![Complements
Radix Complement
Example: Base-10
Example: Base-2
The r's complement of an n-digit number N in base r is defined as
rn
– N for N ≠ 0 and as 0 for N = 0. Comparing with the (r 1) 's
complement, we note that the r's complement is obtained by adding 1
to the (r 1) 's complement, since rn
– N = [(rn
1) – N] + 1.
The 10's complement of 012398 is 987602
The 10's complement of 246700 is 753300
The 2's complement of 1101100 is 0010100
The 2's complement of 0110111 is 1001001](https://image.slidesharecdn.com/numbersystem-class-250710172017-17bbfc0e/85/Number_System-and-Boolean-Algebra-in-Digital-System-Design-26-320.jpg)
Number_System and Boolean Algebra in Digital System Design
- 1. Digital Systems and Binary Numbers
- 2. 1.1 Digital Systems 1.2 Binary Numbers 1.3 Number-base Conversions 1.4 Octal and Hexadecimal Numbers 1.5 Complements 1.6 Signed Binary Numbers 1.7 Binary Codes 1.8 Binary Storage and Registers 1.9 Binary Logic
- 3. Digital Systems and Binary Numbers Digital age and information age Digital computers General purposes Many scientific, industrial and commercial applications Digital systems Telephone switching exchanges Digital camera Electronic calculators, PDA's Digital TV Discrete information-processing systems Manipulate discrete elements of information For example, {1, 2, 3, …} and {A, B, C, …}…
- 4. Analog and Digital Signal Analog system The physical quantities or signals may vary continuously over a specified range. Digital system The physical quantities or signals can assume only discrete values. Greater accuracy t X(t) t X(t) Analog signal Digital signal
- 5. Binary Digital Signal An information variable represented by physical quantity. For digital systems, the variable takes on discrete values. Two level, or binary values are the most prevalent values. Binary values are represented abstractly by: Digits 0 and 1 Words (symbols) False (F) and True (T) Words (symbols) Low (L) and High (H) And words On and Off Binary values are represented by values or ranges of values of physical quantities. t V(t) Binary digital signal Logic 1 Logic 0 undefine
- 6. Decimal Number System Base (also called radix) = 10 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } Digit Position Integer & fraction Digit Weight Weight = (Base) Position Magnitude Sum of “Digit x Weight” Formal Notation 1 0 -1 2 -2 5 1 2 7 4 10 1 0.1 100 0.01 500 10 2 0.7 0.04 d2*B2 +d1*B1 +d0*B0 +d-1*B-1 +d-2*B-2 (512.74)10
- 7. Octal Number System Base = 8 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 } Weights Weight = (Base) Position Magnitude Sum of “Digit x Weight” Formal Notation 1 0 -1 2 -2 8 1 1/8 64 1/64 5 1 2 7 4 5 *82 +1 *81 +2 *80 +7 *8-1 +4 *8-2 =(330.9375)10 (512.74)8
- 8. Binary Number System Base = 2 2 digits { 0, 1 }, called binary digits or “bits” Weights Weight = (Base) Position Magnitude Sum of “Bit x Weight” Formal Notation Groups of bits 4 bits = Nibble 8 bits = Byte 1 0 -1 2 -2 2 1 1/2 4 1/4 1 0 1 0 1 1 *22 +0 *21 +1 *20 +0 *2-1 +1 *2-2 =(5.25)10 (101.01)2 1 0 1 1 1 1 0 0 0 1 0 1
- 9. Hexadecimal Number System Base = 16 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F } Weights Weight = (Base) Position Magnitude Sum of “Digit x Weight” Formal Notation 1 0 -1 2 -2 16 1 1/16 256 1/256 1 E 5 7 A 1 *162 +14 *161 +5 *160 +7 *16-1 +10 *16-2 =(485.4765625)10 (1E5.7A)16
- 10. The Power of 2 n 2n 0 20 =1 1 21 =2 2 22 =4 3 23 =8 4 24 =16 5 25 =32 6 26 =64 7 27 =128 n 2n 8 28 =256 9 29 =512 10 210 =1024 11 211 =2048 12 212 =4096 20 220 =1M 30 230 =1G 40 240 =1T Mega Giga Tera Kilo
- 11. Addition Decimal Addition 5 5 5 5 + 0 1 1 1 1 Carry
- 12. Binary Addition Column Addition 1 0 1 1 1 1 1 1 1 1 0 + 0 0 0 0 1 1 1 1 1 1 1 1 1 = 61 = 23 = 84
- 13. Binary Subtraction Borrow a “Base” when needed 0 0 1 1 1 0 1 1 1 1 0 − 0 1 0 1 1 1 0 = (10)2 2 2 2 2 1 0 0 0 1 = 77 = 23 = 54
- 14. Binary Multiplication Bit by bit 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 x
- 16. Number Base Conversions Decimal (Base 10) Octal (Base 8) Binary (Base 2) Hexadecimal (Base 16) Evaluate Magnitude Evaluate Magnitude Evaluate Magnitude
- 17. Decimal (Integer) to Binary Conversion Divide the number by the ‘Base’ (=2) Take the remainder (either 0 or 1) as a coefficient Take the quotient and repeat the division Example: (13)10 Quotient Remainder Coefficient Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2 MSB LSB MSB LSB 13/ 2 = 6 1 a0 = 1 6 / 2 = 3 0 a1 = 0 3 / 2 = 1 1 a2 = 1 1 / 2 = 0 1 a3 = 1
- 18. Decimal (Fraction) to Binary Conversion Multiply the number by the ‘Base’ (=2) Take the integer (either 0 or 1) as a coefficient Take the resultant fraction and repeat the division Example: (0.625)10 Integer Fraction Coefficient Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2 MSB LSB MSB LSB 0.625 * 2 = 1 . 25 0.25 * 2 = 0 . 5 a-2 = 0 0.5 * 2 = 1 . 0 a-3 = 1 a-1 = 1
- 19. Decimal to Octal Conversion Example: (175)10 Quotient Remainder Coefficient Answer: (175)10 = (a2 a1 a0)8 = (257)8 175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2 Example: (0.3125)10 Integer Fraction Coefficient Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8 0.3125 * 8 = 2 . 5 0.5 * 8 = 4 . 0 a-2 = 4 a-1 = 2
- 20. Binary − Octal Conversion 8 = 23 Each group of 3 bits represents an octal digit Octal Binary 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1 Example: ( 1 0 1 1 0 . 0 1 )2 ( 2 6 . 2 )8 Assume Zeros Works both ways (Binary to Octal & Octal to Binary)
- 21. Binary − Hexadecimal Conversion 16 = 24 Each group of 4 bits represents a hexadecimal digit Hex Binary 0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 A 1 0 1 0 B 1 0 1 1 C 1 1 0 0 D 1 1 0 1 E 1 1 1 0 F 1 1 1 1 Example: ( 1 0 1 1 0 . 0 1 )2 ( 1 6 . 4 )16 Assume Zeros Works both ways (Binary to Hex & Hex to Binary)
- 22. Octal − Hexadecimal Conversion Convert to Binary as an intermediate step Example: ( 0 1 0 1 1 0 . 0 1 0 )2 ( 1 6 . 4 )16 Assume Zeros Works both ways (Octal to Hex & Hex to Octal) ( 2 6 . 2 )8 Assume Zeros
- 23. Decimal, Binary, Octal and Hexadecimal Decimal Binary Octal Hex 00 0000 00 0 01 0001 01 1 02 0010 02 2 03 0011 03 3 04 0100 04 4 05 0101 05 5 06 0110 06 6 07 0111 07 7 08 1000 10 8 09 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F
- 24. 1.5 Complements There are two types of complements for each base-r system: the radix complement and diminished radix complement. Diminished Radix Complement - (r-1)’s Complement Given a number N in base r having n digits, the (r–1)’s complement of N is defined as: (rn –1) – N Example for 6-digit decimal numbers: 9’s complement is (rn – 1)–N = (106 –1)–N = 999999–N 9’s complement of 546700 is 999999–546700 = 453299 Example for 7-digit binary numbers: 1’s complement is (rn – 1) – N = (27 –1)–N = 1111111–N 1’s complement of 1011000 is 1111111–1011000 = 0100111 Observation: Subtraction from (rn – 1) will never require a borrow Diminished radix complement can be computed digit-by-digit For binary: 1 – 0 = 1 and 1 – 1 = 0
- 25. Complements 1’s Complement (Diminished Radix Complement) All ‘0’s become ‘1’s All ‘1’s become ‘0’s Example (10110000)2 (01001111)2 If you add a number and its 1’s complement … 1 0 1 1 0 0 0 0 + 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1
- 26. Complements Radix Complement Example: Base-10 Example: Base-2 The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r 1) 's complement, we note that the r's complement is obtained by adding 1 to the (r 1) 's complement, since rn – N = [(rn 1) – N] + 1. The 10's complement of 012398 is 987602 The 10's complement of 246700 is 753300 The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001
- 27. Complements 2’s Complement (Radix Complement) Take 1’s complement then add 1 Toggle all bits to the left of the first ‘1’ from the right Example: Number: 1’s Comp.: 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 + 1 OR 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0
- 28. Complements Subtraction with Complements The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:
- 29. Complements Example 1.5 Using 10's complement, subtract 72532 – 3250. Example 1.6 Using 10's complement, subtract 3250 – 72532. There is no end carry. Therefore, the answer is – (10's complement of 30718) = 69282.
- 30. Complements Example 1.7 Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y ; and (b) Y X, by using 2's complement. There is no end carry. Therefore, the answer is Y – X = (2's complement of 1101111) = 0010001.
- 31. Complements Subtraction of unsigned numbers can also be done by means of the (r 1)'s complement. Remember that the (r 1) 's complement is one less then the r's complement. Example 1.8 Repeat Example 1.7, but this time using 1's complement. There is no end carry, Therefore, the answer is Y – X = (1's complement of 1101110) = 0010001.
- 33. 1.9 Binary Logic Definition of Binary Logic Binary logic consists of binary variables and a set of logical operations. The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0, Three basic logical operations: AND, OR, and NOT.
- 34. Binary Logic Truth Tables, Boolean Expressions, and Logic Gates x y z x y z 0 0 0 0 1 0 1 0 0 1 1 1 x y z 0 0 0 0 1 1 1 0 1 1 1 1 x z 0 1 1 0 AND OR NOT x y z z = x • y = x y z = x + y z = x = x’ x z
- 36. Binary Logic Logic gates Example of binary signals 0 1 2 3 Logic 1 Logic 0 Un-define Figure 1.3 Example of binary signals
- 37. Binary Logic Logic gates Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.4 Symbols for digital logic circuits Fig. 1.5 Input-Output signals for gates
- 38. Binary Logic Logic gates Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.6 Gates with multiple inputs
- 39. DeMorgan’s Theory • DeMorgan’s Theorems are basically two sets of rules or laws developed from the Boolean expressions for AND, OR and NOT using two input variables, A and B. • These two rules or theorems allow the input variables to be negated and converted from one form of a Boolean function into an opposite form.
- 46. Few more logic gates The NAND gate and NOR gate can be called the universal gates since the combination of these gates can be used to accomplish any of the basic operations. Hence, NAND gate and NOR gate combination can produce an inverter, an OR gate or an AND gate. What is a NAND gate? The NAND gate or “NotAND” gate is the combination of two basic logic gates, the AND gate and the NOT gate connected in series.
- 48. What is a NOR Gate? • NOR gate is the inverse of an OR gate, and its circuit is produced by connecting an OR gate to a NOT gate. Just like an OR gate, a NOR gate may have any number of input probes but only one output probe.
- 52. • A NOT gate is made by joining the inputs of a NAND gate together. Since a NAND gate is equivalent to an AND gate followed by a NOT gate, joining the inputs of a NAND gate leaves only the NOT gate. •
- 53. XOR gate • is a digital logic gate that gives a true (1 or HIGH) output when the number of true inputs is odd.
- 54. XNOR gate • A high output (1) results if both of the inputs to the gate are the same. If one but not both inputs are high (1), a low output (0) results. Hence the gate is sometimes called an "equivalence gate” The algebraic notation used to represent the XNOR operation is •