ece 331 – digital system design number systems and conversion, binary arithmetic, and...

ECE 331 – Digital System Design

Number Systems and Conversion,Binary Arithmetic,

andRepresentation of Negative Numbers

(Lecture #9)

The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,

and were used with permission from Cengage Learning.

Fall 2010 ECE 331 - Digital System Design 2

52

What does this number represent? Consider the “context” within which the number is used.


1011001.101

What is the value of this number? Consider the base (or radix) of the number.


Number Systems


Number Systems

R is the radix or base of the number system Must be a positive number R digits in the number system: [0 .. R-1]

Important number systems for digital systems: Base 2 (binary): [0, 1] Base 8 (octal): [0 .. 7] Base 16 (hexadecimal): [0 .. 9, A, B, C, D, E,

F]


Number Systems

Positional Notation

N = [a4a

3a

2a

1a

0.a

-1a

-2a

-3]

R

N = numeric valueai = ith position in the numberR = radix or base of the number


Number Systems

Power Series Expansion

D = an x R4 + a

n-1 x R3 + … + a

0 x R0

+ a-1

x R-1 + a-2 x R-2 + … a

-m x R-m

D = decimal valueai = ith position in the numberR = radix or base of the number


Number Systems

Examples:

Decimal

953.7810 = 9x102 + 5x101 + 3x100 + 7x10-1 + 8x10-2

Binary

1011.112 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2

= 8 + 0 + 2 + 1 + 1/2 + 1/4

= 11.7510


Number Systems

Examples:

Octal347.218 = 3x82 + 4x81 + 7x80 + 2x8-1 + 1x8-2 = ?

HexadecimalE61A.D716 = 14x163 + 6x162 + 1x161 + 10x160 + 13x16-1 + 7x16-2

= ?


Number Systems

Base Position in Power Series ExpansionR 4 3 2 1 0 -1 -2 -3

Decimal 1010 10000 1000 100 10 1 0.1000 0.0100 0.0010

Binary 22 16 8 4 2 1 0.5000 0.2500 0.1250

Octal 88 4096 512 64 8 1 0.1250 0.0156 0.0020

Hexadecimal 1616 65536 4096 256 16 1 0.0625 0.0039 0.0002

104 103 102 101 100 10-1 10-2 10-3

24 23 22 21 20 2-1 2-2 2-3

84 83 82 81 80 8-1 8-2 8-3

164 163 162 161 160 16-1 16-2 16-3


Conversion


Use repeated division to convert a decimal integer to any other base.

Conversion of a Decimal Integer


Conversion of a Decimal Integer

Example:

Convert the decimal number 57 to a binary (R=2) number and an octal (R=8) number.

57 / 2 = 28: rem = 1 = a0

28 / 2 = 14: rem = 0 = a1

14 / 2 = 7: rem = 0 = a2

7 / 2 = 3: rem = 1 = a3

3 / 2 = 1: rem = 1 = a4

1 / 2 = 0: rem = 1 = a5

5710

= 1110012

57 / 8 = 7: rem = 1 = a0

7 / 8 = 0: rem = 7 = a1

5710

= 718


Use repeated multiplication to convert a decimal fraction to any other base.

Conversion of a Decimal Fraction



Example:

Convert the decimal number 0.625 to a binary (R=2) number and an octal (R=8) number.

0.625 * 2 = 1.250: a-1 = 1

0.250 * 2 = 0.500: a-2 = 0

0.500 * 2 = 1.000: a-3 = 1

0.62510

= 0.1012

0.625 * 8 = 5.000: a0 = 5

0.62510

= 0.58



Example:

Convert the decimal number 0.7 to binary.

0.7 * 2 = 1.4: a-1 = 1

0.4 * 2 = 0.8: a-2 = 0

0.8 * 2 = 1.6: a-3 = 1

0.6 * 2 = 1.2: a-4 = 1

0.2 * 2 = 0.4: a-5 = 0

0.4 * 2 = 0.8: a-6 = 0

0.710

= 0.1 0110 0110 0110 ...2

process begins repeating here!

In some cases, conversion results in a repeating fraction.


Conversion of a Mixed Decimal Number

Conversion of a mixed decimal number is implemented as follows:

Convert the integer part of the number using repeated division.

Convert the fractional part of the decimal number using repeated multiplication.

Combine the integer and fractional components in the new base.


Conversion of a Mixed Decimal Number

Exercise:

Convert 48.562510

to binary.Confirm the results using the Power Series

Expansion.


Conversion Conversion between any two bases, A and B,

can be carried out directly using repeated division and repeated multiplication.

Base A → Base B However, it is generally easier to convert base A

to its decimal equivalent and then convert the decimal value to base B.

Base A → Decimal → Base B

Power Series Expansion Repeated Division, Repeated Multiplication


Conversion

Conversion between binary and octal can be carried out by inspection.

Each octal digit corresponds to 3 bits 101 110 010 . 011 001

2 = 5 6 2 . 3 1

8

010 011 100 . 101 0012 = 2 3 4 . 5 1

8

7 4 5 . 3 28 = 111 100 101 . 011 010

2

3 0 6 . 0 58 = 011 000 110 . 000 101

2

Is the number 392.248 a valid octal number?


Conversion

Conversion between binary and hexadecimal can be carried out by inspection.

Each hexadecimal digit corresponds to 4 bits 1001 1010 0110 . 1011 0101

2 = 9 A 6 . B 5

16

1100 1011 1000 . 1110 01112 = C B 8 . E 7

16

E 9 4 . D 216

= 1110 1001 0100 . 1101 00102

1 C 7 . 8 F16

= 0001 1100 0111 . 1000 11112

Note that the hexadecimal number system requires additional characters to represent its 16 values.


Number SystemsBase: 10 2 8 16

What is the value of 12?


Binary Arithmetic


Binary Addition

0 0 1 1+ 0 + 1 + 0 + 1 0 1 1 10

Sum Carry Sum


Binary Addition

Examples:

01011011+ 01110010

00111100+ 10101010

10110101+ 01101100


Binary Subtraction

0 10 1 1- 0 - 1 - 0 - 1 0 1 1 0

Difference

Borrow


Binary Subtraction

Examples:

01110101- 00110010

00111100- 10101100

10110001- 01101100


Binary Arithmetic

Single-bit Addition Single-bit Subtraction

s

0

1

1

0

c

0

0

0

1

x y

0

0

1

1

0

1

0

1

Carry Sum

d

0

1

1

0

x y

0

0

1

1

0

1

0

1

Difference

What logic function is this?

What logic function is this?


Binary Multiplication

0 0 1 1x 0 x 1 x 0 x 1 0 0 0 1

Product



Examples:

0110x 1010

1011x 0110

1001x 1101


When doing binary multiplication, a common way to avoid carries greater than 1 is to add in the partial products one at a time as illustrated by the following example:

1111 multiplicand1101 multiplier1111 1st partial product

00000 2nd partial product (01111) sum of first two partial products 111100 3rd partial product (1001011) sum after adding 3rd partial product (pp) 1111000 4th partial product 11000011 final product (sum after adding 4th pp)



Representation of Negative Numbers


10011010

What is the value of this number? Is it positive or negative? If negative, what representation are we using?

Fall 2010 34

bn 1– b1 b0

Magnitude

MSB

(a) Unsigned number

bn 1– b1 b0

MagnitudeSign

(b) Signed number

bn 2–

0 denotes1 denotes

+– MSB

Unsigned and Signed Binary Numbers


Unsigned Binary Numbers

For an n-bit unsigned binary number, all n bits are used to represent the

magnitude of the number.

** Cannot represent negative numbers.


Unsigned Binary Numbers

For an n-bit binary number

0 <= D <= 2n – 1 where D = decimal equivalent value

For an 8-bit binary number: 0 <= D <= 28 – 1 28 = 256

For a 16-bit binary number: 0 <= D <= 216 – 1 216 = 65536


Signed Binary Numbers

For an n-bit signed binary number, n-1 bits are used to represent the

magnitude of the number;

the leftmost bit (MSB) is, generally, used to indicate the sign of the number.

0 = positive number1 = negative number


Signed Binary Numbers

Three representations for signed binary numbers:

1. Sign and Magnitude2. 1's Complement3. 2's Complement


Sign and Magnitude

For an n-bit signed binary number, The MSB (leftmost bit) is the sign bit. The remaining n-1 bits represent the

magnitude.

- (2n-1 - 1) <= D <= + (2n-1 – 1) Includes a representation for -0 and +0.

The design of arithmetic circuits for Sign and Magnitude binary numbers is difficult.


Sign and Magnitude

Example:

What is the Sign and Magnitude representation for the following decimal values, using 8 bits:

+ 97- 68


Sign and Magnitude

Example:

Can the following decimal numbers be represented using 8-bit Sign and Magnitude representation?

- 212 - 127+128+255


Questions?

ece 331 – digital system design number systems and conversion, binary arithmetic, and...

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