1
Number System
Lecture 2
2
Introduction to Number Systems
• We are all familiar with the decimal number
system (Base 10). Some other number
systems...
3
Characteristics of Numbering Systems
1) The digits are consecutive.
2) The number of digits is equal to the size of
the ...
4
Significant Digits
Decimal: 82347
Most significant digit Least significant digit
Binary: 101010
Most significant digit L...
5
Binary Number System
• Also called the “Base 2 system”
• The binary number system is used to model
the series of electri...
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...
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 t...
8
Decimal to Binary Conversion
• The easiest way to convert a decimal
number to its binary equivalent is to use the
Divisi...
9
Division Algorithm
Convert 67 to its binary equivalent:
6710 = x2
Step 1: 67 / 2 = 33 R 1 Divide 67 by 2. Record quotien...
10
Binary to Decimal Conversion
• The easiest method for converting a binary number
to its decimal equivalent is to use th...
11
Multiplication Algorithm
Convert (10101101)2 to its decimal equivalent:
Binary 1 0 1 0 1 1 0 1
Positional Values
xxxxxx...
12
Octal Number System
• Also known as the Base 8 System
• Uses digits 0 - 7
• Readily converts to binary
• Groups of thre...
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 ...
14
Octal to Decimal Conversion
Convert 6538 to its decimal equivalent:
6 5 3
xxx
82 81 80
384 + 40 + 3
42710
Positional Va...
15
Octal to Binary Conversion
Each octal number converts to 3 binary digits
To convert 6538 to binary, just
substitute cod...
16
Hexadecimal Number System
• Base 16 system
• Uses digits 0-9 &
letters A,B,C,D,E,F
• Groups of four bits
represent each...
17
Decimal to Hexadecimal Conversion
Convert 83010 to its hexadecimal equivalent:
830 / 16 = 51 R 14
51 / 16 = 3 R 3
3 / 1...
18
Hexadecimal to Decimal Conversion
Convert 3B4F16 to its decimal equivalent:
Hex Digits
3 B 4 F
xxx
163 162 161 160
1228...
19
Binary to Hexadecimal Conversion
• The easiest method for converting binary to
hexadecimal is to use a substitution cod...
20
Convert 0101011010101110011010102 to hex
using the 4-bit substitution code :
0101 0110 1010 1110 0110 1010
Substitution...
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...
22
Complement
• Complement is the negative equivalent of a number.
• If we have a number N then complement of N will
give ...
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 co...
24
Example
• Suppose N = (3675)10
• So we can find two complements of this number.
The 10’s complement and the 9’s complem...
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 g...
26
Example:
• (110100101)2 1’s complement 001011010
• (110100101)2 2’s complement 001011011
27
Use of Complement
• Complement is used to perform subtraction using addition
• Mathematically A-B = A + (-B)
• So we ca...
28
Complementary Arithmetic
• 1’s complement
• Switch all 0’s to 1’s and 1’s to 0’s
Binary # 10110011
1’s complement 01001...
29
Complementary Arithmetic
• 2’s complement
• Step 1: Find 1’s complement of the number
Binary # 11000110
1’s complement ...
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 complem...
31
Thank You
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Lecture 2 ns

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Lecture 2 ns

  1. 1. 1 Number System Lecture 2
  2. 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. 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. 4 Significant Digits Decimal: 82347 Most significant digit Least significant digit Binary: 101010 Most significant digit Least significant digit
  5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26 Example: • (110100101)2 1’s complement 001011010 • (110100101)2 2’s complement 001011011
  27. 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. 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. 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. 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. 31 Thank You
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