ECE 8053 Introduction to Computer Arithmetic e: http://www.ece.msstate.edu/classes/ece8053/fall_ Course & Text Content: Part 1: Number Representation Part 2: Addition/Subtraction Part 3: Multiplication Part 4: Division Part 5: Real Arithmetic (Floating-Point) Part 6: Function Evaluation Part 7: Implementation Topics
21
Embed
ECE 8053 Introduction to Computer Arithmetic (Website: Course & Text Content: Part 1: Number Representation.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Critical Path is 3 gates34=12(12)(0.2ns)=2.4ns2.4ns > 0.59nsMust Use Faster Adder!!!!
Roman Numeral SystemSymbolic Digitssymbol value
I 1 V 5 X 10 L 50 C 100 D 500 M 1000
RULES:• If symbol is repeated or lies to the right of another higher-valued symbol, value is additive XX=10+10=20 CXX=100+10+10=120• If symbol is repeated or lies to the left of a higher-valued symbol, value is subtractive XXC = - (10+10) + 100 = 80 XLVIII = -(10) + 50 + 5 + 3 = 48
Weighted Positional Number System
Example: Arabic Number System (first used by Chinese)
symbol (digit)
value (in 1’s position)
0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
Addition Paradigms• right to left serial 1 147865 +30921 178786
• right to left, parallel 147865 +30921 177786 001000 178786
•random 461325 147865 +30921 177786 001000 178786
Binary Number System• n-ordered sequence:
• each xi{0,1} is a BInary digiT (BIT)
• magnitude of n is important• sequence is a short-hand notation• more precise definition is:
1 2 2 1 0n nx x x x x
11 2 2 1 0
1 2 2 1 00
2 2 2 2 2 2n
n n in n i
i
x x x x x x
• This is a radix-polynomial form
Number System• A Number System is defined if the following exist
•Example: The binary number system
1. 2. 3. Addition operator defined by addition table4. Multiplication operator defined by multiplication table
1. A digit set2. A radix or base value3. An addition operation4. A multiplication operation
{0,1}ix 2
+ 0 1 0 0 1 1 1 10
0 1 0 0 0 1 0 1
Number System Observations• Cardinality of digit set (2) is equal to radix value• Addition operator XOR, Multiplication is AND
•How many integers exist?Mathematically, there are an infinite number, In computer, finite due to register length
minX
maxX
min max[ , ]X X
smallest representable number
largest representable number
range of representable numbers[-inclusive; (-exclusive interval bounds
•When ALU produces a result >Xmax or <Xmin, incorrect result occurs