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CHAPTER 1 : BASES AND NUMBER REPRESENTATION

1.1 : Real Numbers And The Decimal Number System
1.2 : The Binary Number System
1.3 : Conversion From Decimal To Binary
1.4 : The Octal And Hexadecimal Systems
1.5 : Arithmetic In Non-Decimal Bases

1.1 : REAL NUMBERS AND THE DECIMAL NUMBER SYSTEM

 Numbers can be represented using systems similar to the familiar decimal system but using a base other than 10 .

 We investigate the representation of numbers using different number bases , paying particular attention to the number systems used in computing .

Number Base

A fundamental counting group consists of a number of various symbols .
Counting Groups Symbols Of The Group No. Of Symbols
Binary 0,1 2
Octal 0,1,2,3,4,5,6,7 8
Denary(Decimal) 0,1,2,3,4,5,6,7,8,9 10
Hexadecimal 0,1,2,3,4,5,6,7,8,9,A(10),B(11),C(12),D(13),E(14),F(15) 16

Decimal System

 The decimal system is an example of a positional number system , because each digit has a place value that depends on its position in relation to the decimal point.

 Actually , every decimal number can be display in addition form using the column system .

Column System
Enable us to represent any conceivable number by combining numerals and zeros .









Example 1 :
4235.24 is derived from this column system .

Numerals 4 2 3 5 2 4
Power Of Magnification 103 102 101 100 10-1 10-2
Value 1000 100 10 1 0.1 0.01

4235.24 = 4 x 103 + 2 x 102 + 3 x 101 + 5 x 100 + 2 x 10-1 + 4 x 10-2
= 4000 + 200 + 30 + 5 + 0.2 + 0.04
= 4235.24

Example 2 :
3200451 =






1.2 : THE BINARY NUMBER SYSTEM

 The binary system is the positional number system that uses 2 as the base .

 The binary system uses the 2 binary digits ( or bits ) 0 and 1 .

 The place values of the digits in a binary number are powers of 2 .

 We can use column system to convert a number in other bases to its equivalence in denary .

 Method to convert binary to denary
1 ) Write down the place values of each bit ( binary digit )
2 ) Multiply the place values by every bit
3 ) Take the sum of products

Example 1 :
Convert 1110.012 to decimal .

Numerals 1 1 1 0 0 1
Power Of Magnification 23 22 21 20 2-1 2-2
Value 8 4 2 1 0.5 0.25

1110012 = 1 x 23 + 1 x 22 + 1 x 21 + 0 x 20 + 0 x 2-1 + 1 x 2-2
= 8 + 4 + 2 + 0 + 0 + 0.25
= 14.25

Example 2 :
Convert ( 110.000110 )2 to decimal real numbers .
Solution :







1.3 : CONVERSION FROM DECIMAL TO BINARY

 To convert a denary number ( integer ) to binary , we do a repeated division by the desired base ( 2 ) until a quotient 0 is obtained .

 Multiply fraction part by 2 continuously until the fraction becomes zero or the degree of accuracy is satisfied .

Example 1:
Convert 75.4375 DENARY to BINARY
Solution :








Example 2 :
If , find the values of k and x.
Solution :






Example 3 :
Convert ( 4.75 )10 to binary .
Solution :





Example 4 :
Convert ( 6.1 )10 to binary . ( with 6 digits after the point )
Solution :








1.4 : THE OCTAL AND HEXADECIMAL SYSTEMS

Octal Number System
Octal system is a base 8 system which uses the eight digits 0 to 7 to represent any number

Convert Octal To Decimal And Otherwise

Example 1 :
Express the number 7478 in decimal .
Solution:






Example 2:
Convert the octal number 155061.13 into denary form.
Solution:





Example 3 :
Convert 275.4375 denary to octal .
Solution :








Convert Octal To Binary And Otherwise
No. of symbols in octal = 2no. of binary bits

Example 4 :
Convert 514.78 to binary .
Solution :





Example 5 :
Convert 110111 Binary to Octal.
Solution :





Example 6 :
Convert 21.6738 to binary .
Solution :





Example 7 :
Convert 10100011.101112 to octal .
Solution :



Example 8 :
Convert the following numbers to OCTAL (by showing all the working)
(i)
(ii)
Solution :








Hexadecimal Numbers
Hexadecimal system has a base of 16 which can be represented by digits 0 to 9 and the A to F.

Hexadecimal Decimal Interconversion

Example 9 :
Convert 4BEEF816 to decimal .
Solution :





Example 10 :
Convert 985.78125 DENARY to HEXADECIMAL.
Solution :







Example 11 :
Convert 110111 Denary to Hexadecimal.
Solution :









Hexadecimal Binary Interconversion
No. of symbols in hexadecimal = 2no. of binary bits

Example 12:
Convert 57DE.C916 (by showing all the working) into:
(i) decimal.
(ii) binary.
Solution:



Example 13 :
Convert 10111101001.1100012 to hexadecimal .
Solution :






1.5 : ARITHMETICS IN NON-DECIMAL BASES

Binary Operation

Binary Addition
Binary addition table :
+ 0 1
0 0 1
1 1 10

Rules for binary addition :
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 , with carry of 1 to add to next column
1 + 1 + 1 = 1 , with carry of 1 to add to next column

Example 1 :
i ) 10102 + 1112 =







ii ) 11011.012 + 101.11012 =










Example 2:
Rewrite each term of the expression
27.148 + 100111.01112 + 27B.3C16
into binary and perform the arithmetic . Present the final answer in octal and hexadecimal.
Solution:




















Example 3:
Rewrite each term of the expression
C0016 + 1F16
into binary and perform the arithmetic. Present the final answer in octal and decimal.
Solution:
















Binary Subtraction
Rules for binary subtraction :
0 – 0 = 0
1 – 0 = 1
1 – 1 = 0
0 – 1 = 1 , with borrow of 1 from the next column

Example 4 :
i ) What is the value of X (in binary) when .





ii ) 1101.101002 – 11.101112 =






Example 5:
Rewrite each of the terms of the expression
8F3.A16 – 712.468 + 56.12510 – 101010.0012
in binary, and simplify the expression.
Solution:




















Example 6:
Perform the following arithmetic in binary,
10011.1116 + 111108 – 10
Hence, convert your final answer into
( i ) octal,
( ii ) hexadecimal,
( iii ) decimal.
Solution:


















Example 7:
Perform the following arithmetic in binary and give your final answer in denary.
1101001.10112 + 725.468 + 2D.A516 – 25810
Solution:
















Binary Multiplication
Example 8:
i. 11.012 x 101.12 =









ii. 11012 x 11002 =










iii. 410 x 310 =











iv. 1010 x 1410 =










Binary Division
Example 9:
i. 10100012  112 =













ii. 11101112  10012 =














iii. 111.000012  1.012 =