Number systems and codes

1
Number systems and codes

• Number systems
• Ordered set of symbols or digits, with defined
  relations for ? ? ?
• Radix (base) r
• Total of symbols allowed in system
• Decimal 10 digits 09 radix 10
• Binary 2 digits 0, 1 radix 2

2
Number systems and codes

• Positional notation
• N (an-1 an-2 --- a0 . a-1 --- a-m)
• Algebraic value of N ?
• N1 (1357.46)10
• N1 evaluated as ?

3
Number systems and codes

• N a polynomial of radix r
• For radix r, digit set is 0, --- ?
• Hexadecimal r 16
• Digit set 0, --- ?
• Why has arithmetic evolved around radix 10?
• Which radix do you think is the most efficient?

4

3500 BC, Egyptians, nonpositional 2500 BC,
Sumerians, positional, base 60 Today
5
Numeric conversions
binary to octal 10111011001 --gt 10 111 011 001
--gt 2731 (substitution) binary to hex
10111011001 --gt 101 1101 1001 --gt
5D9 (substitution) binary to decimal 10111011001
--gt 1 (1024) 0 (512) 1 (256) 1 (128) 1
(64) 0 (32) 1 (16) 1 (8) 0 (4) 0
(2) 1 (1) 1497 (summation)
hex to binary 5D9 --gt 0101 1101 1001 --gt
10111011001 (substitution) hex to octal 5D9
--gt 0101 1101 1001 --gt 010 111 011 001 --gt
2731 (substitution) hex to decimal 5D9 --gt 5
(256) 13 (16) 9 (1) 1497 (summation)
octal to binary 2731 --gt 010 111 011 001 --gt
10111011001 (substitution) octal to hex
2731 --gt 010 111 011 001 --gt 0101 1101 1001 -gt
5D9 (substitution) octal to decimal 2731 --gt 2
(512) 7 (64) 3 (8) 1 (1) 1497 (summation)
6
Base conversions

• (10111)2 (?)10
• 20 21 22 24 23
• (274)8 (?)10
• (A)rS (?)rD

7
Base conversions

• If source radix rS lt destination radix rD then
• Every digit in the source number system is also a
  digit in the destination number system simply
  evaluate polynomial in destination number system
• If source radix rS gt destination radix rD
• (97)10 ( ? )2
• N dn-1rSn-1 d0rS0
• yp-1rDp-1 y0rD0
• Q How do we find the yp-1y0 coefficients?

8
Base conversions

• Q1
• N (yp-1rDp-2 y1)rD y0
• ? N modulo rD y0
• Q1 modulo rD y1
• Q2 modulo rD y2
• When do we stop?

9
Base conversions

• Q N i 0
• While (Q ? 0) do
• digit i Q mod rD
• Q Q/rD
• End-while
• Division is expensive.
• Question is there a way to avoid division
  (replace it with , ?, ? and possibly some table
  lookups?

10
Converting fractions

• f d-1rS-1 d-2rS-2 y-1rD-1
  y-2rD-2
• frD y-1 y-2rD-1
• Integer fractional
• ? ?frD? y-1 ?n? floor
• ?f1rD? y-2
• .................. When do we stop?