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Lecture Notes on Number Systems
1. Question: Which is the bigger number?
5 3 2 (ans 5)
Alternative question: Which is “five”?
five V cinq 101 (ans none)
The point: These are all names of the abstract notion of ‘fiveness’. We can invent LOTS of other systems to name numbers. NUMBER != NUMERAL
2. Positional (Polynomial) Number Systems Example: decimal: 346 = 3100 + 410 + 6*
general: “ radix R” or “base R”:
Integers
and
Reals
- The “d’s” are generally 0, 1, ... R
• Common notation: dddR
Do some examples: binary , ternary, octal , hexadecimal.
3. Conversion Radix R - to - Decimal: Compute the expansion
d (^) n d (^) n – 1 ... d 0 d (^) i × R i i = 0
n
dn d (^) n – 1 ... d 0 • d (^) – 1 d (^) – 2 ... d (^) – m d (^) i × R i i =– m
n
Radix Point
n d (^) n R n ... d 0 R 0 = + +
Decimal - to - Radix R
Algorithm:
- Divide succesively by R.
- Note remainder at each step. (order of digits is lsb to msb)
Radix R 1 to Radix R (^2)
R 1 to decimal to R (^2)
4. Systems of Special Interest Binary base 2 Easy for computers Octal base 8 Easier for people Hexadecimal base 16 Easier for people
Show conversion between all of these
Emphasize hex numbering 0 ... 9ABCDEF
5. Integer Representation in Real Computers Issues: Fixed Size, Negative Number Representation A. Fixed Size “The XXX is a 16-bit (vs 32-bit) computer ...” If “word” size is n bits
there are 2 n^ possible bit patterns so only 2n^ possible distinct numbers
Implications:
- Cannot represent all possible numbers!!
- Must use some of these bit patterns (half?) to represent negative numbers.
B. Negative Number Representation i. sign magnitude (the obvious one)
- reserve one bit for the sign
- rest of bits are interpreted directly as the number
n R
--- d (^) n R n – 1 ... d 1 R 0 = + + ,with remainder d 0
0 n-
Negation: Complement means flip the bit, so flip every bit and add one Example (R=2, n=8): 9510 0010111 flip 1101000 add 1 1101001 (-95) Note that “re-negating” gets us back to the original.
Notes:
- One zero (this is good)
- Arithmetic easy (we’ll see)
iii. excess B Represent k (whether + or -) as k + B
Note: allows asymetric range.
6. Arithmetic on Positional Systems A. General Arithmetic We all know how to do arithmetic in radix 10 using carries
Same in other radices, just use a different table. Example: Our hero, binary.
B. Sign magnitude
- Add the operands, then check the signs and the magnitudes of the operands to deter- mine the sign of the result.
- It’s a pain and expensive to do in hardware, lots of comparisons and control required.
0 B-1 B R^ n
-B -1^0 R n-B-
<sum,carry>
0,0 1,0 2, 1,0 2,0 3,0 0,
9,
1,0 0, 0,1 1,
0,0 1, 1,0 0,
Carry in 1
C. 2’s complement
- Addition and subtraction are easy. Add the two numbers and throw away the carry.
- Don’t need to worry about the signs of the operands.
Examples : Assume 4-bit computer, first bit is sign bit.
D. Overflow i. Demonstration
We call this OVERFLOW. The number “13” is not representable in a 4-bit, 2’s complement computer.
Question: Does the problem go away if we have more bits? Answer: Nope, not in general, though we could safely add 6+7 with more bits.
ii. Detection.
- Examine the signs of the operands and the result.
- If signs of operands are different, cannot have overflow.
- If sign of operands are the same and the sign of the result is different, overflow!!
- These checks are easy to do in hardware. 7. Representation of Other Scalars A. Alphanumeric characters 8-bit byte 7-bit ASCII code 1 ‘a’ = 61 16 ‘A’ = 41 16 ‘;’ = 3B ‘z’= 7A 16 ‘Z’ = 5A 16 ‘ ‘ = 2016 B. Real numbers i. Fixed Point
- Given n bit number, radix point is “fixed” at some bit f
- So f “integer” bits and n-f “fractional” bits
- ASCII -- American Standard Code for Information Interchange.
OY!! That’s Negative??!!
- single representation of zero -- exponent and fraction are all 0
8. IMPORTANT!!
Question : What’s the value of “0110010110111001”? Answer: You CANNOT tell. There is no notion of TYPE at the bit level in computers. These notions are realized at higher levels.
Table 1: IEEE Floating Point Standard.
Single Double
Mantissa bits(incl hidden) 24 53
sign bits 1 1
exponent bits 8 11
Bias/Excess 127 1023
Max Exponent 127 1023
Min Exponent -126 -
