Numbers Jan 13 2019 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 2n cases A byte is 8 bits therefore 28 256 cases 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 Everything is a number Everything in the computer is stored as a pattern of bits Numbers are stored as a pattern of bits Computers use the binary number system Characters are stored as a pattern of bits Binary distinctions are easy for hardware to work with 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 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 Counting in three systems Dec 0 1 2 3 4 5 6 7 8 9 10 Bin 0 1 10 11 100 101 110 111 1000 1001 1010 Hex 0 1 2 3 4 5 6 7 8 9 A Dec Bin Hex 11 12 13 14 15 16 17 18 19 20 1011 1100 1101 1110 1111 10000 10001 10010 10011 10100 B C D E F 10 11 12 13 14 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 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 101101011100101000001011 5 5 3 4 5 0 1 3 One hexadecimal digit equals four binary digits 101101011100101000001011 B 5 C A 0 B 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 077 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 Integer types There are four integer types byte occupies one byte surprise short occupies two bytes Can hold numbers in range 32768 to 32767 int occupies four bytes Can hold numbers in range 128 to 127 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 123456789L A lowercase L can be used but it s a bad idea 123456789l Floating point literals Floating point literals are written with a decimal point 8 5 7 923 5 000 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 105 float literals are written with an F suffix Examples 8 5F 7 923F 5 000F 4 32E5F If you don t have the F suffix type double is assumed 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 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 7139946 illegal Widening and narrowing byte short int long float double 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 Casts You can convert cast one numeric type to another When you widen no explicit cast is necessary When you narrow an explicit cast is required But it doesn t hurt 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 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 The fifth integer type The primitive type char refers to a single two byte Unicode character You can use characters in arithmetic they will automatically be converted to int There is no good reason this should be a numeric type but characters were numbers in C 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 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 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 …
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