Transcript Lecture #10 - the GMU ECE Department

ECE 301 – Digital Electronics
Number Systems and Conversion,
Binary Arithmetic,
and
Representation 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.
52

What does this number represent?

Consider the “context” in which it is used.
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1011001.101

What is the decimal value of this number?

Consider the base (or radix) of this number.
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Number Systems
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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:
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
Base 2 (binary)
[0, 1]

Base 8 (octal)
[0 .. 7]

Base 16 (hexadecimal)
[0 .. 9, A .. F]
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Number Systems
Positional Notation
radix point
[a4a3a2a1a0.a-1a-2a-3]R
ai = ith position in the number
R = radix or base of the number
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Number Systems
Power Series Expansion
D = an x R4 + an-1 x R3 + … + a0 x R0
-1
-2
-m
+ a-1 x R + a-2 x R + … a-m x R
D = decimal value
ai = ith position in the number
R = radix or base of the number
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Number Systems: Example
Decimal
2
1
0
927.4510 = 9 x 10 + 2 x 10 + 7 x 10 +
-1
-2
4 x 10 + 5 x 10
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Number Systems: Example
Binary
3
2
1
0
1101.1012 = 1 x 2 + 1 x 2 + 0 x 2 + 1 x 2 +
-1
-2
-3
1x2 +0x2 +1x2
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Number Systems: Example
Octal
2
1
0
326.478 = 3 x 8 + 2 x 8 + 6 x 8 +
-1
-2
4x8 +7x8
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Number Systems: Example
Hexadecimal
2
1
0
E5A.2B16 = 14 x 16 + 5 x 16 + 10 x 16 +
-1
-2
2 x 16 + 11 x 16
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Conversion between Number Systems
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Conversion of a Decimal Integer
Use repeated division to convert a decimal
integer to any other base.
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Conversion of a Decimal Integer
Example:
Convert the decimal number 57 to binary and
to octal:
57 / 2 = 28: rem = 1 = a0
57 / 8 = 7: rem = 1 = a0
28 / 2 = 14: rem = 0 = a1
7 / 8 = 0: rem = 7 = 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 = 718
5710 = 1110012
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Conversion of a Decimal Fraction
Use repeated multiplication to convert a
decimal fraction to any other base.
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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
0.625 * 8 = 5.000: a0 = 5
0.62510 = 0.58
0.500 * 2 = 1.000: a-3 = 1
0.62510 = 0.1012
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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
In some cases, conversion
results in a repeating fraction.
0.6 * 2 = 1.2: a-4 = 1
0.2 * 2 = 0.4: a-5 = 0
0.4 * 2 = 0.8: a-6 = 0
process begins repeating here!
0.710 = 0.1 0110 0110 0110 ...2
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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.
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Conversion of a Mixed Decimal Number
Example:
Convert 48.562510 to binary.
Confirm the results using the Power Series Expansion.
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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
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Repeated Division, Repeated Multiplication
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Conversion between Bases

Conversion between binary and octal can be
carried out by inspection.

Each octal digit corresponds to 3 bits





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101 110 010 . 011 0012 = 5 6 2 . 3 18
010 011 100 . 101 0012 = 2 3 4 . 5 18
7 4 5 . 3 28 = 111 100 101 . 011 0102
3 0 6 . 0 58 = 011 000 110 . 000 1012
Is the number 392.248 a valid octal number?
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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 01012 = 9 A 6 . B 516

1100 1011 1000 . 1110 01112 = C B 8 . E 716

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.


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Number Systems
Base:
10
2
8
16
What is the
value of 12?
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Binary Arithmetic
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Binary Addition
0
+ 0
0
0
+ 1
1
Sum
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1
+ 0
1
1
+ 1
10
Carry
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Sum
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Binary Addition: Examples
+
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01011011
01110010
+
10110101
01101100
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+
00111100
10101010
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Binary Subtraction
Borrow
-
0
0
0
10
- 1
1
1
- 0
1
1
- 1
0
Difference
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Binary Subtraction: Examples
-
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01110101
00110010
-
10110001
01101100
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00111100
10101100
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Binary Arithmetic
Single-bit Addition
A
0
0
1
B
Carry
0
0
1
0
0
0
1
1
1
Single-bit Subtraction
Sum
0
1
1
0
A
0
0
1
B
Difference
0
0
1
1
0
1
1
1
0
What logic function is this?
What logic function is this?
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Binary Multiplication
0
x 0
0
0
x 1
0
1
x 0
0
1
x 1
1
Product
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Binary Multiplication: Examples
x
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1011
0110
x
1001
1101
ECE 301 - Digital Electronics
x
0110
1010
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Representation of Negative Numbers
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10011010

What is the decimal value of this number?

Is it positive or negative?

If negative, what representation are we using?
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Unsigned and Signed Binary Numbers
bn – 1
b1
b0
b1
b0
Magnitude
MSB
Unsigned number
bn – 1
Sign
0 denotes +
1 denotes –
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bn – 2
Magnitude
MSB
Signed number
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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.
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Unsigned Binary Numbers

For an n-bit binary number
0 <= D <= 2n – 1


For an 8-bit binary number:


where D = decimal equivalent value
28 = 256
For a 16-bit binary number:

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0 <= D <= 28 – 1
0 <= D <= 216 – 1
216 = 65536
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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 number
1 = negative number
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Signed Binary Numbers
Representations for signed binary numbers:
1. Sign and Magnitude
2. 1's Complement
3. 2's Complement
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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.
- (2

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n-1
– 1) <= N <= + (2
n-1
– 1)
Includes a representation for +0 and -0
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Sign and Magnitude: Example
What is the Sign and Magnitude representation for the
following decimal values, using 8 bits?
+ 97
- 68
- 97
+ 68
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Sign and Magnitude: Example
Can the following decimal numbers be represented
using 8-bit Sign and Magnitude representation?
- 212
- 127
+128
+255
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Questions?
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