Number Systems: Positive Integers

1
Department of Computer and Information
Science,School of Science, IUPUI
CSCI 230
Information Representation Positive Integers
Dale Roberts, Lecturer IUPUI droberts_at_cs.iupui.edu
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Information Representation

• Computer use a binary systems
• Why binary?
• Electronic bi-stable environment
• on/off, high/low voltage
• Bit each bit can be either 0 or 1
• Reliability
• With only 2 values, can be widely separated,
  therefore clearly differentiated
• drift causes less error
• Example

Digital v.s, Analog
1 0 0 1 0 1 0 1
1 0 0 0 0 0 0 1
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• Binary Representation in Computer System
• All information of diverse type is represented
  within computers in the form of bit patterns.
• e.g., text, numerical data, sound, and images
• One important aspect of computer design is to
  decide how information is converted ultimately to
  a bit pattern
• Writing software also frequently requires
  understanding how information is represented
  along with accuracies

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Number Systems

• Decimal Number System
• Base is 10 or D or Dec
• Ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Each place is weighted by the power of 10
• Example
• 1234.2110 or 1234.21D
• 1 x 103 2 x 102 3 x 101 4 x 100 2 x 10
  -1 1 x 10 -2
• 1000 200 30 4 0.2 0.01

1,000
100
1
10
10
1
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Binary Number System

• Binary Number System
• Base is 2 or b or B or Bin
• Two symbols 0 and 1
• Each place is weighted by the power of 2
• Example
• 10112 or 1011 B
• 1 x 23 0 x 22 1 x 21 1 x 20
• 8 0 2 1
• 1110
• 11 in decimal number system is 1011 in binary
  number system

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Conversion between Decimal and Binary

• Conversion from decimal number system to binary
  system
• Question represent 3410 in the binary number
  system
• Answer using the divide-by-2 technique
  repeatedly
• If we write the remainder from right to left
• 3410 ? 1 x 25 0 x 24 0 x 23 0 x 22 1x 21
  0 x 20
• ? 1000102

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Practice Exercises

• 13D (?) B
• 23D (?) B
• 72D (?) B

Blocks 512 256 128 64 32 16 8 4 2 1
32 16 8 4 2 1
2
8
8
1
1101B
4
1
4
16
2
8
4
1
16
2
8
32
4
1
64
16
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Conversion between Binary and Decimal

• Conversion from binary number system to decimal
  system
• Example check if 1000102 is 3410
• using the weights appropriately
• 1000102 ? 1 x 25 0 x 24 0 x 23 0 x 22 1 x
  21 0 x 20
• ? 32 0 0 0 2
  0
• ? 3410

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Practice Exercises

• Ex 0101B? ( ? ) D
• Ex 1100B ? ( ? ) D
• Ex 0101 1100B ? ( ? ) D

Bit 4 23 8 Bit 3 22 4 Bit 2 21 2 Bit 1 20 1
0 1 0 1
4
1

5D

• 1 0 0
• 8 4 2 1 12D

0 1 0 1 1 1 0 0 128
64 32 16 8 4
2 1 92D
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Binary Arithmetic on Integers

• Addition

0
1
1
1 0
Carry bit

• Example find binary number of a b
• If a 13D , b 5D
• If a 15D, b 10D

1 1 0 1b
0 1 0 1b
1
0b
0
0
1
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Binary Arithmetic on Integers

• Multiplication

0
0
0
1
Example if a 100001b , b 101b , find a x b
? 33D
? 5D
1 0 0 0 0 1
0 0 0 0 0 0 0
1 0 0 0 0 1 0 0
? 165D
1 0 1 0 0 1 0 1b
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Hexadecimal Number System

• Hexadecimal Number System
• Base 16 or H or Hex
• 16 symbols
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(10), B(11),
  C(12), D(13), E(14), F(15)
• Hexadecimal to Decimal
• (an-1an-2a1a0)16 (an-1 x 16n-1 an-2 x 16n-2
  a1 x 161 a0 x 160 )D
• Example (1C7)16 (1 x 162 12 x 161 7 x 160
  )10 (256 192 7)10 (455)10
• Decimal to Hexadecimal Repeated division by 16
• Similar in principle to generating binary codes
• Example (829)10 (? )16
• Stop, since quotient 0

Divide-by-16 Quotient Remainder Hexadecimal digit
829 / 16 51 / 16 3 / 16 51 3 0 13 3 3 Lower digit D Second digit 3 Third digit 3
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Hexadecimal Conversions

• Hexadecimal to Binary
• Expand each hexadecimal digit to 4 binary bits.
• Example (E29)16 (1110 0010 1001)2
• Binary to Hexadecimal
• Combine every 4 bits into one hexadecimal digit
• Example (0101 1111 1010 0110)2
  (5FA6)16

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Octal Number System

• Octal Number System
• Base 8 or o or Oct
• 8 symbols 0, 1, 2, 3, 4, 5, 6, 7
• Octal to Decimal
• (an-1an-2a1a0)8 (an-1 x 8n-1 an-2 x 8n-2
  a1 x 81 a0 x 80 )10
• Example (127)8 (1 x 82 2 x 81 7 x 80 )10
  (64 16 7)10 (87)10
• Decimal to Octal
• Repeated division by 8 (similar in principle to
  generating binary codes)
• Example (213)10 (? )8
• Stop, since quotient
  0
• Hence, (213)10
  (325)8

Divide-by -8 Quotient Remainder Octal digit
213 / 8 26 / 8 3 / 8 26 3 0 5 2 3 Lower digit 5 Second digit 2 Third digit 3
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Octal Conversions

• Octal to Binary
• Expand each octal digit to 3 binary bits.
• Example (725)8 (111 010 101)2
• Binary to Octal
• Combine every 3 bits into one octal digit
• Example (110 010 011)2 (623)8

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Practice Exercises

• 1)  Convert the following binary numbers to
  decimal numbers
• (a)     0011 B
• (b)     0101 B
• (c)     0001 0110 B
• (d)     0101 0011 B
• 2) Convert the following decimal numbers to
  binary
• (a)     21 D
• (b)     731 D
• (c)     1,023 D

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Practice Exercises

• 3)    Convert the following binary numbers to
  hexadecimal numbers
• (a)     0011 B
• (b)     0101 B
• (c)     0001 0110 B
• (d)     0101 0011 B
• (a)     21 D
• (b)     731 D
• (c)     1,023 D
• 4.) Perform the following binary additions and
  subtractions. Show your work without using
  decimal numbers during conversion.
• (a)     111 B 101 B
• (b)     1001 B 11 B

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Acknowledgements

• These slides where originally prepared by Dr.
  Jeffrey Huang, updated by Dale Roberts.