Upload
yuri-santana
View
26
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Number Systems and Conversion, Binary Arithmetic, and Representation of Negative Numbers (Lecture #10). ECE 331 – Digital System Design. The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition , by Roth and Kinney, - PowerPoint PPT Presentation
Citation preview
ECE 331 – Digital System Design
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 331 - Digital System Design 2
52
What does this number represent? Consider the “context” in which it is used.
Spring 2011 ECE 331 - Digital System Design 3
1011001.101
What is the decimal value of this number? Consider the base (or radix) of this number.
Spring 2011 ECE 331 - Digital System Design 5
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]
Spring 2011 ECE 331 - Digital System Design 6
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
Spring 2011 ECE 331 - Digital System Design 7
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
Spring 2011 ECE 331 - Digital System Design 8
Number Systems: Example
Decimal
927.4510 = 9 x 102 + 2 x 101 + 7 x 100 +4 x 10-1 + 5 x 10-2
Spring 2011 ECE 331 - Digital System Design 9
Number Systems: Example
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
Spring 2011 ECE 331 - Digital System Design 10
Number Systems: Example
Octal
326.478 = 3 x 82 + 2 x 81 + 6 x 80 +4 x 8-1 + 7 x 8-2
Spring 2011 ECE 331 - Digital System Design 11
Number Systems: Example
Hexadecimal
E5A.2B16 = 14 x 162 + 5 x 161 + 10 x 160 +2 x 16-1 + 11 x 16-2
Spring 2011 ECE 331 - Digital System Design 13
Use repeated division to convert a decimal integer to any other base.
Conversion of a Decimal Integer
Spring 2011 ECE 331 - Digital System Design 14
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
Spring 2011 ECE 331 - Digital System Design 15
Use repeated multiplication to convert a decimal fraction to any other base.
Conversion of a Decimal Fraction
Spring 2011 ECE 331 - Digital System Design 16
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
Spring 2011 ECE 331 - Digital System Design 17
Conversion of a Decimal Fraction
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.
Spring 2011 ECE 331 - Digital System Design 18
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.
Spring 2011 ECE 331 - Digital System Design 19
Conversion of a Mixed Decimal Number
Example:
Convert 48.562510 to binary.
Confirm the results using the Power Series Expansion.
Spring 2011 ECE 331 - Digital System Design 20
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
Spring 2011 ECE 331 - Digital System Design 21
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?
Spring 2011 ECE 331 - Digital System Design 22
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.
Spring 2011 ECE 331 - Digital System Design 23
Number SystemsBase: 10 2 8 16
What is the value of 12?
Spring 2011 ECE 331 - Digital System Design 25
Binary Addition
0 0 1 1+ 0 + 1 + 0 + 1 0 1 1 10
Sum Carry Sum
Spring 2011 ECE 331 - Digital System Design 26
Binary Addition: Examples
01011011+ 01110010
00111100+ 10101010
10110101+ 01101100
Spring 2011 ECE 331 - Digital System Design 27
Binary Subtraction
0 10 1 1- 0 - 1 - 0 - 1 0 1 1 0
Difference
Borrow
Spring 2011 ECE 331 - Digital System Design 28
Binary Subtraction: Examples
01110101- 00110010
00111100- 10101100
10110001- 01101100
Spring 2011 ECE 331 - Digital System Design 29
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
Spring 2011 ECE 331 - Digital System Design 30
Binary Multiplication
0 0 1 1x 0 x 1 x 0 x 1 0 0 0 1
Product
Spring 2011 ECE 331 - Digital System Design 31
Binary Multiplication: Examples
0110x 1010
1011x 0110
1001x 1101
Spring 2011 ECE 331 - Digital System Design 33
10011010 What is the decimal value of this number? Is it positive or negative? If negative, what representation are we using?
Spring 2011 ECE 331 - Digital System Design 34
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
Spring 2011 ECE 331 - Digital System Design 35
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.
Spring 2011 ECE 331 - Digital System Design 36
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
Spring 2011 ECE 331 - Digital System Design 37
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
Spring 2011 ECE 331 - Digital System Design 38
Signed Binary Numbers
Representations for signed binary numbers:
1. Sign and Magnitude2. 1's Complement3. 2's Complement
Spring 2011 ECE 331 - Digital System Design 39
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)
Spring 2011 ECE 331 - Digital System Design 40
Sign and Magnitude: Example
What is the Sign and Magnitude representation for the following decimal values, using 8 bits?
+ 97- 68- 97+ 68
Spring 2011 ECE 331 - Digital System Design 41
Sign and Magnitude: Example
Can the following decimal numbers be represented using 8-bit Sign and Magnitude representation?
- 212 - 127+128+255