The document discusses different numeral systems including binary, decimal, octal and hexadecimal. It provides steps for converting numbers between these numeral systems, whether it be from decimal to other bases or vice versa. Specifically, it explains dividing or multiplying by the base to convert to/from decimal, and breaking into bits and grouping when converting between octal and hexadecimal.

1. Lecture 1
Data Types Part 2
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2. Content Lecture 1 Part2
 Numeral Systems
 Converting
 Converting Decimal to Binary
 Converting Decimal to Octal
 Converting Decimal to Hexadecimal
 Converting Binary, Octal and Hexadecimal to Decimal
 Converting between Octal and Hexadecimal
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3. Content Lecture 1 Part2
 Representation of Data in Computer
 Bit
 Nibble
 Byte
 Bit Operators
 Bit-AND
 Bit-OR
 Bit-XOR
 Bit-Not
 Bit left shift
 Bit right shift
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4. Numeral Systems
 Numeral systems are a form of a writing representation of
mathematical numbers
 A numeral system represents uniquely a given set of
numbers
 The most common used numeral system are
 Binary (base 2)
 Decimal (base 10)
 Octal (base 8)
 Hexadecimal (base 16)
 To indicate the used system you place the base after the
number as subscript
 Example: Decimal number (654321)10
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5. Numeral Systems
 Binary
 The binary numeral system is the number system to base 2
 It uses only the digit 0 and 1
 Internally used by all modern computers
 Example: (01001101)2
 Octal
 The octal numeral system is the number system to base 8
 It uses the digits 0 to 7
 Used for example for file permissions under Unix systems
(chmod)
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6. Numeral Systems
 It does not need an extra symbol
 Many programming languages use a single 0 to indicate a octal
number (020)
 Example: (123456)10 = (361100)8
 Hexadecimal
 A positional numeral system with base 16
 It uses 16 symbols commonly 0–9 and A, B, C, D, E, F (or a–f)
 Primary use: a human friendly representation of binary numbers
 Example:
(123)10 = (7B)16
(471508)10 = (731D4)16
 Many programming languages use a 0x to indicate a hexadecimal
number (0x12)
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7. Converting
 Converting means to transform a number from one
numeral system to the other
 Converting will be done between these numeral systems:
 Decimal
 Binary
 Hexadecimal
 Octal
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8. Converting Decimal to Binary
Proceeding
 Dividing the decimal number with 2 (base of binary
number system)
 Note the remainder separately as the first digit from the
right
 Continually repeat the process of dividing until the
quotient is zero
 The remainders are noted separately after each step
 Finally write down the remainders in reverse order
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9. Converting Decimal to Binary
Example 1
Convert (26)10 to Binary
26/2 = 13 + 0  13 * 2 + 0 = 26
13/2 = 6 + 1  6 * 2 + 1 = 13
6/2 = 3 + 0  3 * 2 + 0 = 6
3/2 = 1 + 1  1 * 2 + 1 = 3
1/2 = 0 + 1  0 * 2 + 1 = 1
 (1 1 0 1 0)2
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10. Converting Decimal to Binary
Example 2
Convert (381)10 to Binary
381/2 = 190 + 1  190 * 2 + 1 = 381
190/2 = 95 + 0  95 * 2 + 0 = 190
95/2 = 47 + 1  47 * 2 + 1 = 95
47/2 = 23 + 1  23 * 2 + 1 = 47
23/2 = 11 + 1  11 * 2 + 1 = 23
11/2 = 5 + 1  5 * 2 + 1 = 11
5/2 = 2 + 1  2 * 2 + 1 = 5
2/2 = 1 + 0  1 * 2 + 0 = 2
1/2 = 0 + 1  0 * 2 + 1 = 1
 (1 0 1 1 1 1 1 0 1)2
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11. Converting Decimal to Octal
Proceeding
 Dividing the decimal number with 8 (base of octal number
system)
 Note the remainder separately as the first digit from the
right
 Continually repeat the process of dividing until the
quotient is zero
 The remainders are noted separately after each step
 Finally write down the remainders in reverse order
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12. Converting Decimal to Octal
Example 1
Convert (26)10 to Octal
26/8 = 3 + 2  3 * 8 + 2 = 26
3/8 = 0 + 3  0 * 8 + 3 = 3
 (32)8
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13. Converting Decimal to Octal
Example 2
Convert (384729)10 to Octal
384729/8 = 48091 + 1  48091*8 + 1 = 384729
48091/8 = 6011 + 3  6011*8 + 3 = 48091
6011/8 = 751 + 3  751*8 + 3 = 6011
751/8 = 93 + 7  93*8 + 7 = 751
93/8 = 11 + 5  11*8 + 5 = 93
11/8 = 1 + 3  1*8 + 3 = 11
1/8 = 0 + 1  0*8 + 1 = 1
 (1357331)8
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14. Converting Decimal to
Hexadecimal
Proceeding
 Dividing the decimal number with 16 (base of hexadecimal
number system)
 Note the remainder separately as the first digit from the
right
 If it exceeds 9 convert it into the hexadecimal letter (10 to
A, 11 to B, 12 to C, 13 to D, 14 to E, 15 to F)
 Continually repeat the process of dividing until the
quotient is zero
 The remainders are noted separately after each step
 Finally write down the remainders in reverse order
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15. Converting Decimal to
Hexadecimal
Example 1
Convert (26)10 to Hexadecimal
26/16 = 1 + A  1 * 16 + 10 = 26
1/16 = 0 + 1  0 * 16 + 1 = 1
 (1A)16
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16. Converting Decimal to
Hexadecimal
Example 2
Convert (330382)10 to Hexadecimal
330382/16 = 20648 + E  20648*16 + 14 = 330382
20648/16 = 1290 + 8  1290*16 + 8 = 20648
1290/16 = 80 + A  80*16 + 10 = 1290
80/16 = 5 + 0  5*16 + 0 = 80
5/16 = 0 + 5  0*16 + 5 = 5
 (50A8E)16
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17. Converting Binary, Octal and
Hexadecimal to Decimal
Proceeding
 Each digit of the binary, octal or hexadecimal number is to
be multiplied by its weighted position
 That means:
 binary  decimal: use 2 as weight
 octal  decimal: use 8 as weight
 hexadecimal  decimal: use 16 as weight
 Each of the weighted values is added to get the decimal
number
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18. Converting Binary, Octal and
Hexadecimal to Decimal
Example 1 Convert Binary to Decimal
Therefore the weight is 2
Convert (11010)2 to Decimal
Binary 1 1 0 1 0
Weight 24 23 22 21 20
1*24 + 1*23 + 0*22 + 1*21 + 0*20
Sum 16 + 8 + 0 + 2 + 0 = (26)10
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19. Converting Binary, Octal and
Hexadecimal to Decimal
Example 2 Convert Binary to Decimal
Therefore the weight is 2
Convert (110001)2 to Decimal
Binary 1 1 0 0 0 1
Weight 25 24 23 22 21 20
1*25 + 1*24 + 0*23 + 0*22 + 0*21 + 1*
Sum 32+ 16 + 0 + 0 + 0 + 1 = (49)10
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20. Converting Binary, Octal and
Hexadecimal to Decimal
Example 1 Convert Octal to Decimal
Therefore the weight is 8
Convert (32)8 to Decimal
Octal 3 2
Weight 81 80
3*81 + 2*80
24 + 2
Sum 24 + 2 = (26)10
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21. Converting Binary, Octal and
Hexadecimal to Decimal
Example 2 Convert Octal to Decimal
Therefore the weight is 8
Convert (157)8 to Decimal
Octal 1 5 7
Weight 82 81 80
1*82 + 5*81 + 7*80
64 + 40 + 7
Sum 64 + 40 + 7 = (111)10
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22. Converting Binary, Octal and
Hexadecimal to Decimal
Example 1 Convert Hexadecimal to Decimal
Therefore the weight is 16
Convert (1A)16 to Decimal
Hex 1 A
Weight 161 160
1*161 + 10* 160
Sum 16 + 10 = (26)10
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23. Converting Binary, Octal and
Hexadecimal to Decimal
Example 2 Convert Hexadecimal to Decimal
Therefore the weight is 16
Convert (D47)16 to Decimal
Hex D 4 7
Weight 162 161 160
D*162 + 4* 161 + 7* 160
13*256 + 4*16 + 7*1
3328 + 64 + 7
Sum 3328 + 64 + 7 = (3399)10
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24. Converting Octal and
Hexadecimal to binary
 Octal numbers are 0 to 7
 Each octal numbers is represented by 3 bits
 In binary it follows:
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111
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25. Converting Octal and
Hexadecimal to binary
 Hexadecimal numbers are 0 to 9, A to F
 Each hexadecimal numbers is represented by 4 bits
 In binary it follows:
0 0000 8 1000
1 0001 9 1001
2 0010 10 = A 1010
3 0011 11 = B 1011
4 0100 12 = C 1100
5 0101 13 = D 1101
6 0110 14 = E 1110
7 0111 15 = F 1111
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26. Converting Between Octal and
Hexadecimal
Proceeding
 Converting each octal digit to a 3-bit binary form
 Combine all the 3-bit binary numbers
 Segregating the binary numbers into 4-bit binary form by
starting from the first number from the right bit (Last
Significant Bit/LSB) towards the last number on the left bit
(Most Significant Bit/MSB)
 Converting these 4-bit blocks into their respective
hexadecimal symbols
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27. Converting Between Octal and
Hexadecimal
Example 1 Convert Octal to Hexadecimal
Convert (27)8 to Hexadecimal
Octal 2 7
3-bit binary form 010 111
010111
4-bit binary form 0001 0111
1 7
 (27)8 = (17)16
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28. Converting Between Octal and
Hexadecimal
Example 2 Convert Octal to Hexadecimal
Convert (472)8 to Hexadecimal
Octal 4 7 2
3-bit binary form 100 111 010
100111010
4-bit binary form 0001 0011 1010
1 3 A
 (472)8 = (13A)16
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29. Converting Between Octal and
Hexadecimal
Example 1 Convert Hexadecimal to Octal
Convert (17)16 to Octal
Hex 1 7
4-bit binary form 0001 0111
00 010 111
3-bit binary form 010 111
2 7
 (17)16 = (27)8
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30. Converting Between Octal and
Hexadecimal
Example 2 Convert Hexadecimal to Octal
Convert (3A5)16 to Octal
Hex 3 A 5
4-bit binary form 0011 1010 0101
3-bit binary form 001 110 100 101
1 6 4 5
 (3A5)16 = (1645)8
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31. General Remarks
 All conversions follow a certain pattern
 If you convert from decimal you always use division by the base
of the other system
 decimal  binary : divided by 2
 decimal  octal: divided by 8
 decimal  hexadecimal: divided by 16
 If you convert to decimal you always use multiplication
 binary  decimal: use 2 as weight
 octal  decimal: use 8 as weight
 hexadecimal  decimal: use 16 as weight
 To convert between hexadecimal and octal you always convert
into bits first
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32. General Remarks
 How can you convert from binary into octal or hexadecimal
and vice versa?
 Solutions:
 Convert from binary into decimal and from decimal
into octal or hexadecimal or vice versa
 Convert directly the binary bits in 3- bit for octal and
4-bits for hexadecimal
 Example 110101010
110 101 010
6 5 2
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33. Representation of Data in the
Computer
 The data in Computer Systems are presented through a
binary or two's complement numbering system
 The used units are:
 Bit
 Nibble
 Byte
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34. Representation of Data in the
Computer
Bit
 A bit or binary digit is the smallest unit of data in computing
 A bit representing only two different value interpreted by 0 and 1
 You can represent an infinite number of items with only one bit
 true/false
 red/blue
 male/female
 +/-,
 on/off
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35. Representation of Data in the
Computer
Nibble
 A collection of 4 bits is called nibble
 Also called "semioctet" or "quartet" in a networking or
telecommunication context
 Nibbles represent binary coded decimal (BCD) and
hexadecimal number
 You can display up to 24 = 16 distinct values
 Nibbles are used for binary coding of real numbers
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36. Representation of Data in the
Computer
Bytes
 A byte is a collection of 8 bits and is the smallest addressable
data item on the microprocessor
 The bits of a byte are normally numbered from 0 to 7 by
Bit 0: lower order bit or least significant bit
Bits 0 – 3: low order nibble
Bits 4 – 7: high order nibble
Bit 7: high order bit or most significant bit
 A byte can represent 28 = 256 different values. It can represent
number values from 0 … 255 or signed number from -128 to 127
 The most important use for a byte is holding a character (see
Char type)
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37. Representation of Data in the
Computer
Example
Nibbles
8 = 1000
15 = 1111
Bytes
32 = 00100000
5 = 00000101
18 = 00010010
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38. Bit operators
Bit operators are a powerful and a very machine close
concept.
The following bit operators exist:
& Bit-AND
| Bit-OR
^ Bit-XOR
~ Bit-NOT (1 complement)
<< Bit left shift
>> Bit right shift
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39. Bit-AND
 A combination of two integer numbers with Bit-And leads
to:
 If the bit is 1 in each number it is also 1 in the result
 If the bit is different or 0 in both numbers than the bit is
0 also in the result
Examples
1001 & 101110 &
0101 010101
0001 000100
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40. Bit-OR
 A combination of two integer numbers with Bit-OR leads
to:
 If the bit is 1 in one of the numbers the bit is also 1 in
the result
 Otherwise it is 0
Examples
1001 | 101110 |
0101 010101
1101 111111
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41. Bit-XOR (exclusive OR)
 A combination of two integer numbers with Bit-XOR leads
to:
 If the bit is 1 in only one of the numbers than the bit is
also 1 in the result
 Otherwise it is 0
Examples
1001 ^ 101110 ^
0101 010101
1100 111011
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42. Bit-NOT
 Using this unary bit operator Bit-NOT leads to:
 All bits are set which are not set before
 All bits that have been set before are not set in the result
 That means 0 change to 1 and 1 change to 0
Example
~1001 ~010101
0110 101010
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43. Bit left shift
 Bit left shift leads to that all bits in the integer number are
shifted to the left according to a given number
 All bits over the left border are deleted and on the right
border Zeros are added
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0 0 1 0 1 1 1 0
0 1 0 1 1 1 0 0 Zero
added
Shifted by 1 bit to left

44. Bit left shift
Example (8-bit register)
00001001 << 2
00100100
10010101 << 2
01010100
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45. Bit right shift
 Bit right shift leads to that all bits in the integer number
are shifted to the right according to a given number
 All bits over the right border are deleted and on the left
border Zeros are added
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0 0 1 0 1 1 1 0
0 0 0 1 0 1 1 1
Shifted by 1 bit to right
Zero
added

46. Bit right shift
Example (8-bit register)
00001001 >> 2
00000010
10010101 >> 2
00100101
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47. Use of Bit Operators
 To set User Permissions like write, read or both rights
 Set or unset a bit (to 1 or to 0)
 Checking if a specific bit is set or not
 low-level programming:
 writing device drivers
 low-level graphics
 Decoding
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USER-ID Read Write Delete
1234D Yes No No 100
3645E No Yes No 010
3812F No Yes Yes 011
1048G Yes Yes No 110

48. Any
questions?
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