its nearest integer. This integer is returned as a double value.
double rint(double x)
x is rounded to its nearest integer. If x is equally close to two integers, the even one is returned as a double.
int round(float x)
Return (int)Math.floor(x+0.5).
long round(double x)
Return (long)Math.floor(x+0.5).

11. Rounding Methods Examples
Math.ceil(2.1) returns 3.0
Math.ceil(2.0) returns 2.0
Math.ceil(-2.0) returns –2.0
Math.ceil(-2.1) returns -2.0
Math.floor(2.1) returns 2.0
Math.floor(2.0) returns 2.0
Math.floor(-2.0) returns –2.0
Math.floor(-2.1) returns -3.0
Math.rint(2.1) returns 2.0
Math.rint(2.0) returns 2.0
Math.rint(-2.0) returns –2.0
Math.rint(-2.1) returns -2.0
Math.rint(2.5) returns 2.0
Math.rint(-2.5) returns -2.0
Math.round(2.6f) returns 3
Math.round(2.0) returns 2
Math.round(-2.0f) returns -2
Math.round(-2.6) returns -3

12. min, max, and abs max(a, b)and min(a, b) abs(a) random() Examples:
Returns the maximum or minimum of two parameters.
abs(a)
Returns the absolute value of the parameter.
random()
Returns a random double value in the range [0.0, 1.0).
Examples:
Math.max(2, 3) returns 3
Math.max(2.5, 3) returns 3.0
Math.min(2.5, 3.6) returns 2.5
Math.abs(-2) returns 2
Math.abs(-2.1) returns 2.1

13. The random Method
Generates a random double value greater than or equal to 0.0 and less than 1.0 (0 <= Math.random() < 1.0).
Examples:
In general,

14. Case Study: Computing Angles of a Triangle
Write a program that prompts the user to enter the x- and y-coordinates of the three corner points in a triangle and then displays the triangle’s angles.
Note: distance between two points (x1, y1) and (x2, y2) can be computed using the formula
√((𝑥2−𝑥1)2+(𝑦2−𝑦1)2)

15. 1 import java.util.Scanner;
public class ComputeAngles {
public static void main(String[] args) {
Scanner input = new Scanner(System.in);
// Prompt the user to enter three points
System.out.print("Enter three points: ");
double x1 = input.nextDouble();
double y1 = input.nextDouble();
double x2 = input.nextDouble();
double y2 = input.nextDouble();
double x3 = input.nextDouble();
double y3 = input.nextDouble();
16 // Compute three sides
17 double a = Math.sqrt((x2 - x3) * (x2 - x3)
(y2 - y3) * (y2 - y3));
19 double b = Math.sqrt((x1 - x3) * (x1 - x3)
(y1 - y3) * (y1 - y3));

16. 20 double c = Math.sqrt((x1 - x2) * (x1 - x2)
(y1 - y2) * (y1 - y2));
// Compute three angles
double A = Math.toDegrees(Math.acos((a * a - b * b - c * c)
/ (-2 * b * c)));
double B = Math.toDegrees(Math.acos((b * b - a * a - c * c)
/ (-2 * a * c)));
double C = Math.toDegrees(Math.acos((c * c - b * b - a * a)
/ (-2 * a * b)));
// Display results
System.out.println("The three angles are " +
Math.round(A * 100) / " " +
Math.round(C * 100) / 100.0);
Math.round(B * 100) / " " + }

17. Output Enter three points: 1 1 6.5 1 6.5 2.5
The three angles are

18. Caution
The Math class is used in the program, but not imported, because it is in the java. lang package. All the classes in the java.lang package are implicitly imported in a Java program.
The program prompts the user to enter three points (line 6). This prompting message is not clear. You should give the user explicit instructions on how to enter these points as follows:
System.out.print("Enter the coordinates of three points separated " + "by spaces like x1 y1 x2 y2 x3 y3: ");

19. Character Data Type char letter = 'A'; (ASCII)
A character data type represents a single character
A string literal must be enclosed in quotation marks (" "). A character literal is a single character enclosed in single quotation marks (' '). Therefore, "A" is a string, but 'A' is a character.
char letter = 'A'; (ASCII)
char numChar = '4'; (ASCII)
char letter = '\u0041'; (Unicode)
char numChar = '\u0034'; (Unicode)
Four hexadecimal digits.

20. Character Data Type
NOTE: The increment and decrement operators can also be used on char variables to get the next or preceding Unicode character. For example, the following statements display character b.
char ch = 'a';
System.out.println(++ch);

21. Unicode Format
Java characters use Unicode, a 16-bit encoding scheme established by the Unicode Consortium to support the interchange, processing, and display of written texts in the world’s diverse languages. Unicode takes two bytes, preceded by \u, expressed in four hexadecimal numbers that run from '\u0000' to '\uFFFF'. So, Unicode can represent characters.
Unicode \u03b1 \u03b2 \u03b3 for three Greek letters

22. ASCII Code for Commonly Used Characters

23. Escape Sequences for Special Characters
Can You Write the following?
System.out.println("He said "Java is fun"");
Compilation error

24. Escape Sequences for Special Characters
Examples :
System.out.println("He said \"Java is fun\"");
The output is:
He said "Java is fun“
System.out.println("\\ is a backslash character");
\ is a backslash character

25. Appendix B: ASCII Character Set
ASCII Character Set is a subset of the Unicode from \u0000 to \u007f

26. ASCII Character Set, cont.
ASCII Character Set is a subset of the Unicode from \u0000 to \u007f

27. Casting between char and Numeric Types
A char can be cast into any numeric type, and vice versa.
int i = 'a'; // Same as int i = (int)'a';
char c = 97; // Same as char c = (char)97;
When an integer is cast into a char, only its lower 16 bits of data are used; the other part is ignored
char ch = (char)0XAB0041; // The lower 16 bits hex code 0041 is // assigned to ch
System.out.println(ch); // ch is character A

28. Casting between char and Numeric Types
When a floating-point value is cast into a char, the floating-point value is first cast into an int, which is then cast into a char.
char ch = (char)65.25; // Decimal 65 is assigned to ch System.out.println(ch); // ch is character A
When a char is cast into a numeric type, the character’s Unicode is cast into the specified numeric type.
int i = (int)'A'; // The Unicode of character A is assigned to i
System.out.println(i); // i is 65

29. Casting between char and Numeric Types
All numeric operators can be applied to char operands.
A char operand is automatically cast into a number if the other operand is a number or a character.
If the other operand is a string, the character is concatenated with the string.

30. Casting between char and Numeric Types
Example:
i = '2' + '3';
// (int)'2' is 50 and (int)'3' is 51
System.out.println("i is " + i);
// i is 101
int j = 2 + 'a';
// (int)'a' is 97
System.out.println("j is " + j);
// j is 99 System.out.println(j + " is the Unicode for character " + (char)j);
// 99 is the Unicode for character c
System.out.println("Chapter " + '2');

31. Output:
i is 101
j is 99
99 is the Unicode for character c
Chapter 2

32. Comparing and Testing Characters
if (ch >= 'A' && ch <= 'Z')
System.out.println(ch + " is an uppercase letter");
else if (ch >= 'a' && ch <= 'z')
System.out.println(ch + " is a lowercase letter");
else if (ch >= '0' && ch <= '9')
System.out.println(ch + " is a numeric character");

33. Methods in the Character Class

34. Methods in the Character Class
Example:
System.out.println("isDigit('a') is " + Character.isDigit('a'));
System.out.println("isLetter('a') is " + Character.isLetter('a'));
System.out.println("isLowerCase('a') is " + Character.isLowerCase('a'));
System.out.println("isUpperCase('a') is " + Character.isUpperCase('a'));
System.out.println("toLowerCase('T') is " + Character.toLowerCase('T'));
System.out.println("toUpperCase('q') is " + Character.toUpperCase('q'));

35. Output: isDigit('a') is false isLetter('a') is true
isLowerCase('a') is true
isUpperCase('a') is false
toLowerCase('T') is t
toUpperCase('q') is Q

36. The String Type
The char type only represents one character. To represent a string of characters, use the data type called String. For example,
String message = "Welcome to Java";
String is actually a predefined class in the Java library just like the System class and Scanner class. The String type is not a primitive type. It is known as a reference type. Any Java class can be used as a reference type for a variable.

37. Simple Methods for String Objects

38. Simple Methods for String Objects
Strings are objects in Java. The methods in the preceding table can only be invoked from a specific string instance. For this reason, these methods are called instance methods. A non-instance method is called a static method. A static method can be invoked without using an object. All the methods defined in the Math class are static methods. They are not tied to a specific object instance. The syntax to invoke an instance method is
referenceVariable.methodName(arguments).

39. Getting String Length String message = "Welcome to Java";
System.out.println("The length of " + message + " is "
+ message.length());
Output :
The length of Welcome to Java is 15

40. Getting Characters from a String
String message = "Welcome to Java";
System.out.println("The first character in message is "
+ message.charAt(0));

41. Caution
The s.charAt(index) method can be used to retrieve a specific character in a string s,where the index is between 0 and s.length()–1.
Attempting to access characters in a string s out of bounds is a common programming error. To avoid it, make sure that you do not use an index beyond s.length() – 1.
For example, s.charAt(s.length()) would cause a
StringIndexOutOfBoundsException compilation error

42. Converting Strings
"Welcome".toLowerCase() returns a new string, welcome.
"Welcome".toUpperCase() returns a new string, WELCOME.
The trim() method returns a new string by eliminating whitespace characters from both ends of the string.
The characters ' ',\t,\f,\r, or \n are known as whitespace characters.
For example,
"\t Good Night \n".trim() returns a new string Good Night
" Welcome ".trim() returns a new string, Welcome.

43. String Concatenation // Three strings are concatenated
String s3 = s1.concat(s2); or String s3 = s1 + s2;
// Three strings are concatenated
String message = "Welcome " + "to " + "Java";
// String Chapter is concatenated with number 2
String s = "Chapter" + 2; // s becomes Chapter2
// String Supplement is concatenated with character B
String s1 = "Supplement" + 'B'; // s1 becomes SupplementB

44. String Concatenation System.out.println("i + j is " + (i + j));
If i = 1 and j = 2,
System.out.println("i + j is " + i + j);
The output is "i + j is 12"
System.out.println("i + j is " + (i + j));
The output is "i + j is 3"

45. Reading a String from the Console
The next() method reads a string that ends with a whitespace character.
The nextLine() method reads an entire line of text. The nextLine() method reads a string that ends with the Enter key pressed.
Scanner input = new Scanner(System.in);
System.out.print("Enter three words separated by spaces: ");
String s1 = input.next();
String s2 = input.next();
String s3 = input.next();
System.out.println("s1 is " + s1);
System.out.println("s2 is " + s2);
System.out.println("s3 is " + s3);

46. Output Enter three words separated by spaces: Welcome to Java
s1 is Welcome
s2 is to
s3 is Java

47. Reading a String from the Console
Scanner input = new Scanner(System.in); System.out.println("Enter a line: ");
String s = input.nextLine();
System.out.println("The line entered is " + s);

48. Output Enter a line: Welcome to Java
The line entered is Welcome to Java

49. Reading a Character from the Console
Scanner input = new Scanner(System.in);
System.out.print("Enter a character: ");
String s = input.nextLine();
char ch = s.charAt(0);
System.out.println("The character entered is " + ch);

50. Output
Enter a character: Welcome to Java
The character entered is W

51. Comparing Strings
Run

52. Obtaining Substrings

53. Finding a Character or a Substring in a String

54. Finding a Character or a Substring in a String
int k = s.indexOf(' ');
String firstName = s.substring(0, k);
String lastName = s.substring(k + 1);

55. Conversion between Strings and Numbers
int intValue = Integer.parseInt(intString);
double doubleValue = Double.parseDouble(doubleString);
String s = number + "";

56. Problem: Guessing Birthday
The program can guess your birth date. Run to see how it works.
GuessBirthday

57. Mathematics Basis for the Game
19 is in binary. 7 is 111 in binary. 23 is in binary

58. Case Study: Converting a Hexadecimal Digit to a Decimal Value
Write a program that converts a hexadecimal digit into a decimal value.

59. Case Study: Revising the Lottery Program Using Strings
A problem can be solved using many different approaches. This section rewrites the lottery program in Listing 3.7 using strings. Using strings simplifies this program.

60. Formatting Output Use the printf statement.
System.out.printf(format, items);
Where format is a string that may consist of substrings and format specifiers. A format specifier specifies how an item should be displayed. An item may be a numeric value, character, boolean value, or a string. Each specifier begins with a percent sign.

61. Frequently-Used Specifiers
Specifier Output Example
%b a boolean value true or false
%c a character 'a'
%d a decimal integer
%f a floating-point number
%e a number in standard scientific notation e+01
%s a string "Java is cool"