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Number Systems: Positive Integers

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Binary Number System. Binary Number System. Base is 2 or b' or B' or Bin' Two symbols: 0 and 1 ... Hexadecimal Number System. Hexadecimal Number System. Base ... – PowerPoint PPT presentation

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Title: 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
2
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
3
  • 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

4
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
5
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

6
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

7
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
8
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

9
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
10
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
11
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
12
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
13
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

14
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
15
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

16
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

17
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

18
Acknowledgements
  • These slides where originally prepared by Dr.
    Jeffrey Huang, updated by Dale Roberts.
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