Numbers

Numbers
Integer Types
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
– 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
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
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
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
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 32767
– 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 a L suffix: 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 a 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;
Widening and narrowing
• You can always assign a
narrower value to a wider
variable
byte
short
int
long
float
double
– This is called widening
• You can do something
special to assign a wide
variable to a narrower
variable
– This is called narrowing
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
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 End
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