Number Systems

Number Systems

PrehistoryUnary, or marks: /

/////// = 7/////// + ////// = /////////////

Grouping lead to Roman Numerals:VII + V = VVII = XII

Better, Arabic Numerals:7 + 5 = 12 = 1 x 10 + 2

Arabic Numerals

345 is really 3 x 100 + 4 x 10 + 5 x 1 3 x 102 + 4 x 101 + 5 x 100

3 is the most significant symbol (carries the most weight)

5 is the least significant symbol (carries the least weight)

Digits (or symbols) allowed: 0-9Base (or radix): 10

•Base 10 is a special case of positional number system•First used over 4000 years ago in Mesopotamia(Iraq)

•Base 60 (Sexagesimal)•Digits: 0..59 (written differently)•5,4560 = 5 x 60 + 45 x 1 = 34510

•Positional number systems a great advance in mathematics•Why?

•Try multiplication in (non-positional) Roman numerals(!):

XXXIII (33 in decimal) XII (12 in decimal)---------XXXIIIXXXIIICCCXXX-----------CCCXXXXXXXXXIIIIIICCCLXXXXVICCCXCVI = 396

•The Mesopotamians wouldn’t have had this problem.

•There are many ways to “represent” a number•Representation does not affect computation result

LIX + XXXIII = LXXXXII (Roman)59 + 33 = 92 (Decimal)

•Representation affects difficulty of computing results•Computers need a representation that works with fastelectronic circuits

Binary: positional numbers work great with 2-state devices

•Digits (symbols) allowed: 0, 1•Binary Digits, or bits

•Base (radix): 2•10012 is really

•1 x 23 + 0 x 22 + 0 X 21 + 1 X 20

•910

•110002 is really•?•?

•Computers usually multiply Arabic numerals by convertingto binary, multiplying and converting back (much as us with Roman numerals)

Octal number system:

Digits (symbols): 0 – 7Base (radix): 83458 is really 3 x 82 + 4 x 81 + 5 x 80

192 + 32 + 5 22910

10018 is really ? ? ?

In C, octal numbers are represented with a leading 0 (0345 or 01001).

Hexadecimal number system:

Digits (symbols) allowed: 0 – 9, a – fBase (radix): 16

Hex Decimal

a 10

b 11

c 12

d 13

e 14

f 15

•A316 is really:•A x 161 + 3 x 160

•160 + 3•16310

•3E816 is really:•3 x 162 + E x 161 + 8 x 160

•3 x 256 + 14 x 16 + 8 x 1•768 + 224 + 8•100010

•10C16 is really:•?•?•?•?

•In C, hex numbers are represented with a leading 0x(0xa3 or 0x10c).

For any positional number system:•Base (radix): b•Digits (symbols): 0..b – 1•Sn-1Sn-2….S2S1S0

•

•Use sum to transform any base to decimal

Value = Σ(Sibi) n-1

i=0

Decimal Binary

•Divide decimal value by 2 until the value is 0 (see book)•Know your powers of two and subtract

… 256 128 64 32 16 8 4 2 1•Example: 42

•What is the biggest power of two that fits?•What is the remainder?•What fits?•What is the remainder?•What fits? What is the binary representation?

Binary Octal

•Group into 3’s starting at least significant symbol•Add leading 0’s if needed (why not trailing?)

•Write 1 octal digit for each group•Example:

100 010 111 (binary) 4 2 7 (octal)

10 101 110 (binary) 2 5 6 (octal)

Octal Binary•Write down the 3-bit binary code for each octal digit

Binary Hex

•Group into 4’s starting al least significant symbol•Adding leading 0’s if needed

•Write 1 hex digit for each group•Example:

1001 1110 0111 0000 9 e 7 0

0001 1111 1010 0011 1 f a 3

Hex Binary•Write down the 4 bit binary code for each hex digit•Example:

3 9 c 8 0011 1001 1100 1000

Hex Octal•Do it in 2 steps, hex binary octal

Decimal Hex•Do it in 2 steps, decimal binaryhex

Why use hex and octal?

Negative Integers

Most humans precede number with “-” (e.g., -2000) Accountants, however, use

parentheses: (2000)

Sign-magnitudeExample: -1000 in hex? 100010 = 3 x 162 + e x 161 + 8 x 160

-3E816

Mesopotamians used positional fractions•Sqrt(2) = •1.24,51,1060 = 1 x 600 + 24 x 60-1 + 51 x 60-2 +

10 x 60-3

• = 1.414222•Most accurate approximations until the Renaissance•What is 3E.8F16?

•How about 10.1012?

f f . . . f f f . f f f n-1 n-2 2 1 0 -1 -2 -3

Binary point

2-1 = .52-2 = .252-3 = .1252-4 = .0625

Converting decimal to binary fractions

•Consider left and right of the decimal point separately.•The stuff to the left can be converted to binary as before.•Use the following algorithm to convert the fraction:

Fraction Fraction x 2 Digit left of decimal point

0.8 1.6 1 most significant (f-1)

0.6 1.2 1

0.2 0.4 0

0.4 0.8 0

0.8 (it must repeat from here!!)•Different bases have different repeating fractions.

•0.810 = 0.110011001100…2 = 0.11002

•Numbers can repeat in one base and not in another.

What is 2.2 in:

•Binary

•Hex