Number Systems

1CMPE12c Gabriel Hugh Elkaim

Number Systems


A Brief History of Numbers

From Gonick, Cartoon Guide to Computer Science


Prehistoric Ledgers


Elaborate Finger Counting


Ancient Number Systems


Positional 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


Positional Number System

• Base 10 is a special case of positional number system

• PNS First used over 4000 years ago in Mesopotamia (Iraq)– Base 60– 0...59 (written as 60 different symbols)– 5,4560 = 5 x 60 + 45 = 34510

• Positional Number Systems are great for algebra

• Why?


Arabic Numerals

• 345 is really– 3 x 102 + 4 x 101 + 5 x 100

– 3 x 100 + 4 x 10 + 5 x 1– 3 is the most significant symbol (carries the

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

least weight)

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


Try multiplication in (non-positional) Roman numerals!

XXXIII (33 in decimal) XII (12 in decimal)---------XXXIIIXXXIIICCCXXX-----------CCCXXXXXXXXXIIIIII-----------CCCLXXXXVI-----------CCCXCVI = 396 in decimal


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 fast electronic circuits• Positional numbers work great with 2-state devices



What ’10’ Means


Number Base Systems


Binary Numbers


The Powers of 2


Equivalent Numbers


Converting Binary to Decimal


•Base (radix): 2•Digits (symbols) allowed: 0, 1

•Binary Digits, or bits•10012 is really

1 x 23 + 0 x 22 + 0 X 21 + 1 X 20

910

•110002 is really1 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 0 x 20

2410

Binary Number System


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

Binary Number System

So if the computer is all binary how does it multiply 5 by 324 when I type it in the calculator program?


Octal Number System• Base (radix): 8• Digits (symbols): 0 – 7• 3458 is really

– 3 x 82 + 4 x 81 + 5 x 80

– 192 + 32 + 5– 22910

• 10018 is really– 1 x 83 + 0 x 82 + 0 x 81 + 1 x 80

– 512 + 1– 51310

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


Hexadecimal Number System

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

fHex Decimal

a 10

b 11

c 12

d 13

e 14

f 15


A316 is really:A x 161 + 3 x 160

160 + 316310

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

3 x 256 + 14 x 16 + 8 x 1768 + 224 + 8100010


Some Examples of converting hex numbers to decimal


10C16 is really:1 x 162 + 0 x 161 + C x 160

1 x 256 + 12 x 16256 + 19244810

In C, hex numbers are represented with a leading “0x” (for example “0xa3” or “0x10c”).



For any positional number system

•Base (radix): b•Digits (symbols): 0 … b – 1•Sn-1Sn-2….S2S1S0

Use summation to transform any base to decimal

Value = Σ (Sibi) n-1

i=0



More PNS fun

• 21203 =

• 4035 =

• 2717 =

• 3569 =

• 11102 =

• 2A612 =

• BEEF16 =

=6910

=10310

=4110

=29410

=1410

=41410

=4887910


Decimal Binary Conversion• Divide decimal value by 2 until the value is 0 • 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?


02

1

2

2

2

5

2

10

2

21

2

43

2

86

2

172

2

345345

10101

1001

b10101100134510

Decimal Binary Conversion


Decimal Binary Conversion

• 12810=

• 31010=

• 2610=


Binary Octal Conversion• 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• Examples:

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

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


Octal Binary Conversion

It is simple, just write down the 3-bit binary code for each octal digit

Octal Binary

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111


Binary Hex Conversion

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

• Write 1 hex digit for each group• Examples:

1001 1110 0111 0000 9 e 7 0

0001 1111 1010 0011 1 f a 3


Hex Binary Conversion

Again, simply write down the 4 bit binary code for each hex digit

Example: 3 9 c 8 0011 1001 1100 1000


Conversion TableDecimal Hexadecimal Octal Binary

0 0 0 0000

1 1 1 0001

2 2 2 0010

3 3 3 0011

4 4 4 0100

5 5 5 0101

6 6 6 0110

7 7 7 0111

8 8 10 1000

9 9 11 1001

10 A 12 1010

11 B 13 1011

12 C 14 1100

13 D 15 1101

14 E 16 1110

15 F 17 1111


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

Decimal Hex•Do it in 2 steps, decimal binary hex

So why use hex and octal and not just binary and decimal?


Negative Integers

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

parentheses: (2000) or color 2000• Sign-magnitude format• Example: -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

= 1.41422210

Most accurate approximation until the Renaissance


fn-1 fn-2 … f2 f1 f0 f-1 f-2 f-3 … fm-1

Radix point

Generalized Representation

For a number “f” with ‘n’ digits to the left and ‘m’ to the right of the decimal place

Position is the power


Fractional Representation

• What is 3E.8F16?

• How about 10.1012?

= 3 x 161 + E x 160 + 8 x 16-1 + F x 16-2

= 48 + 14 + 8/16 + 15/256

= 1 x 21 + 0 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3

= 2 + 0 + 1/2 + 1/8


More PNS Fractional Fun

• 21.0123 =

• 4.1335 =

• 22.617 =

• A.3A12 =


Converting Decimal 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 table/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.

For 0.810 to binary


What is 2.210 in:

•Binary

•Hex

Number Systems

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