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Digital Design Number Systems

Creditss adapted from J.F. Wakerly,

Digital Design, 4/e, Prentice Hall, 2006 C.H.

Roth, Fundamentals of Logic Design, 5/e, Thomson,

2004

Positional Number Systems

- A number is represented by a string of digits,

where each digit position has an associated

weight and it has the following form - dp-1dp-2 d1d0 . d-1d-2 d-n
- The value of the number is given by

Binary Numbers

- The general form of a binary number of pn binary

digits (bits) is - bp-1bp-2 b1b0 . b-1b-2 b-n
- and its value is

Octal and Hexadecimal Numbers

- The octal number system uses radix 8, while the

hexadecimal number system uses radix 16 - The octal and hex number systems are useful for

representing multibit numbers

Conversion from Binary to Decimal

- Method summation

Example 101110110012 1 210 0 29 1 28

1 27 1 26 0 25

1 24 1 23 0 22 0 21 1 20

149710

Conversion from Decimal to Binary

- Method successive divisions
- Example

EXAMPLE convert 5310 to binary

EXAMPLE convert .625ten to binary

EXAMPLE convert 0.710 to binary.

EXAMPLE convert 231.34 to base 7.

Addition of Binary Numbers

EXAMPLE Add 1310 and 1110 in binary.

Subtraction of Binary Numbers

EXAMPLES

Representation of Negative Numbers

- Signed-Magnitude Representation
- 10ten -10ten
- 001010two

101010two - The number zero has two representations (0 and

-0) - An n-bit signed-magnitude number lies within the

range -(2n-1 - 1) through (2n-1 - 1) - To add signed-magnitude numbers we must examine

the signs of the addends to determine what to do

- Radix Complement Representation
- Diminished Radix Complement Representation

Representing Numbers

- Key observation Numbers are just strings of

symbols. The meaning (value) we assign to each

string instance (pattern) is up to us. If the

string is n symbols (digits) long and each symbol

can take up to different r instances (radix) then

we can form rn different patterns. - Common sense characteristics of a system

number - Assign a different value to each different

pattern - Split the patterns equally between positive

numbers and negative numbers - The mechanic of doing arithmetic operations

should be as simple as possible

Complement Number Systems

- While the signed-magnitude system negate a number

by changing its sign, a complement number system

negates a number by taking its complement. - Radix-complement RepresentationThe complement of

an n-digit number D is obtained by subtracting it

from rn - rn D ((rn-1)-D) 1
- Diminished Radix-complement RepresentationIn a

diminished radix-complement system the complement

of an n-digit number D is obtained by subtracting

it from rn-1

Complement Number Systems

Complement Number Systems

Complement Number Systems

- Once we know how to compute the diminished-radix

complement of a number, computing the

radix-complement is very simple - radix complement diminished-radix complement

1

0

1

9

8

2

3

7

4

6

5

C2 Number System

- For binary numbers, the radix complement is

called twos complement (C2). - The MSB of a number in this system serves as the

sign bit. - Negative numbers have MSB equal to 1
- Positive numbers have MSB equal to 0
- The range of representable numbers is (2n-1)

through (2n-1-1) - Zero has only one representation

Twos Complement Number System

C1 Number System

- For binary numbers, the diminished-radix

complement is called ones complement (C1). - The MSB of a number in this system serves as the

sign bit. - Negative numbers have MSB equal to 1
- Positive numbers have MSB equal to 0
- The range of representable numbers is (2n-1-1)

through (2n-1-1) - Zero has two representations positive zero (00

00) and negative zero (11 11)

Summary of Signed Number Systems

C1 Number System

- In the C1 number system to negate an n-bit number

all we have do is to flip (invert) all the bits

C2 Number System

- In the C2 number system to negate an n-bit number

requires two steps - invert all bits of the number (i.e. take

the C1 of the number) and then - add 1

Playing with the C2 notation

- The sum of a number and its inverted

representation must be 111.111two, which in C2

represent 1

C2 EXAMPLES

C2 sign extension

- As far as m gt n, it is possible to convert n-bit

numbers into m-bit numbers, but some care is

needed - copy the most significant bit (the sign bit) into

the other bits 0010 0000 0010 1010 1111

1010 - This procedure is referred as sign extension

C2 Addition and Subtraction

1. Addition of 2 positive numbers, sum lt 2n 1.

2. Addition of 2 positive numbers, sum 2n 1

C2 Addition and Subtraction

3. Addition of positive and negative numbers

(negative number has greater

magnitude).

4. Addition of positive and negative numbers

(positive number has

greater magnitude).

C2 Addition and Subtraction

5. Addition of two negative numbers, sum 2n

1.

6. Addition of two negative numbers, sum gt 2n

1.

Detecting overflow

- Overflow occurs when the value affects the sign

bit - adding two positives yields a negative
- adding two negatives gives a positive
- subtract a negative from a positive and get a

negative - subtract a positive from a negative and get a

positive - No overflow when adding a positive and a negative

number - No overflow when subtracting two numbers of same

sign - Consider the operations A B, and A B
- Can overflow occur if B is 0
- Can overflow occur if A is 0

cannot occur !

can occur ! (for A-B if B-2n-1)

Binary Codes for Decimal Numbers

Gray Code

Character Codes

N-cubes and Hamming distance

Traversing a 3-cube in Gray code order

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