
Understanding Numbering Systems
You can use devices such as computers, cell phones, microwave ovens, and automobiles without understanding how they work internally. Likewise, you can write many Java programs without understanding how the data items they use are represented internally. After you learn how data items are stored, however, you gain a deeper understanding about computer programming in general and Java in particular. You also can more easily troubleshoot some types of problems that arise in your programs. The numbering system you know best is the decimal numbering system, which is based on 10 digits, 0 through 9. When you use the decimal system, no other symbols are available; if you want to express a value larger than 9, you must use multiple digits from the same pool of 10, placing them in columns. Decimal numbers are also called base 10 numbers. When you use the decimal system, you analyze a multicolumn number by mentally assigning place values to each column. The value of the rightmost column is 1, the value of the next column to the left is 10, the next column’s value is 100, and so on; you multiply the column value by 10 as you move to the left. There is no limit to the number of columns you can use; you simply add them to the left as you need to express higher values. For example, Figure B1 shows how the value 305 is represented in the decimal system. You simply multiply the digit in each column by the value of the column, and then add the values together. 100s 10s 1s 3 0 5 Value is 3 X 100 = 300 0 X 10 = 0 5 X 1 = 5  305 The binary numbering system works in the same way as the decimal numbering system, except that it uses only two digits, 0 and 1. When you use the binary system and you want to express a value greater than 1, you must use multiple columns because no single symbol represents any value other than 0 or 1. Instead of each new column to the left being 10 times greater than the previous column, each new binary column is only two times the value of the previous column. Binary numbers are called base 2 numbers. For example, Figure shows how the decimal number 9 is represented in the binary system. Notice that both the binary and decimal systems allow you to create numbers with 0 in one or more columns. As with the decimal system, the binary system has no limit to the number of columns—you can use as many as it takes to express a value. For example, Figure B3 shows that the decimal number 50 requires six binary system columns. 8s 4s 2s 1s 1 0 0 1 Value is 1 X 8 = 8 0 X 4 = 0 0 X 2 = 0 1 X 1 = 1  9 32s 16s 8s 4s 2s 1s 1 1 0 0 1 0 Value is 1 X 32 = 32 1 X 16 = 16 0 X 8 = 0 0 X 4 = 0 1 X 2 = 2 0 X 1 = 0  50 A computer stores every piece of data it uses as a set of 0s and 1s. Each 0 or 1 is known as a bit, which is short for binary digit. Every computer uses 0s and 1s because all its values are stored as electronic signals that are either on or off. This twostate system is most easily represented using just two digits. 
Representing Numeric Values
In Chapter 2, you learned that a floatingpoint number contains decimal positions. The term floating point comes from the fact that the decimal point can be at any location in the stored value, allowing a much larger range of possible values to be stored in the same amount of memory. For example, assume that a computer could store only four digits and that the decimal point had to fall after the first two. The positive values that could be stored would then range from 00.00 through 99.99. However, if the decimal point could fall anywhere, the values could range from .0000 through 9999. Computers use more storage for each value, and store negative values as well, but the principle is the same. Because of the binary nature of computers, representing floatingpoint numbers is imprecise. For example, suppose you want to represent the value 1/10 (0.10). You could try using each of the following techniques: • If you use two bits to store the value, only four combinations are available (00, 01, 10, and 11), so they can only represent 0/4, 1/4, 2/4 (or 1/2), and 3/4. None of these is exactly 1/10, but 0/4 is the closest. • Suppose you use three bits. This allows twice as many combinations, or eight, and the closest to 1/10 is 1/8. The approximation is closer than with two bits, but still not exact. • Suppose you use four bits, which allows 16 combinations. The closest value to 1/10 is 2/16. This value is no closer to 1/10 than you could achieve with three bits. • Suppose you use eight bits. Now, there are 256 bit combinations from 0/256 through 255/256. The value of 26/256, at 0.1015625, is closer than any of the other values so far, but it’s still not exact. • No matter how many bits you add to the representation, doubling the number of combinations each time, you can never express 0.1 exactly. Although you cannot store 0.1 exactly, you can still display it. For example, the following two lines of code display 0.1 as expected: double oneTenth = 0.1; System.out.println(oneTenth); When Java displays a floatingpoint number, it always displays at least one digit after the decimal. After that, it uses only as many digits as necessary to distinguish the number from the nearest floatingpoint value it can represent. However, when you use 0.1 in an arithmetic statement, the imprecision becomes evident. Figure B4 shows a simple program that declares two variables named oneTenth and threeTenths; the variables contain the values 0.1 and 0.3, respectively.import java.util.Scanner; public class FloatingPointTest { public static void main(String[] args) { double oneTenth = 0.1; double threeTenths = 0.3; System.out.println(oneTenth + oneTenth + oneTenth); System.out.println(oneTenth + oneTenth + oneTenth == threeTenths); } }For many purposes, you do not care about the small imprecisions generated by floatingpoint calculations, but sometimes they can make a difference. For example, several popular movies have used the idea that small amounts of extra money can be sliced off bank balances when compounding interest and then siphoned to a criminal’s account. Many programmers recommend that you use the Java class BigDecimal when working with monetary or scientific values where precision is important. Additionally, be aware that when you test two floatingpoint values for equivalency, you might not get the expected results. When precision is not an issue, but betterlooking output is important, you can format the output to eliminate the small imprecisions that occur far to the right of the decimal point. Appendix C teaches you many techniques for formatting output to a desired number of decimal places. 
Representing Character Values
The characters used in Java are represented in Unicode, which is a 16bit coding scheme for characters. For example, the letter A actually is stored in computer memory as a set of 16 zeros and ones as 0000 0000 0100 0001 (a space is inserted after each set of four digits for readability). Because 16digit numbers are difficult to read, programmers often use a shorthand notation called the hexadecimal numbering system, or base 16. The hexadecimal system uses 16 values, 0 through 9 and A through F, to represent the decimal values 0 through 15. In hexadecimal shorthand, binary 0000 becomes 0, binary 0100 becomes 4, and binary 0001 becomes 1, so the letter A is represented in hexadecimal as 0041. You tell the compiler to treat the fourdigit hexadecimal 0041 as a single character by preceding it with the \u escape sequence. Therefore, each of the following declarations stores the character A: char letter = 'A'; char letter = '\u0041'; char letter = 65; The options that use hexadecimal and decimal values are more difficult and confusing to use than the first method, so it is not recommended that you store letters of the alphabet using numeric values. However, you can produce some interesting output using the Unicode format. For example, the sequence ‘\u0007’ produces a belllike noise if you send it to output. Letters from foreign alphabets that use characters instead of letters (Greek, Hebrew, Chinese, and so on) and other special symbols (foreign currency symbols, mathematical symbols, geometric shapes, and so on) are available using Unicode, but not on a standard keyboard, so it may be important that you know how to use Unicode characters. For more information about Unicode, go to www.unicode.org. In the United States, the most widely used character set traditionally has been ASCII (American Standard Code for Information Interchange). The ASCII character set contains 128 characters. You can create any Unicode character by adding eight 0s to the beginning of its ASCII character equivalent. This means that the decimal value of any ASCII character is the same as that of the corresponding Unicode character. For example, B has the value 66 in both character sets. The decimal values are important because they allow you to show nonprintable characters, such as a carriage return, in decimal codes. Also, the numeric values of the coding schemes are used when a computer sorts numbers and strings. When you sort characters in ascending order, for example, numbers are sorted first (because their Unicode values begin with decimal code 48), followed by capital letters (starting with decimal 65) and then lowercase letters (starting with decimal 97). Chapter 2 contains a list of Unicode values for some commonly used characters. For a complete list, see www.unicode.org/charts. There you will find Greek, Armenian, Hebrew, Tagalog, Cherokee, and a host of other character sets. Unicode also contains characters for mathematical symbols, geometric shapes, and other unusual characters. The ASCII character set is more limited than Unicode; it contains only letters and symbols used in the English language.