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
q Numbers can be represented using systems similar to the familiar decimal system but using a base other than 10 .
q 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
q 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.
q 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
q The binary system is the positional number system that uses 2 as the base .
q The binary system uses the 2 binary digits ( or bits ) 0 and 1 .
q The place values of the digits in a binary number are powers of 2 .
q We can use column system to convert a number in other bases to its equivalence in denary .
q 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
q To convert a denary number ( integer ) to binary , we do a repeated division by the desired base ( 2 ) until a quotient 0 is obtained .
q 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 =
