1. Speaker: Fuw-Yi Yang 楊伏夷 伏夷非征番, 道德經 察政章(Chapter 58) 伏者潛藏也
數位系統  Digital Systems 
Department of Computer Science and Information Engineering, Chaoyang University of Technology
朝陽科技大學資工系
Speaker: Fuw-Yi Yang 楊伏夷
伏夷非征番,
道德經 察政章(Chapter 58) 伏者潛藏也
道紀章(Chapter 14) 道無形象, 視之不可見者曰夷
Fuw-Yi Yang

2. Text Book: Digital Design 4th Ed.
Chap 1 Digital Systems and Binary Numbers
1.1 Digital Systems
1.2 Binary Numbers-- Table 1.1
1.3 Number-Base Conversions-- Example 1.1~1.4
1.4 Octal and Hexadecimal Numbers -- Table 1.2
1.5 Complements -- Example 1.5~1.8
1.6 Signed Binary Numbers -- Table 1.3
1.7 Binary Codes -- Table 1.4~1.7
1.8 Binary Storage and Registers -- Figure 1.1~1.2
1.9 Binary Logic -- Table 1.8, Figure 1.3~1.6
Fuw-Yi Yang

3. Text Book: Digital Design 4th Ed. Chap 1 1.2 Binary Numbers
In general, a number expressed in a base-r system has
coefficients multiplied by powers of r:
anrn+an-1rn-1+…+a1r1+a0+a-1r-1+a-2r-2 +…+a-mr-m
r is called base or radix.
Fuw-Yi Yang

4. In general, a number expressed in a base-r system has
coefficients multiplied by powers of r:
anrn+an-1rn-1+…+a1r1+a0+a-1r-1+a-2r-2 +…+a-mr-m
r is called base or radix.
Fuw-Yi Yang

5. Text Book: Digital Design 4th Ed. Chap 1 1.2 Binary Numbers
是否注意到指數為負之值呢?
Ex: 2-1, 2-3 ,2-4
Fuw-Yi Yang

6. Text Book: Digital Design 4th Ed. Chap 1 1.3 Number-Base Conversions
Example1.1 Convert decimal 41 to binary, (41)10 = (?)2
(41)D= (?)B
Example1.2 (153)10 = (?)8
Example1.3 (0.6875)10 = (?)2
Example1.4 (0.513)10 = (?)8
Fuw-Yi Yang

7. Text Book: Digital Design 4th Ed.
Chap Octal and Hexadecimal Numbers
See Table 1.2
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8. Text Book: Digital Design 4th Ed.
Chap Octal and Hexadecimal Numbers
See Table 1.2
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9. Text Book: Digital Design 4th Ed. Chap 1 1.5 Complements
Diminished Radix Complement
Given a number N in base r having n digits, the (r - 1)’s
complement of N is defined as (rn - 1) - N.
The 1’s complement of is
Radix Complement
Given a number N in base r having n digits, the r’s
complement of N is defined as
rn - N for N 0 and as 0 for N = 0 .
The 2’s complement of is
Fuw-Yi Yang

10. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
The subtraction of two n-digit unsigned numbers M - N in
base r can be done as follows:
1. M + (rn - N ), note that (rn - N ) is r’s complement of N.
2. If M  N, the sum will produce an end carry rn, which can be discarded; what is left is the result M - N.
3. If M < N, the sum does not produce an end carry and is equal to rn - (N - M), which is r’s complement of
(N - M). Take the r’s complement of the sum and place a negative sign in front.
Fuw-Yi Yang

11. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
Example 1.5 Using 10’s complement,
subtract
1. M = 72532, N = 3250, 10’s complement of N = 96750 2.
3. answer: 69282
Fuw-Yi Yang

12. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
Example 1.6 Using 10’s complement,
subtract
1. M = 3250, N = 72532, 10’s complement of N = 27468 2.
3. answer: -( ) =
Fuw-Yi Yang

13. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
Example 1.7 Using 2’s complement,
subtract
1. M = ,
N = , 2’s complement of N = 2.
3. answer:
Fuw-Yi Yang

14. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
Example 1.7' Using 2’s complement,
subtract
1. M = ,
N = , 2’s complement of N = 2.
3. answer: - ( ) =
Fuw-Yi Yang

15. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
Example 1.8 Using 1’s complement,
subtract
1. M = ,
N = , 1’s complement of N = 2.
3. answer: (rn carry, call end-around carry)
Fuw-Yi Yang

16. Text Book: Digital Design 4th Ed.
Chap Complements— Subtraction with
Complements
Example 1.8' Using 1’s complement,
subtract
1. M = ,
N = , 1’s complement of N = 2.
3. answer: - ( ) =
Fuw-Yi Yang

17. Text Book: Digital Design 4th Ed. Chap 1 1.6 Signed Binary Numbers
Next table shows signed binary numbers
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18. Text Book: Digital Design 4th Ed. Chap 1 1.6 Signed Binary Numbers
Arithmetic addition
Arithmetic subtraction
See next table
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19. Text Book: Digital Design 4th Ed. Chap 1 1.6 Signed Binary Numbers
Arithmetic addition with comparison:
The addition of two numbers in the signed magnitude system follows the rules of ordinary arithmetic.
If the signed are the same, we add the two magnitudes and give the sum the common sign.
If the signed are different, we subtract the smaller magnitude from the larger and give the difference the sign of the larger magnitude. EX. (+25) + (-38) = -( ) = -13
Fuw-Yi Yang

20. Text Book: Digital Design 4th Ed. Chap 1 1.6 Signed Binary Numbers
Arithmetic addition without comparison:
The addition of two signed binary numbers with negative numbers represented in signed 2’s complement form is obtained from the addition of the two numbers, including their signed bits. A carry out of the signed bit position is discarded (note that the 4th case).
See examples in next page.
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21. Text Book: Digital Design 4th Ed. Chap 1 1.6 Signed Binary Numbers
Arithmetic addition without comparison:
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22. Text Book: Digital Design 4th Ed. Chap 1 1.7 Binary Codes
BCD (Binary-Coded Decimal) Code Table 1.4
Decimal codes Table 1.5
(4 different Codes for the Decimal Digits)
Gray code Table 1.6
ASCII character code Table 1.7
Error Detecting code
See next tables 1.4~1.7
Fuw-Yi Yang

23. Text Book: Digital Design 4th Ed. Chap 1 1.7 Binary Codes
BCD Code
Decimal codes
Gray code
ASCII character code
Error Detecting code
See next tables
Fuw-Yi Yang

24. Text Book: Digital Design 4th Ed. Chap 1 1.7 Binary Codes
BCD Code
Decimal codes
Gray code
ASCII character code
Error Detecting code
See next tables
Fuw-Yi Yang

25. Text Book: Digital Design 4th Ed. Chap 1 1.7 Binary Codes
BCD Code
Decimal codes
Gray code
ASCII character code
Error Detecting code
See next tables
Fuw-Yi Yang

26. Text Book: Digital Design 4th Ed. Chap 1 1.7 Binary Codes
BCD Code
Decimal codes
Gray code
ASCII character code
Error Detecting code
See next tables
Fuw-Yi Yang

27. Text Book: Digital Design 4th Ed.
Chap Binary Storage and Registers
A register is a group of binary cells.
A register with n cells can store any discrete quantity of information that contains n bits.
A digital system is characterized by its registers and the components that perform data processing.
In digital systems, a register transfer operation is a basic operation that consists of a transfer of binary information from one set of registers into another set of registers.
Fuw-Yi Yang

28. Text Book: Digital Design 4th Ed.
Chap Binary Storage and Registers
A register is a group of binary cells.
A register with n cells can store any discrete quantity of information that contains n bits.
A digital system is characterized by its registers and the components that perform data processing.
In digital systems, a register transfer operation is a basic operation that consists of a transfer of binary information from one set of registers into another set of registers.
Fuw-Yi Yang

29. Text Book: Digital Design 4th Ed.
Chap Binary Storage and Registers
A register is a group of binary cells.
A register with n cells can store any discrete quantity of information that contains n bits.
A digital system is characterized by its registers and the components that perform data processing.
In digital systems, a register transfer operation is a basic operation that consists of a transfer of binary information from one set of registers into another set of registers.
Fuw-Yi Yang

30. Text Book: Digital Design 4th Ed. Chap 1 1.9 Binary Logic
Binary logic consists of binary variables and a set of
logical operations.
There are three basic logical operations:
AND, OR, and NOT.
AND, OR, NOT: Table 1.8
Binary signals: Figure 1.3
Logic circuit: Figures 1.4~1.6
Ex 1.35, 1.36
Fuw-Yi Yang

31. Binary logic consists of binary variables and a set of
logical operations.
There are three basic logical operations:
AND, OR, and NOT.
See truth table 1.8 next page.
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