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DCAM Part 66 B 1.1 Level 1Module 5.2 - Numbering SystemNumbering
System : Binary, octal and hexadecimal; Demonstration of
conversions between the decimal and binary, octal and hexadecimal
systems and vice versa.
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Level 1 :A familiarisation with the principal element of the
subject.Objectives: the applicant should be able to give a simple
description of the whole subject, using common words and
examples.The applicant should be able to use typical terms.
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5.2 Numbering SystemIntroduction:- knowledge of Numbering
Systems is fundamental to understanding computers and their
operation use to count objects or perform mathematical
calculations- Each is a set of symbols and characters referred as
digits.Positional Notation the standard shorthand form writing
numbers. The value of the particular digit depends on; * the digit
value.* the position of the digit within the number.e.g. 3721
standard shorthand.Three thousand seven hundred and twenty one
standard longhand.The digit at far right Least Significant Digit
(LSD).The digit at far left Most Significant Digit (MSD).
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Numbering SystemBase Has a base which is equal to the number of
digits. A subscript is added to a number to indicate its
base.e.g.1012 - indicates the number 101 is a base of 2 or binary
number.The value of largest digit of a numbering system is:- One
less than the base The value of smallest digit of a numbering
system is:- ZeroEach digit is multiplied by the base raised to the
appropriate power for the digit position.e.g. decimal No. 3721 is
equal to:(3 X 103) + (7 X 102) + (2 X 101) + (1 X 100) 3000 + 700 +
20 + 1 3721
1031021011003721thousandshundredstensunits
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Binary Number System a simplest number system employing
positional number. has a base of 2. two binary digits (BITS) used
are 0 and 1.IN A DIGITAL COMPUTER:Only two distinct states.All
inputs to a digital computer must be converted to a series of 1s
and 0s (binary) before the computer can make use of the
data.Conversion from binary to decimal is straight forward and
easily performed using positional notation.EXAMPLE: 1.Positional
notation16 + 4 + 2 + 1 =2310
2726252423222120Weight Value 1286432168421Base 10
Value00010111Binary Number to be Converted 16421Equivalent base 10
Number
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EXAMPLE: 2.Positional notationBinary Number SystemIn example 2,
a binary number representing a fraction is shown.Add together the
base 10 values for each bit position containing a 18 + 4 + 0.5 +
0.125 + 0.0625 = 12.687510
232221202-12-22-32-4Weight Value 84210.50.250.130.06Base 10
Value11001011Binary Number to be Converted 84 0.5 0.13
0.06Equivalent base 10 Number
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Decimal Number System has a base of 10. Most familiar, used for
everyday counting. mathematical calculations. contains ten digits
from 0 to 9, with 9 the largest digit.DECIMAL
DIGITS987654321POSITION NOTATIONDecimal point600000 + 500000 + 8000
+ 900 + 10 + 2 + 3/10 + 3/100658912.3310
10510410310210110010-110-2Weight Value 65891233Number
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Decimal to Binary Conversion to repeatedly divide the decimal by
the base number. by keeping track of the remainders.- The new
numbering base equivalent is obtained. Case of DECIMAL to BINARY
conversions, The decimal number is divided by the base number 2.
The first remainder obtained is the least significant digit (LSD).-
The last remainder is the most significant digit (MSD).
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Decimal to Binary Conversion
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Octal Number System has a base of 8 the weight value of each BIT
position (80, 81, 82..) and the base 10 equivalent are shown.To
convert 4522 (base 8) to base 10, multiply EACH total octal digit
by its corresponding base 10 value, then add together the computed
base 10 values.2048 + 320 + 16 + 2 = 238610 45228 = 238610
8483828180Weight Value 4,096 5126481Base 10 Value04522Octal
Number to be Converted 2,048 320162Equivalent base 10 Number
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Decimal to Octal Conversion Decimal to Octal conversions can
also be accomplished by successive division. The decimal number is
repeatedly divided by base 8 and again the remainders are used for
decimal to octal equivalent number.
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Binary to Octal Conversion - three-bit position represent eight
combinations (000 thru 111). - octal no. can be substituted for a
3-bit binary numbers. - binary no. separated into groups of 3-bits
from right (LSD) to (MSD) at the left. - Each group of 3-bits is
replaced by an octal equivalent. - Octal to binary is the reverse
procedure.
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HEXADECIMAL NUMBER SYSTEM- another system often used in
microcomputers. - has base 16 which requires 16 digits. - Digits
used are 0 through 9 and A through F. - A thru F - equivalent
decimal numbers of 10 thru 15 respectively.HEXADECIMAL DIGITS F E D
C B A 9 8 7 6 5 4 3 2 1 015 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 This
system is called alphanumeric number system since numbers and
letters are used to represent the digits.
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40,960 + 2,048 + 240 + 5 = 4325310 A8F516 = 4325310HEXADECIMAL
NUMBER SYSTEM- Refer positional notation and convert A8F5 (base 16)
to base 10 equivalent shown. Often hexadecimal numbers are written
with an H following the hexadecimal number.
163162161160Weight Value 4,096 256161Base 10
ValueA8F5Hexadecimal Number to be Converted 40,9602,048
2405Equivalent base 10 Number
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HEXADECIMAL CONVERSION- decimal to Hexadecimal conversion can be
done by successive division. - the decimal number is divided by
base number 16. if the remainder is greater than 9, it should be
changed to the hexadecimal equivalent of the remainder.
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e.g., if the remainder is 10 it should be changed to A, if the
remainder is 11 it should be changed to B and so on, up to 15 which
is F.HEXADECIMAL CONVERSION
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BINARY TO HEXADECIMAL CONVERSION used as a shorthand notation
for binary numbers. in binary, 4-bit positions are necessary to
obtain 16 combination numbers (0000 thru 1111). the binary numbers
is separated into groups of four beginning at LSD and preceding to
the left. - Each group of four bits is replaced by hexadecimal
equivalent.
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BINARY TO HEXADECIMAL CONVERSIONIn forming the 4 - bit grouping,
0s may be required to complete the first (MSD) group.
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BINARY CODED DECIMAL NUMBER SYSTEM A method of representing
decimal numbers in digital computers is known as Binary Coded
Decimal (BCD).DECIMAL TO BCD 7 3 8 10
0111 0011 1000 DECIMAL TO BCD 1001 0100 0110
9 4 6 10
232221202322212023222120WEIGHT
VALUE011101101001BCD769DECIMAL102101100WEIGHT VALUE