ESSENTIAL of (CS/IT/IS) class 03-04 (NUMERIC SYSTEMS)
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The document discusses various numbering systems used in digital logic, including decimal, binary, octal, and hexadecimal systems. It explains the basics of these systems, including conversion methods between them, and highlights operations such as addition, subtraction, and multiplication in binary arithmetic. Additionally, it covers the concepts of signed numbers and their representation in the binary system.
![SEIYUN UNIVERSITY
College of Applied Science – CS department
Complementary Arithmetic
• 1’s complement
• Switch all 0’s to 1’s and 1’s to 0’s
Binary # 10110011
1’s complement 01001100
• The 2's complement representation of a binary number
X is defined by the equation:
[X]2 = 2n – X
For X = 1010 and n = 4 then:
[X]2 = 24 - 1010
= 10000 - 1010 = 0110](https://image.slidesharecdn.com/ess-class-03-04-240203233523-e72f7ac6/85/ESSENTIAL-of-CS-IT-IS-class-03-04-NUMERIC-SYSTEMS-24-320.jpg)
ESSENTIAL of (CS/IT/IS) class 03-04 (NUMERIC SYSTEMS)
- 1. SEIYUN UNIVERSITY College of Applied Science – CS department Instructor: Dr. Mazin Md. Alkathiri Information technology Department Faculty of Applied Sciences Seiyun University – Yemen Jan 2024
- 2. SEIYUN UNIVERSITY College of Applied Science – CS department 2
- 3. SEIYUN UNIVERSITY College of Applied Science – CS department 3 A digital logic system may well have a numerical computation capability as well as its inherent logical capability. Human beings normally perform arithmetic operations using the decimal number system, but, a digital machine is inherently binary and its numerical calculations are executed using a binary number system. Since the decimal system has ten digits, a ten-state device is required to represent the decimal digits. Ten-state devices are not readily available in the electrical world, however two-state devices such as a transistor which are operating in a switching mode. A number of other systems such as the hexadecimal system are used in conjunction with programmable logic devices. Introduction
- 4. SEIYUN UNIVERSITY College of Applied Science – CS department 4 We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: Decimal Base 10 Binary Base 2 Octal Base 8 Hexadecimal Base 16 Introduction
- 5. SEIYUN UNIVERSITY College of Applied Science – CS department 5 The digits are consecutive. The number of digits is equal to the size of the base. Zero is always the first digit. The base number is never a digit. When 1 is added to the largest digit, a sum of zero and a carry of one results. Characteristics of Numbering Systems
- 6. SEIYUN UNIVERSITY College of Applied Science – CS department Significant Digits Binary: 11101101 Most significant digit Least significant digit Hexadecimal: 1D63A7A Most significant digit Least significant digit
- 7. SEIYUN UNIVERSITY College of Applied Science – CS department Binary Number System • Also called the “Base 2 system” • The binary number system is used to model the series of electrical signals computers use to represent information • 0 represents the no voltage or an off state • 1 represents the presence of voltage or an on state
- 8. SEIYUN UNIVERSITY College of Applied Science – CS department Binary Numbering Scale Base 2 Number Base 10 Equivalent Power Positional Value 000 0 20 1 001 1 21 2 010 2 22 4 011 3 23 8 100 4 24 16 101 5 25 32 110 6 26 64 111 7 27 128
- 9. SEIYUN UNIVERSITY College of Applied Science – CS department Binary Addition 4 Possible Binary Addition Combinations: (1) 0 (2) 0 +0 +1 00 01 (3) 1 (4) 1 +0 +1 01 10 Sum Carry
- 10. SEIYUN UNIVERSITY College of Applied Science – CS department Decimal to Binary Conversion • The easiest way to convert a decimal number to its binary equivalent is to use the Division Algorithm • This method repeatedly divides a decimal number by 2 and records the quotient and remainder • The remainder digits (a sequence of zeros and ones) form the binary equivalent in least significant to most significant digit sequence
- 11. SEIYUN UNIVERSITY College of Applied Science – CS department Division Algorithm Convert (67)10 to its binary equivalent: 6710 = x2 Step 1: 67 / 2 = 33 R 1 Divide 67 by 2. Record quotient in next row Step 2: 33 / 2 = 16 R 1 Again divide by 2; record quotient in next row Step 3: 16 / 2 = 8 R 0 Repeat again Step 4: 8 / 2 = 4 R 0 Repeat again Step 5: 4 / 2 = 2 R 0 Repeat again Step 6: 2 / 2 = 1 R 0 Repeat again Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0 (1 0 0 0 0 1 1)2
- 12. SEIYUN UNIVERSITY College of Applied Science – CS department Binary to Decimal Conversion • The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm • Multiply the binary digits by increasing powers of two, starting from the right • Then, to find the decimal number equivalent, sum those products
- 13. SEIYUN UNIVERSITY College of Applied Science – CS department Multiplication Algorithm Convert (10101101)2 to its decimal equivalent: Binary 1 0 1 0 1 1 0 1 Positional Values x x x x x x x x 20 21 22 23 24 25 26 27 128 + 32 + 8 + 4 + 1 Products (173)10
- 14. SEIYUN UNIVERSITY College of Applied Science – CS department Octal Number System • Also known as the Base 8 System • Uses digits 0 - 7 • Readily converts to binary • Groups of three (binary) digits can be used to represent each octal digit • Also uses multiplication and division algorithms for conversion to and from base 10
- 15. SEIYUN UNIVERSITY College of Applied Science – CS department Decimal to Octal Conversion Convert (427)10 to its octal equivalent: 427 / 8 = 53 R3 Divide by 8; R is LSD 53 / 8 = 6 R5 Divide Q by 8; R is next digit 6 / 8 = 0 R6 Repeat until Q = 0 (653)8
- 16. SEIYUN UNIVERSITY College of Applied Science – CS department Octal to Decimal Conversion Convert (653)8 to its decimal equivalent: 6 5 3 x x x 82 81 80 384 + 40 + 3 (427)10 Positional Values Products Octal Digits
- 17. SEIYUN UNIVERSITY College of Applied Science – CS department Octal to Binary Conversion Each octal number converts to 3 binary digits To convert (653)8 to binary, just substitute code: 6 5 3 110 101 011
- 18. SEIYUN UNIVERSITY College of Applied Science – CS department Hexadecimal Number System • Base 16 system • Uses digits 0-9 & letters A,B,C,D,E,F • Groups of four bits represent each base 16 digit
- 19. SEIYUN UNIVERSITY College of Applied Science – CS department Decimal to Hexadecimal Conversion Convert (830)10 to its hexadecimal equivalent: 830 / 16 = 51 R14 51 / 16 = 3 R3 3 / 16 = 0 R3 (33E)16 = E in Hex
- 20. SEIYUN UNIVERSITY College of Applied Science – CS department Hexadecimal to Decimal Conversion Convert (3B4F)16 to its decimal equivalent: Hex Digits 3 B 4 F x x x 163 162 161 160 12288 +2816 + 64 +15 (15,183)10 Positional Values Products x
- 21. SEIYUN UNIVERSITY College of Applied Science – CS department Binary to Hexadecimal Conversion • The easiest method for converting binary to hexadecimal is to use a substitution code • Each hex number converts to 4 binary digits
- 22. SEIYUN UNIVERSITY College of Applied Science – CS department Convert (010101101010111001101010)2 to hex using the 4-bit substitution code : 0101 0110 1010 1110 0110 1010 Substitution Code 5 6 A E 6 A (56AE6A)16
- 23. SEIYUN UNIVERSITY College of Applied Science – CS department Substitution code can also be used to convert binary to octal by using 3-bit groupings: 010 101 101 010 111 001 101 010 Substitution Code 2 5 5 2 7 1 5 2 (25527152)8
- 24. SEIYUN UNIVERSITY College of Applied Science – CS department Complementary Arithmetic • 1’s complement • Switch all 0’s to 1’s and 1’s to 0’s Binary # 10110011 1’s complement 01001100 • The 2's complement representation of a binary number X is defined by the equation: [X]2 = 2n – X For X = 1010 and n = 4 then: [X]2 = 24 - 1010 = 10000 - 1010 = 0110
- 25. SEIYUN UNIVERSITY College of Applied Science – CS department • 2’s complement • Step 1: Find 1’s complement of the number Binary # 11000110 1’s complement 00111001 • Step 2: Add 1 to the 1’s complement 00111001 + 1 00111010 Complementary Arithmetic
- 26. SEIYUN UNIVERSITY College of Applied Science – CS department Signed magnitude representation Signed 1's complement representation Signed 2's complement representation Example: Represent (+9) and (-9) in 8 bit-binary number Only one way to represent +9 ==> 0 0001001 Three different ways to represent -9: In signed-magnitude: 1 0001001 In signed-1's complement: 1 1110110 In signed-2's complement: 1 1110111 Need to be able to represent both positive and negative numbers - Following 3 representations Signed Numbers
- 27. SEIYUN UNIVERSITY College of Applied Science – CS department Complement: Signed magnitude: Complement only the sign bit Signed 1's complement: Complement all the bits including sign bit Signed 2's complement: After the first (1) of the positive number complement the bits including its sign bit. Signed Numbers
- 28. SEIYUN UNIVERSITY College of Applied Science – CS department Singed Binary Numbers In computers, both positive and negative numbers are represents only with binary digits; The left most bit (sign bit) in the number represents sign of the number; The sign bit is 0 for the positive numbers, and 1 for negative numbers.
- 29. SEIYUN UNIVERSITY College of Applied Science – CS department Singed Binary Numbers Addition and subtraction of 2's complement numbers: Addition and subtraction in the 2's complement system are both carried out as additions. Example 1: Addition of two 8-bit positive numbers.
- 30. SEIYUN UNIVERSITY College of Applied Science – CS department Operations on Binary system The rules for the addition of two single-bit numbers as following : A B Sum Carry 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 11 + 7 18 Carry Sum
- 31. SEIYUN UNIVERSITY College of Applied Science – CS department The rules for the subtraction of two single-bit numbers as following : A B Difference Borrow 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 12 - 3 9 Operations on Binary system Borrow Deff.
- 32. SEIYUN UNIVERSITY College of Applied Science – CS department The rules for the Multiplication of two single-bit numbers as following : Operations on Binary system A B Product 0 0 0 0 1 0 1 0 0 1 1 1 1110 x 1011 1110 1110 0000 1110 10011010
- 33. SEIYUN UNIVERSITY College of Applied Science – CS department Conversion of decimal numbers to any Radix number Conversion of decimal to any radix number accomplished in two steps: Step 1: Convert the Integer part – by successive division method; Step 2: Convert the Fraction part - by successive multiplication method.
- 34. SEIYUN UNIVERSITY College of Applied Science – CS department Conversion of decimal numbers to any Radix number 1. successive division for integer part:
- 35. SEIYUN UNIVERSITY College of Applied Science – CS department Conversion of decimal numbers to any Radix number
- 36. SEIYUN UNIVERSITY College of Applied Science – CS department 2. successive multiplication for Fraction part: Conversion of decimal numbers to any Radix number
- 37. SEIYUN UNIVERSITY College of Applied Science – CS department Conversion of decimal numbers to any Radix number
- 38. SEIYUN UNIVERSITY College of Applied Science – CS department Fractions
- 39. SEIYUN UNIVERSITY College of Applied Science – CS department 3.14579 .14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. 11.001001... Convert Decimal to Binary (3.14579)10 = (?)2 Fractions
- 40. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal The rules for the Addition of two numbers as following : (3 4)8 + (4 2)8 (7 6)8 (5 6)8 + (6 3)8 (1 4 1)8
- 41. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal
- 42. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal
- 43. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal
- 44. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal
- 45. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal
- 46. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Octal
- 47. SEIYUN UNIVERSITY College of Applied Science – CS department F E D C B A 9 8 7 6 5 4 3 2 1 0 + F E D C B A 9 8 7 6 5 4 3 2 1 0 0 10 F E D C B A 9 8 7 6 5 4 3 2 1 1 11 10 F E D C B A 9 8 7 6 5 4 3 2 2 12 11 10 F E D C B A 9 8 7 6 5 4 3 3 13 12 11 10 F E D C B A 9 8 7 6 5 4 4 14 13 12 11 10 F E D C B A 9 8 7 6 5 5 15 14 13 12 11 10 F E D C B A 9 8 7 6 6 16 15 14 13 12 11 10 F E D C B A 9 8 7 7 17 16 15 14 13 12 11 10 F E D C B A 9 8 8 18 17 16 15 14 13 12 11 10 F E D C B A 9 9 19 18 17 16 15 14 13 12 11 10 F E D C B A A 1A 19 18 17 16 15 14 13 12 11 10 F E D C B B 1B 1A 19 18 17 16 15 14 13 12 11 10 F E D C C 1C 1B 1A 19 18 17 16 15 14 13 12 11 10 F E D D 1D 1C 1B 1A 19 18 17 16 15 14 13 12 11 10 F E E 1E 1D 1C 1B 1A 19 18 17 16 15 14 13 12 11 10 F F Operation in Hexa The rules for the Addition of two numbers as following : (3 5 A B 2)16 (1 A 6 7 5)16 (5 0 1 2 7)16 1 1 1 carry
- 48. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 49. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 50. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 51. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 52. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 53. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 54. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 55. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa
- 56. SEIYUN UNIVERSITY College of Applied Science – CS department Operation in Hexa