number systems and arithmetic. introduction to numbering systems we are all familiar with the...
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Number SystemsNumber Systemsand Arithmeticand Arithmetic
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Introduction to Numbering Introduction to Numbering SystemsSystems
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
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Significant DigitsSignificant Digits
Binary: 11101101
Most significant digit Least significant digit
Hexadecimal: 1D63A7A
Most significant digit Least significant digit
Rightmost digit is LSB and leftmost is MSB
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Binary Number SystemBinary 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
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Binary Numbering ScaleBinary Numbering Scale
Base 2 Number
Base 10 Equivalent
PowerPositional
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
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Decimal to Binary ConversionDecimal 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
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Division AlgorithmDivision AlgorithmConvert 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
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Binary to Decimal ConversionBinary 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
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Multiplication AlgorithmMultiplication Algorithm
Convert (10101101)2 to its decimal equivalent:
Binary 1 0 1 0 1 1 0 1
Positional Values
xxxxxxxx2021222324252627
128 + 32 + 8 + 4 + 1Products
17310
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Octal Number SystemOctal 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
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Decimal to Octal ConversionDecimal to Octal Conversion
Convert 42710 to its octal equivalent:
427 / 8 = 53 R3 Divide by 8; R is LSD
53 / 8 = 6 R5 Divide Q by 8; R is next digit
6 / 8 = 0 R6 Repeat until Q = 0
6538
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Octal to Decimal ConversionOctal to Decimal Conversion
Convert 6538 to its decimal equivalent:
6 5 3xxx
82 81 80
384 + 40 + 3
42710
Positional Values
Products
Octal Digits
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Octal to Binary ConversionOctal 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
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Hexadecimal Number SystemHexadecimal Number System
Base 16 system Uses digits 0-9 &
letters A,B,C,D,E,F Groups of four bits
represent eachbase 16 digit
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Decimal to Hexadecimal Decimal to Hexadecimal ConversionConversion
Convert 83010 to its hexadecimal equivalent:
830 / 16 = 51 R14
51 / 16 = 3 R3
3 / 16 = 0 R3
33E16
= E in Hex
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Hexadecimal to Decimal Hexadecimal to Decimal ConversionConversion
Convert 3B4F to its decimal equivalent:
Hex Digits 3 B 4 Fxxx
163 162 161 160
12288 +2816 + 64 +15
15,18310
Positional Values
Products
x
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Convert 0101011010101110011010102 to hex
using the 4-bit substitution code :
0101 0110 1010 1110 0110 1010
Substitution CodeSubstitution Code
5 6 A E 6 A
56AE6A16
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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 CodeSubstitution Code
2 5 5 2 7 1 5 2
255271528
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Binary to Hexadecimal Binary to Hexadecimal ConversionConversion
The easiest method for converting binary to hexadecimal is to use a substitution code
Each hex number converts to 4 binary digits
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Representation of fractional Representation of fractional numbersnumbers
convert 0.1011 to decimal
= ½ + 0 + 1/8 + 1/16
= 0.6875 (decimal)
2 ) 111011.101 to decimal
= 1x32 + 1x16 + 1x8 + 0x4 + 1x2 + 1x1 + ½ + 0x1/4 + 1x1/8
= 59.625 (decimal)
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convert 59.625 to binary (59) – 111011 0.625= 0.625x2 = 1.25 // 1 is MSB 0.25 x 2 = 0.5 0.5 x 2 = 1.0 – stop when fractional part is
zero= 101Thus 59.625 = 111011.101
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convert (F9A.BC3) to decimal
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convert (F9A.BC3) to decimal
= 15x256 + 9x16 + 10x1 + 11/16 + 12/256 + 3/4096
= (3994.7351074)