Number System (PDF)

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```					  EEB 2023
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Number Systems
A Review

Antonakos, Appendix F, pp.605
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Objectives
I   Examine the decimal number systems
N Decimal to binary
N Binary to decimal
I   Examine the hexadecimal number system
I   Examine the octal number system
I   Floating point numbers
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Outcomes
I   To explain the necessity of decimal, binary,
I   To be able to convert decimal numbers to binary
numbers and vice versa
I   To be able to convert binary numbers to
I   To be able to convert binary numbers to octal
numbers and vice versa

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Counting in Decimal
0      1          2    3    4    5    6    7    8       9
10     11         12   13   14   15   16   17   18      19
20     21         22   23   24   25   26   27   28      29
30     31         32   33   34   35   36   37   38      39
40     41         42
50

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Counting in Binary

0   0   0
0   0   1
0   1   0
0   1   1
1   0   0

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Converting Binary to Decimal
I Example: Convert 11001010 to decimal
Step 1 Write down the binary weightings

Step 2 Put binary number into weightings table

Step 3 Add the bits which have a 1.
128 + 64 + 8 + 2 = 202
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Converting Decimal to Binary
I Example: Convert 231 to binary
Step 1 Write out the weightings of eight number

Step 2 Subtract each weighting in turn from the
given number, starting with the largest
weighting
I If subtraction leads to a +ve result record a 1

I If subtraction leads to a -ve result record a 0

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231                       7
-128       1*          -   4   1
-    103               -       3
- 64       1           -   2   1
39                       1
- 32       1           -   1   1
7                       0
- 16       0       The binary number can be read off
7               from top to bottom:
- 8        0                            11100111
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Another example
143                         7
-128            1*       -   4   1
-     15                 -       3
- 64            0        -   2   1
15                         1
- 32            0        -   1   1
15                         0
- 16            0
15                 The binary number can be read off
- 8             1    from top to bottom:
10001111
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Exercise
I   Convert 11111000 to decimal.
I   Convert 12010 to binary
ANS:
I 248

I 1111000

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Two’s Complement
Step 1: Invert all bits (1’s complement)

Step 2: Add 1 to the complemented number

Example: Obtain the 2’s complement of -21

+21 = 0 0 0 1 0 1 0 1
11101010          invert (1’s complement)
-21 = 1 1 1 0 1 0 1 1

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Example: Add decimal numbers +6 and -4

Step 1: Obtain 2’s complement of -4
+4 = 0 0 0 0 0 1 0 0
11111011         1’s complement
-4= 11111100

Step 2: +6 - 4 =    00000110
11111100 +
100000010
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Exercise
I    Add +9-4 using 2’s complement.

Step 1: Obtain 2’s complement of -4
+4 = 0 0 0 0 0 1 0 0
11111011         1’s complement
-4= 11111100

Step 2: +9 - 4 =      00001001
11111100 +
100000101
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101010101111000011111110001000010000010101010101011111100011101011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I   Base 16
I   Symbols used are: 0 - 9, A - F
N   0123456789ABCDEF
I   For an 8-bit binary number, the 8 bits are
divided in to two groups of 4 bits.
N   Each group of 4 bits can be represented by a
hex symbol.

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Decimal   Hex   Binary
0         0     0000
1         1     0001
2         2     0010
3         3     0011
4         4     0100
5         5     0101
6         6     0110
7         7     0111
8         8     1000
9         9     1001
10        A     1010
11        B     1011
12        C     1100
13        D     1101
14        E     1110
15        F     1111

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Example 1: Convert 01001110 to hexadecimal
0100 1110
4      E

Example 2: Convert hex B804 to its binary equivalent

B    8   0     4
1011 1000 0000 0100

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0 1 2 3 4 5 6 7 8 9 A B C                 D E F
10 11 12 13 14 15 16 17 18 19 1A 1B 1C    1D 1E 1F
20 21 22 23 24                2A 2B 2C          2F

90                                           9F
A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 AA AB AC AD AE AF
F0 F1                                        FF

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Exercise
I   Convert 3D16 to binary
ANS:
I 00111101

I 5516

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Octal Number System
I    Base 8 system
I    Symbols used are 0 - 7
N   01234567
I    Three binary bits are used to represent one
octal symbol

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Octal to Binary
A sequence of 8 bits can be represented as follows;
Example: Convert an 8-bit binary number to its octal
equivalent
01001110

01 001   110      binary
1  1     6       octal

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SAQ
I   Find as many interpretations as possible for the
binary word
10110111

I   Unsigned integer: 18310
I   2complements signed integer: -7310
I   Floating point fraction: depends on how many of
the eight bits are used for the mantissa and how
many for the exponent.

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