CS310-CH21 - APPB - NumberSystems

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Number Systems
Binary, Octal, Decimal, Hexadecimal
Chaminade University
Department of Computer Science
 Prof. Martins
Computer Organization 
& Architecture 
Created by Authors Modified by P. Martins
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Number Systems
A number system is a way of counting things. It's a way of identifying the quantity of something.
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The Decimal Number System
Our counting system is based on the number 10 (10 fingers). 
The main principle of the decimal system is that 10 is considered as a new unit from which point counting starts again. Ten tens is again a new unit. The multiples of 10 are counted by the same systems as 1 to 9. 
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The Decimal Number System
10 digits:
0 1 2 3 4 5 6 7 8 9 
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The Decimal Number System
(132)10 = 100 + 30 + 2
= 1*102 + 3* 101 + 2 * 100
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The Binary System
“There are 10 types of people in the world: those who know binary and those who do not “
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Power of the base
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The Binary System
Two digits:
0, 1 (computers use transistors)
Consists of two possible states such as On - Off, Yes - No, True - False, Zero - Non Zero.
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Counting from 0 to 15
Decimal Binary
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The Binary Number System
(101)2
= 1 * 22 + 0 * 21 + 1*20
= 4 + 0 + 1 = (5)
Conversão de binario para decimal
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The Octal Number System
The octal, or base 8, number system is a common system used with computers. Because of its relationship with the binary system, it is useful in programming some types of computers. 
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The Octal Number System
Eight digits:
0,1,2,3,4,5,6,7
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Counting from 0 to 15
Decimal Octal
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The Octal Number System
(123)8 = 
1 * 82 + 2 * 81 + 3 * 80
= 64 + 16 + 3 = 83 decimal
Conversão octal para decimal
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Power of the base
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The Hexadecimal System
16 digits
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
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Byte Representation
A byte of memory can store a number in the range 00 to FF Hex 
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Remarks
A single hexadecimal number requires 4 units of binary numbers. This makes it reasonably easy to convert between these two numbering systems. E.g.
1 = 0001
9 = 0101
A = 1010 
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Counting from 0 to 15
decimal
hexadecimal
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The Hexadecimal System
(23A)16 = 
2 * 162 + 3 * 161 + 10 * 160
= 2 * 256 + 48 + 10 = 512 + 48 + 10 = 512 + 58 = 570
Conversão para decimal
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Power of the base
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MSD and LSD
When determining the most and least significant digits in an octal number, use the same rules that you used with the other number systems. The digit farthest to the left of the radix point is the MSD, and the one farthest right of the radix point is the LSD. 
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Converting Decimal to Octal
Convert 42710 to its Octal equivalent
427/8=> Q=53, R = 3
53/8 => Q=6, R = 5 6538
6/8 => Q=0, R = 6
 
Q = Quotient R = Remainder
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Converting Octal do Decimal
Convert 6538 to its Decimal equivalent
Octal Digits 6 5 3
 x x x
Positional Value 82 81 80 
Sum over the product 384 + 40 + 3
 42710
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Converting Octal to Binary
Convert 2678 to Binary
2 6 7
010 110 111
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Converting Decimal to Hexa
Convert 83010 to its Hex equivalent
830/16 q =51, r =14 
51 /16 q =3, r =3
3/16 q =0, r=3
						
 
33E16 
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Converting Hex to Decimal
Convert 3B4F16 to its Decimal equivalent
Hex digits 3 B 4 F
 x x x x
Positional Value 163 162 161 160
Product 12288 + 2816 + 64 + 15
					15,18310
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Converting Binary to Hex
Convert 10010010011011102 to Hex 
 1001 0010 0110 1110
 9 2 6 14 (E in Hex)
				926E16
0000 = 0 0100 = 4 1000 = 8 1100 = 12
0001 = 1 0101 = 5 1001 = 9 1101 = 13
0010 = 2 0110 = 6 1010 = 10 1110 = 14
0011 = 3 0111 =7 1011 = 11 1111 = 15
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Converting Hex to Octal
Use an intermediate conversion step:
	Hex Binary Octal
 or
 Hex Decimal Octal
 Octal Binary Hex
 or
 Octal Decimal Hex
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Convert the following numbers
(25)10 = ?2
(110101)2 = ?10
(13)10 = ?8
(43)16 = ?10
(96)10 = ?16
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Answers
2510 = 110012
1101012 = 5310
1310 = 158
4316 = 6710
9610 = 6016
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Summary
Use “4 bit Substitution Code” for Hex Binary 
Use “3 bit Substitution Code” for Octal Binary
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Appendix - Fractions
Converting binary fractions to decimal
Converting decimal fractions to binary
Converting hexadecimal fractions to decimal
Converting decimal fractions to hexadecimal
Converting hexadecimal fractions to binary
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Converting Fractional values
Fractional values are represented with negative powers of the radix.
Example:
1001.101 = 23 + 20 + 2-1 + 2-3 = (9.625)10
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Converting binary fractions to decimal
(10.011)2 = ( ? )10
 (10)2 = (2)10
.011 = 0 * 2-1 + 1 * 2-2 + 1 * 2-3
 = 0 + 1/4 + 1/8
 = 0 + 0.25 + 0.125
 = 0.375
Therefore (10.011)2 = (2.375)10
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Converting decimal fractions to binary
(3.1875)10 = ( ? )2
0.1875 * 2 = 0.3750 
0.3750 * 2 = 0.7500 
0.7500 * 2 = 1.5000 
0.5000 * 2 = 1.0000 
0.0000 * 2 = 0.0000 
Therefore (3.1875)10 = (11.0011)2
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Converting hexadecimal fractions to decimal
(C.3)16 = ( ? )10
(C)16 = (12)10
(0.3)16 = ( ? )10
3 * 16-1 = 0.1875 
Therefore, (C.3)16 = (12.1875)10
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Converting decimal fractions to hexadecimal
(15.125)10 = ( ? )16
(15)10 = (F)16
(0.125)10 = ( ? )16 but
(0.125)10 = (.0010)2 = (.2)16 therefore,
(15.125)10 = (F.2)16
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Converting hexadecimal fractions to binary
(F.C)16 = ( ? )2
(F)16 = (1111)2
(.C)16 = ( ? )2 but we know that (C)16 = (1100)2
Then (.C)16 = (.1100)2
Therefore, (F.C)16 = (1111.1100)2
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The PDP-8 minicomputers broke down instructions into four 3-bit words, using an octal digit to represent each 3-bit group 
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Note: you cannot convert 125 decimal to hexadecimal and use the result
As the fractional part. That is only allowed in binary conversions.
Multiply 0.125 x 2 = 0.250 > 0 
 0.250 x 2 = 0.500 > 0
 0.500 x 2 = 1.000 > 1
 0.000 x 2 = 0.000 > 0 therefore 0.125 decimal = .0010 binary = .2 hexa