Transcript Number systems and codes Number systems Radix (base) r

Number systems and codes
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Number systems
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Ordered set of symbols or digits, with defined
relations for +   
Radix (base) r
Total # of symbols allowed in system
Decimal: 10 digits {0…9}; radix = 10
Binary: 2 digits {0, 1}; radix = 2
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Number systems and codes
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Positional notation
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N = (an-1 an-2 --- a0 . a-1 --- a-m)
Algebraic value of N = ?
N1 = (1357.46)10
N1 evaluated as ?
Number systems and codes
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N = a polynomial of radix r
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Hexadecimal r = 16
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For radix r, digit set is = {0, --- ?}
Digit set = {0, --- ?}
Why has arithmetic evolved around radix
10?
Which radix do you think is the “most
efficient”?
3500 BC, Egyptians, nonpositional
2500 BC, Sumerians, positional, base 60
Today:
Numeric conversions
binary to octal: 10111011001 --> 10 111 011 001 --> 2731
(substitution)
binary to hex: 10111011001 --> 101 1101 1001 --> 5D9
(substitution)
binary to decimal: 10111011001 --> 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 --> 0101 1101 1001 --> 10111011001
hex to octal: 5D9 --> 0101 1101 1001 --> 010 111 011 001 --> 2731
hex to decimal: 5D9 --> 5 (256) + 13 (16) + 9 (1) = 1497
(substitution)
(substitution)
(summation)
octal to binary: 2731 --> 010 111 011 001 --> 10111011001
octal to hex: 2731 --> 010 111 011 001 --> 0101 1101 1001 -> 5D9
octal to decimal: 2731 --> 2 (512) + 7 (64) + 3 (8) + 1 (1) = 1497
(substitution)
(substitution)
(summation)
Base conversions
(10111)2 = (?)10
20 + 21 + 22 + 24 = 23
(274)8 = (?)10
(A)rS = (?)rD
Base conversions
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If source radix rS < destination radix rD then
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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 > destination radix rD
(97)10 = ( ? )2
N = dn-1rSn-1 + …+d0rS0
= yp-1rDp-1 + …+y0rD0
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Q: How do we find the yp-1…y0 coefficients?
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?
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?
Converting fractions
f = d-1rS-1 + d-2rS-2 + …
f•rD = y-1 + y-2rD-1
= y-1rD-1 + y-2rD-2 + …
Integer fractional
 f•rD = y-1
f1•rD = y-2
..................
n = floor
When do we stop?