1. ECE 331 – Digital System Design
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.

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

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

6. ECE 331 - Digital System Design
Number Systems
Positional Notation
[a4a3a2a1a0.a-1a-2a-3]R
ai = ith position in the number
R = radix or base of the number
radix point
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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. Number Systems: Example
Decimal
= 9 x x x 100 +
4 x x 10-2
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ECE Digital System Design

9. Number Systems: Example
Binary
= 1 x x x x 20 +
1 x x x 2-3
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ECE Digital System Design

10. Number Systems: Example
Octal
= 3 x x x 80 +
4 x x 8-2
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ECE Digital System Design

11. Number Systems: Example
Hexadecimal
E5A.2B16 = 14 x x x 160 +
2 x x 16-2
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ECE Digital System Design

12. Conversion between Number Systems
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ECE Digital System Design

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

15. Conversion of a Decimal Fraction
Use repeated multiplication to convert a decimal fraction to any other base.
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ECE Digital System Design

16. Conversion of a Decimal Fraction
Example:
Convert the decimal number 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 =
0.625 * 8 = 5.000: a0 = 5
= 0.58
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ECE 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 =
In some cases, conversion
results in a repeating fraction.
process begins repeating here!
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ECE 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.
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ECE Digital System Design

19. Conversion of a Mixed Decimal Number
Example:
Convert to binary.
Confirm the results using the Power Series Expansion.
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ECE 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
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ECE 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 = = = =
Is the number a valid octal number?
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ECE 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
= 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

23. 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

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

25. ECE 331 - Digital System Design
Binary Addition
Sum
Carry
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ECE Digital System Design

26. Binary Addition: Examples
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ECE Digital System Design

27. ECE 331 - Digital System Design
Binary Subtraction
Difference
Borrow
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ECE Digital System Design

28. Binary Subtraction: Examples
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ECE Digital System Design

29. ECE 331 - Digital System Design
Binary Arithmetic
Single-bit Addition
Single-bit Subtraction
What logic function is this? A B
Carry
Sum 1 A B
Difference 1
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ECE Digital System Design

30. Binary Multiplication
x x x x 1
Product
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ECE Digital System Design

31. Binary Multiplication: Examples
0110
x 1010
1011
x 0110
1001
x 1101
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ECE Digital System Design

32. Representation of Negative Numbers
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ECE Digital System Design

33. ECE 331 - Digital System Design
What is the decimal value of this number?
Is it positive or negative?
If negative, what representation are we using?
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ECE Digital System Design

34. Unsigned and Signed Binary Numbers1 –
Magnitude
MSB
Unsigned number
Sign
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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ECE 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
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ECE 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 number
1 = negative number
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ECE Digital System Design

38. Signed Binary Numbers Representations for signed binary numbers:
1. Sign and Magnitude
2. 1's Complement
3. 2's Complement
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ECE Digital System Design

39. ECE 331 - Digital System Design
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)
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ECE 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
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ECE 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
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ECE Digital System Design

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