1. ECE 331 – Digital System Design
Number Systems and Conversion,
Binary Arithmetic,
and
Representation 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.

2. ECE 331 - Digital System Design52
What does this number represent?
Consider the “context” within which the number is used.
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ECE Digital System Design

3. ECE 331 - Digital System Design
What is the value of this number?
Consider the base (or radix) of the number.
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ECE Digital System Design

4. ECE 331 - Digital System Design
Number Systems
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ECE Digital System Design

5. ECE 331 - Digital System Design
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]
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ECE Digital System Design

6. ECE 331 - Digital System Design
Number Systems
Positional Notation
N = [a4a3a2a1a0.a-1a-2a-3]R
N = numeric value
ai = ith position in the number
R = radix or base of the number
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ECE Digital System Design

7. + a-1 x R-1 + a-2 x R-2 + … a-m x R-m
Number Systems
Power Series Expansion
D = an x R4 + an-1 x R3 + … + a0 x R0
+ a-1 x R-1 + a-2 x R-2 + … a-m x R-m
D = decimal value
ai = ith position in the number
R = radix or base of the number
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ECE Digital System Design

8. ECE 331 - Digital System Design
Number Systems
Examples:
Decimal
= 9x x x x x10-2
Binary
= 1x23 + 0x22 + 1x21 + 1x20 + 1x x2-2
= / /4 =
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ECE Digital System Design

9. ECE 331 - Digital System Design
Number Systems
Examples:
Octal
= 3x82 + 4x81 + 7x80 + 2x x8-2
= ?
Hexadecimal
E61A.D716 = 14x x x x x x16-2
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ECE Digital System Design

10. ECE 331 - Digital System Design
Number Systems
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ECE Digital System Design

11. ECE 331 - Digital System Design
Conversion
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ECE Digital System Design

12. Conversion of a Decimal Integer
Use repeated division to convert a decimal integer to any other base.
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13. 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 =
57 / 8 = 7: rem = 1 = a0
7 / 8 = 0: rem = 7 = a1
5710 = 718
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ECE Digital System Design

14. Conversion of a Decimal Fraction
Use repeated multiplication to convert a decimal fraction to any other base.
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15. Conversion of a Decimal Fraction
Example:
Convert the decimal number 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 =
0.625 * 8 = 5.000: a0 = 5
= 0.58
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16. 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 =
In some cases, conversion
results in a repeating fraction.
process begins repeating here!
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17. 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.
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ECE Digital System Design

18. Conversion of a Mixed Decimal Number
Exercise:
Convert to binary.
Confirm the results using the Power Series Expansion.
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ECE Digital System Design

19. ECE 331 - Digital System Design
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
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ECE Digital System Design

20. ECE 331 - Digital System Design
Conversion
Conversion between binary and octal can be carried out by inspection.
Each octal digit corresponds to 3 bits = = = =
Is the number a valid octal number?
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ECE Digital System Design

21. ECE 331 - Digital System Design
Conversion
Conversion between binary and hexadecimal can be carried out by inspection.
Each hexadecimal digit corresponds to 4 bits
= 9 A 6 . B 516
= C B 8 . E 716
E D 216 =
1 C F16 =
Note that the hexadecimal number system requires additional characters to represent its 16 values.
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ECE Digital System Design

22. ECE 331 - Digital System Design
Number Systems
Base: 10 2 8 16
What is the
value of 12?
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ECE Digital System Design

23. ECE 331 - Digital System Design
Binary Arithmetic
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ECE Digital System Design

24. ECE 331 - Digital System Design
Binary Addition
Sum
Carry
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25. ECE 331 - Digital System Design
Binary Addition
Examples:
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26. ECE 331 - Digital System Design
Binary Subtraction
Difference
Borrow
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27. ECE 331 - Digital System Design
Binary Subtraction
Examples:
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28. ECE 331 - Digital System Design
Binary Arithmetic
Single-bit Addition
Single-bit Subtraction s 1 c x y
Carry
Sum d 1 x y
Difference
What logic function is this?
What logic function is this?
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29. Binary Multiplication
x x x x 1
Product
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ECE Digital System Design

30. Binary Multiplication
Examples:
1011
x 0110
1001
x 1101
0110
x 1010
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31. Binary Multiplication
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 multiplicand
1101 multiplier
st partial product
nd partial product
(01111) sum of first two partial products
rd partial product
( ) sum after adding 3rd partial product (pp)
  th partial product
final product (sum after adding 4th pp)
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32. Representation of Negative Numbers
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33. ECE 331 - Digital System Design
What is the value of this number?
Is it positive or negative?
If negative, what representation are we using?
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34. Unsigned and Signed Binary Numbers1 –
Magnitude
MSB
(a) Unsigned number
Sign
(b) Signed number 2
0 denotes
1 denotes +
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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.
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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
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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 (MSB) is, generally, used to indicate the sign of the number.
0 = positive number
1 = negative number
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ECE Digital System Design

38. Signed Binary Numbers Three representations for signed binary numbers:
1. Sign and Magnitude
2. 1's Complement
3. 2's Complement
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39. ECE 331 - Digital System Design
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.
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ECE Digital System Design

40. ECE 331 - Digital System Design
Sign and Magnitude
Example:
What is the Sign and Magnitude representation for the following decimal values, using 8 bits:
+ 97
- 68
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41. ECE 331 - Digital System Design
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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ECE Digital System Design

42. ECE 331 - Digital System Design
Questions?
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ECE Digital System Design