Numbers

1
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 casesTT,
  TF, FT, FF
• Three bits can distinguish between eight
  casesTTT, 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

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
  digits0, 1, 2, 3, 4, 5, 6, 7
• The decimal (base 10) number system uses ten
  digits0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• The hexadecimal, or hex (base 16) number system
  uses sixteen digits0, 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 isnt, its 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, F is the largest digit
• Hence, 4A6FF 1 4A700

6
Counting in three systems

• Dec Bin Hex
• 0 0 0
• 1 1 1
• 2 10 2
• 3 11 3
• 4 100 4
• 5 101 5
• 6 110 6
• 7 111 7
• 8 1000 8
• 9 1001 9
• 10 1010 A

• Dec Bin Hex
• 11 1011 B
• 12 1100 C
• 13 1101 D
• 14 1110 E
• 15 1111 F
• 16 10000 10
• 17 10001 11
• 18 10010 12
• 19 10011 13
• 20 10100 14

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
• In Java, you can write numbers as octal, decimal,
  or hexadecimal (but not binary)
• Colors are usually specified in hexadecimal
  notationFF0000, 669966,

8
Using octal and hex numbers

• Computers use binary, but the numbers are too
  long and confusing for people--its 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

9
Writing octal and hex integers

• Integers are usually written in decimal
  notation7, 532, -28
• To write a number in octal, just start with a
  zero02, 0657, -077
• ...but dont 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 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 an L suffix 123456789L
• A lowercase L can be used, but its a bad idea
  123456789l

11
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, that is,
  432000
• Example 4.32E-5 means 4.32 x 10-5, that is,
  0.0000432

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
• float literals must be written with an F suffix
• Examples 8.5F -7.923F 5.000F
  4.32E5F
• double occupies eight bytes
• Can hold numbers in the range 1.7E308 to 4.9E-324
• Accuracy is seventeen or eighteen digits
• Examples 8.5 -7.923 5.000 4.32E5

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 doesnt 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

14
Widening and narrowing

• You can always widen (assign a narrower type to a
  wider type)
• double wideint narrowwide narrow
• But if you want to narrow (assign a wider type to
  a narrower type), you have to cast it
• narrow (int)wide

15
Casts

• You can convert (cast) one numeric type to
  another
• When you widen, no explicit cast is necessary
• But it doesnt hurt
• When you narrow, an explicit cast is required
• This requirement is made to help avoid accidental
  loss of precision
• 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 didnt

16
Example casts

• short s 0int i 0double d 0.0
• d i // legald s // legali s
  //legal
• i d // illegals d // illegals i
  // illegal

• i (int) d // legals (short) d //
  legals (short) i // legal
• d 3.7E20i 50000
• // The following give// runtime errorss
  (short) ii (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 charSystem.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 (widened) to the wider
  type
• int narrow 5double widedouble 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
• 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
There are 10 kinds of people in the worldthose
who understand binary, and those who dont.

--Anon.
Real Programmers always confuse Christmas and
Halloween because Oct31 Dec25
--Andrew Rutherford