Lecture 2 ns
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This document provides an overview of different number systems, including binary, octal, decimal, and hexadecimal. It discusses the characteristics of numbering systems, significant digits, and how to convert between different bases using algorithms like division and multiplication. Specifically, it explains how to convert a decimal number to its binary, octal, or hexadecimal equivalent and vice versa. The document also covers binary addition and complements, describing how 1's and 2's complements are used to represent negative numbers in binary and perform subtraction through addition.
Lecture 2 ns
- 2. 2 Introduction to Number Systems • We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: • Binary Base 2 • Octal Base 8 • Hexadecimal Base 16
- 3. 3 Characteristics of Numbering Systems 1) The digits are consecutive. 2) The number of digits is equal to the size of the base. 3) Zero is always the first digit. 4) The base number is never a digit. 5) When 1 is added to the largest digit, a sum of zero and a carry of one results. 6) Numeric values determined by the implicit positional values of the digits.
- 4. 4 Significant Digits Decimal: 82347 Most significant digit Least significant digit Binary: 101010 Most significant digit Least significant digit
- 5. 5 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
- 6. 6 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
- 7. 7 Binary Addition 4 Possible Binary Addition Combinations: (1) 0 (2) 0 +0 +1 00 01 (3) 1 (4) 1 +0 +1 01 10 SumCarry Note that leading zeroes are frequently dropped. SumCarry
- 8. 8 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
- 9. 9 Division Algorithm Convert 67 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 12
- 10. 10 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
- 11. 11 Multiplication Algorithm Convert (10101101)2 to its decimal equivalent: Binary 1 0 1 0 1 1 0 1 Positional Values xxxxxxxx 2021222324252627 128 +0+32 +0+ 8 + 4 +0+ 1Products 17310
- 12. 12 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
- 13. 13 Decimal to Octal Conversion Convert 42710 to its octal equivalent: 427 / 8 = 53 R 3 Divide by 8; R is LSD 53 / 8 = 6 R 5 Divide Q by 8; R is next digit 6 / 8 = 0 R 6 Repeat until Q = 0 6538
- 14. 14 Octal to Decimal Conversion Convert 6538 to its decimal equivalent: 6 5 3 xxx 82 81 80 384 + 40 + 3 42710 Positional Values Products Octal Digits
- 15. 15 Octal to Binary Conversion Each octal number converts to 3 binary digits To convert 6538 to binary, just substitute code: 6 5 3 110 101 011 Code 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
- 16. 16 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
- 17. 17 Decimal to Hexadecimal Conversion Convert 83010 to its hexadecimal equivalent: 830 / 16 = 51 R 14 51 / 16 = 3 R 3 3 / 16 = 0 R 3 33E16 = E in Hex
- 18. 18 Hexadecimal to Decimal Conversion Convert 3B4F16 to its decimal equivalent: Hex Digits 3 B 4 F xxx 163 162 161 160 12288 +2816 + 64 +15 15,18310 Positional Values Products x
- 19. 19 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
- 20. 20 Convert 0101011010101110011010102 to hex using the 4-bit substitution code : 0101 0110 1010 1110 0110 1010 Substitution Code 5 6 A E 6 A 56AE6A16
- 21. 21 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 255271528
- 22. 22 Complement • Complement is the negative equivalent of a number. • If we have a number N then complement of N will give us another number which is equivalent to –N • So if complement of N is M, then we can say M = -N So complement of M = -M = -(-N) = N • So complement of complement gives the original number
- 23. 23 Types of Complement • For a number of base r, two types of complements can be found • 1. r’s complement • 2. (r-1)’s complement • Definition: • If N is a number of base r having n digits then • r’s complement of N = rn – N and • (r-1)’s complement of N = rn-N-1
- 24. 24 Example • Suppose N = (3675)10 • So we can find two complements of this number. The 10’s complement and the 9’s complement. Here n = 4 • 10’s complement of (3675) = 104-3675 = 6325 • 9’s complement of (3675) = 104-3675 -1 = 6324
- 25. 25 Short cut way to find (r-1)’s complement • In the previous example we see that 9’s complement of 3675 is 6324. We can get the result by subtracting each digit from 9. • Similarly for other base, the (r-1)’s complement can be found by subtracting each digit from r-1 (the highest digit in that system). • For binary 1’s complement is even more easy. Just change 1 to 0 and 0 to 1. (Because 1-1=0 and 1- 0=1)
- 26. 26 Example: • (110100101)2 1’s complement 001011010 • (110100101)2 2’s complement 001011011
- 27. 27 Use of Complement • Complement is used to perform subtraction using addition • Mathematically A-B = A + (-B) • So we can get the result of A-B by adding complement of B with A. • So A-B = A + Complement of (B) • Now we can use either r’s complement or (r-1)’s complement
- 28. 28 Complementary Arithmetic • 1’s complement • Switch all 0’s to 1’s and 1’s to 0’s Binary # 10110011 1’s complement 01001100
- 29. 29 Complementary Arithmetic • 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 + 00000001 00111010
- 30. 30 Complementary Arithmetic 1000 8 8 -100 -4 6 14 1000 100 Add 2’s Com 100 1’s complement 011 Discard MSB 1100 2’s complement 100 Result 100 Addition, subtraction, multiplication and division all can be done by only addition.
- 31. 31 Thank You