{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lec2 - 6 6 6 7 1 1 1 7 7 7 8 1-0-7-8 9 1 1-1-6-7 10 1...

This preview shows pages 1–7. Sign up to view the full content.

CSE20 Lecture 2: Number Systems: Binary Numbers, Gray Code, and Negative Numbers CK Cheng 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Number Systems 1. Introduction 2. Binary Numbers 3. Gray code 4. Negative Numbers 5. Residual Numbers 2
2. Binary Numbers b 2 b 1 b 0 Value 0 0 0 0 0 0 1 1 0 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 1 1 1 7 8 4 2 1 0 0 1 1 0 1 0 1 1 0 0 0 3 + 5 = 8 8 4 2 1 0 0 1 1 0 1 1 0 1 0 0 1 3 + 6 = 9 + + Examples : (3) (5) (8) (3) (6) (9) This is a non-redundant number system 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Binary Cont. a b Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 id a b c Carry Sum 0 0 0 0 0 0 1 0 0 1 0 1 2 0 1 0 0 1 3 0 1 1 1 0 4 1 0 0 0 1 5 1 0 1 1 0 6 1 1 0 1 0 7 1 1 1 1 1 2*0 + 0 = 0 0 0 id 0 2*0 + 1 = 0 0 1 id 1 2*1 + 0 = 1 1 0 id 6 2*1 + 1 = 1 1 1 id 7 RULE: 2 x Carry + Sum = a + b + c 4
3. Gray Code reflection Low power (reliability) when the numbers are consecutive in series. The idea is to only change ONE bit at a time. e.g. addresses, analog signals NOTE: Not for arithmetic operations (the rule is too complicated) 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4. Negative Numbers Given a positive integer x, represent the negative integer –x in (b n-1 , …, b 0 ) (i) Signed bit system b n-1 =1: negative, (b n-2 ,…,b 0 )=x. (ii) One's Complement Present 2 n - 1 - x in binary. (iii) Two's Complement Present 2 n – x in binary. Ignore bit b n . 6
4. Negative Numbers NOTE: Back to binary system id b 3 b 2 b 1 b 0 Signed One's Two's 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 2 0 0 1 0 2 2 2 3 0 0 1 1 3 3 3 4 0 1 0 0 4 4 4 5 0 1 0 1 5 5 5 6 0 1 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6 6 6 7 1 1 1 7 7 7 8 1-0-7-8 9 1 1-1-6-7 10 1 1-2-5-6 11 1 1 1-3-4-5 12 1 1-4-3-4 13 1 1 1-5-2-3 14 1 1 1-6-1-2 15 1 1 1 1-7-0-1 (i) Signed bit - x b3: negative (ii) One's Complement 2 n - 1 - x (iii) Two's Complement 2 n - x n is the number of bits (in this case n=4) One's Complement Two's Complement 8 = 16 - 1 - x 8 = 16 - x 9 = 16 - 1 – x Use the above formulas to solve for x when number is negative Two's (b 4 ) b 3 b 2 b 1 b 7 1 1 1 1 6 1 1 1 5 1 1 1 4 1 1 3 1 1 1 2 1 1 1 1 1 1-1 1 1 1 1-2 1 1 1-3 1 1 1-4 1 1-5 1 1 1-6 1 1-7 1 1-8 1 Deriving One’s and Two’s Reverse Derivation 7...
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

lec2 - 6 6 6 7 1 1 1 7 7 7 8 1-0-7-8 9 1 1-1-6-7 10 1...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online