1. Number systems and codes
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

2. Number systems and codes
Positional notation
N = (an-1 an a0 . a a-m)
Algebraic value of N = ?
N1 = ( )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: > > 2731 (substitution)
binary to hex: > > 5D9 (substitution)
binary to decimal: > 1 (1024) + 0 (512) + 1 (256) + 1 (128) + 1 (64) (32) + 1 (16) + 1 (8) + 0 (4) + 0 (2) + 1 (1) = (summation)
hex to binary: 5D9 --> > (substitution)
hex to octal: 5D9 --> > > 2731 (substitution)
hex to decimal: 5D9 --> 5 (256) + 13 (16) + 9 (1) = (summation)
octal to binary: > > (substitution)
octal to hex: > > > 5D9 (substitution)
octal to decimal: > 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 < 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 > destination radix rD
(97)10 = ( ? )2
N = dn-1rSn-1 + …+d0rS0
= yp-1rDp-1 + …+y0rD0
Q: How do we find the yp-1…y0 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 Division is expensive. 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 + …
f•rD = y-1 + y-2rD-1
Integer fractional
 f•rD = y-1 n = floor
f1•rD = y-2
When do we stop? } }