18-May-15 Numbers

2 Bits and bytes A bit is a single two-valued quantity: yes or no, true or false, on or off, high or low, good or bad One bit can distinguish between two cases: T, F Two bits can distinguish between four cases: TT, TF, FT, FF Three bits can distinguish between eight cases: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF In general, n bits can distinguish between 2 n cases A byte is 8 bits, therefore 2 8 = 256 cases

3 Number systems The binary (base 2) number system uses two “binary digits, ” (abbreviation: bits) -- 0 and 1 The octal (base 8) number system uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7 The decimal (base 10) number system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 The hexadecimal, or “hex” (base 16) number system uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

4 Everything is a number? Everything in the computer is stored as a pattern of bits Binary distinctions are easy for hardware to work with Numbers are stored as a pattern of bits Computers use the binary number system Characters are stored as a pattern of bits One byte (8 bits) can represent one of 256 characters So, is everything in the computer stored as a number? No it isn’t, it’s stored as a bit pattern There are many ways to interpret a bit pattern

5 Counting To count up in any number system, add 1 to the rightmost digit if the result is higher than the largest digit, set that digit to zero and carry to the next place repeat addition of 1 and carrying as many times as necessary Example: In hex, 4A6FF + 1 = 4A700

6 Counting in three systems Dec Bin Hex A Dec Bin Hex B C D E F

7 Computers use binary numbers People like to use decimal numbers Computers use binary numbers Java translates decimal numbers into binary The computer does all its arithmetic in binary Java translates binary results back into decimal You occasionally have to use numbers in other number systems Colors are usually specified in hexadecimal notation: #FF0000, #669966,

8 Using octal and hex numbers Computers use binary, but the numbers are too long and confusing for people--it’s easy to lose your place Octal or hex is better for people Translation between binary and octal or hex is easy One octal digit equals three binary digits One hexadecimal digit equals four binary digits B 5 C A 0 B

9 Writing octal and hex integers Integers are usually written in decimal notation: 7, 532, -28 To write a number in octal, just start with a zero: 02, 0657, but don’t use the digits 8 or 9 ! To write a number in hexadecimal, start with 0x or 0X : 0xA, 0X43AB5, -0xFFFF The “digits” A through F can be upper or lower case Uppercase is usually preferred Lowercase is more readable for long numbers

10 Integer types There are four integer types byte – occupies one byte (surprise!) Can hold numbers in range –128 to 127 short – occupies two bytes Can hold numbers in range –32768 to int – occupies four bytes Can hold numbers up to + or – 2 billion long – occupies eight bytes Can hold numbers up to about 19 digits Literals are written with an L suffix: L A lowercase L can be used, but it’s a bad idea: l

11 Floating-point literals Floating-point literals are written with a decimal point: Floating-point numbers may also be written in “scientific notation”– times a power of 10 We use E to represent “times 10 to the” Example: 4.32E5 means 4.32 x 10 5 float literals are written with an F suffix Examples: 8.5F F 5.000F 4.32E5F If you don’t have the F suffix, type double is assumed

12 Floating point types There are two floating-point types float – occupies four bytes Can hold numbers in the range 3.4E38 to 1.4E-45 Accuracy is about nine digits double – occupies eight bytes Can hold numbers in the range 1.7E308 to 4.9E-324 Accuracy is seventeen or eighteen digits

13 Number “width” Numeric types are considered wider or narrower than other numeric types This is based partly on number of bytes occupied Also based on how large a number it can hold Java doesn’t mind if you assign a narrow value to a wide variable: int n = 3; Java is not happy if you assign a wide value to a narrow variable: byte b = ; // illegal

14 Widening and narrowing You can always widen (assign a narrower type to a wider type): double wide; int narrow; wide = narrow; But if you want to narrow (assign a wider type to a narrower type), you have to cast it: narrow = (int)wide; double float long int short byte

15 Casts You can convert (cast) one numeric type to another When you widen, no explicit cast is necessary But it doesn’t hurt When you narrow, an explicit cast is required This requirement is made to help avoid errors Casting tells Java that the value in the wider type will fit in the narrower type Java checks to make sure that the cast works, and gives you an error if it didn’t

16 Example casts short s = 0; int i = 0; double d = 0.0; d = i; // legal d = s; // legal i = s; //legal i = d; // illegal s = d; // illegal s = i; // illegal i = (int) d; // legal s = (short) d; // legal s = (short) i; // legal d = 3.7E20; i = 50000; // The following give // runtime errors: s = (short) i; i = (int) d;

17 The fifth integer type The primitive type char refers to a single, two-byte Unicode character There is no good reason this should be a numeric type......but characters were numbers in C You can use characters in arithmetic (they will automatically be converted to int ) char ch = 'A'; char ch2 = (char) (ch + 1); // cast result back to char System.out.println(ch + " " + ch2 + " " + (ch + 1)); A B 66 To assign a char to a byte, or a byte to a char, you must use a cast

18 Mixed types If you mix numeric types, the narrower type is automatically promoted to the wider type int narrow = 5; double wide; double anotherWide = wide + narrow; Integer division is when you divide one integer type by another; the fractional part is discarded Example: narrow = 19 / 5; // result is 3

19 Math methods Converting a double to an int just discards the fractional part: (int)17.93 is 17 (int) –17.93 is -17 double Math.floor(double) Given a double, returns (as a double ) the largest integral value not greater than the argument Math.floor(17.93) returns 17.0 Math.floor(-17.93) returns –18.0 double Math.ceil(double) Given a double, returns (as a double ) the smallest integral value not smaller than the argument Math.ceil(17.93) returns 18.0 Math.ceil(-17.93) returns –17.0

20 Method parameters When you send a message to an object with a numeric parameter, and the object needs to promote the parameter in order to use a method, it will do so Example: double twice(double n) { return 2.0 * n; } twice(5) returns 10.0 This promotion will only occur if necessary Example 2: double half(double n) { return n / 2; } int half(int n) { return n / 2; } half(25) returns 12

21 The End