ECE 301 – Digital Electronics Number Systems and Conversion, Binary Arithmetic, and Representation...

ECE 301 – Digital Electronics

Number Systems and Conversion,Binary Arithmetic,

andRepresentation of Negative Numbers

(Lecture #10)

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.

Spring 2011 ECE 301 - Digital Electronics 2

52

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


1011001.101

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


Number Systems

Positional Notation

[a4a

3a

2a

1a

0.a

-1a

-2a

-3]

R

ai = ith position in the numberR = radix or base of the number

radix point


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: Example

Decimal

927.4510 = 9 x 102 + 2 x 101 + 7 x 100 +4 x 10-1 + 5 x 10-2



Binary

1101.1012 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 +1 x 2-1 + 0 x 2-2 + 1 x 2-3



Octal

326.478 = 3 x 82 + 2 x 81 + 6 x 80 +4 x 8-1 + 7 x 8-2



Hexadecimal

E5A.2B16 = 14 x 162 + 5 x 161 + 10 x 160 +2 x 16-1 + 11 x 16-2


Conversion between Number Systems


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 binary and to octal:

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 binary and to octal.

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

Convert the integer part of the decimal number using repeated division.

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

Combine the integer and fractional parts in the new base.


Conversion of a Mixed Decimal Number

Example:

Convert 48.562510 to binary.

Confirm the results using the Power Series Expansion.


Conversion between Bases Conversion between any two bases 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 value → Base B

Power Series Expansion Repeated Division, Repeated Multiplication


Conversion between Bases

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 between Bases

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

What logic function is this?

What logic function is this?

A B Difference

0 0 0

0 1 1

1 0 1

1 1 0

A B Carry Sum

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0


Binary Multiplication

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

Product


Binary Multiplication: Examples

0110x 1010

1011x 0110

1001x 1101


Representation of Negative Numbers


10011010 What is the decimal value of this number? Is it positive or negative? If negative, what representation are we using?


bn 1– b1 b0

Magnitude

MSB Unsigned number

bn 1– b1 b0

MagnitudeSign

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 is, generally, used to indicate the sign of the number.

0 = positive number1 = negative number


Signed Binary Numbers

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 leftmost bit is the sign bit. The remaining n-1 bits represent the

magnitude.

Includes a representation for +0 and -0

- (2n-1 – 1) <= N <= + (2n-1 – 1)


Sign and Magnitude: Example

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

+ 97- 68- 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 301 – Digital Electronics Number Systems and Conversion, Binary Arithmetic, and Representation...

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