06. Aptitude Questions  Numbers Thoery  Number Systems  Conversions
Number Systems :
A number system is a format for representing different
numerical values
Any number system calculates the value of a number
being represented by
1)
The digit used to represent and
2)
The place where the digits are present in
the actual number.
The way in which the value of the number is calculated
also depends on the base of the number system.
Definition of Base of a number system :
Base of a number system is the number of digits used to represent the number in
that number system. Base is also called radix.
We use 10 digits to represent the numbers in the
decimal number system, so the base of the decimal number system is 10.
In case a number XYZ
*where X ,Y and Z are digits is represented in some number system, which
has a base “B”, its value would be calculated by the expression : X x B^{2}
+ Y x B + Z
Decimal Number System:
The system generally used by convention is the decimal number system. The base of decimal number system is 10(ten) and the digits use are from 0(zero) to 9(nine)
The system generally used by convention is the decimal number system. The base of decimal number system is 10(ten) and the digits use are from 0(zero) to 9(nine)
In decimal number system , a number can be represented as
47218= 4 x 10^{4} +7 x 10^{3} +2 x 10^{2}
+1 x 10^{1} +8 x 10^{0}
=
40000 + 7000 + 200 +10 +8
Decimal number system is the mostly used number system .
Decimal number system is the mostly used number system .
Binary Number System:
The digits used for the binary system would be 0 and 1. We use only 2 digits in the binary number system , so the base of the binary number system is 2.
The digits used for the binary system would be 0 and 1. We use only 2 digits in the binary number system , so the base of the binary number system is 2.
The binary digits o and 1 are called bits.
In binary number system zero is represented by 0 and 1
is represent by 1. After 1, there is no any digit to represent two. Therefore,
two is written is as 10 , three written as 11. Again the number four is
represented as 100.
For a number represented as 10101 in the binary system
would have a value of
(1 x 2^{0}) + (0 x 2^{1}) + ( 1 x 2^{2})
+ (0 x 2^{3 }) + ( 1 x 2^{4)}
which amounts to 1 +0 + 4 + +0 + 16 =21
Binary number system is very useful in computer and other digital goods.
Binary number system is very useful in computer and other digital goods.
Octal Number System:
The digits use for the octal system would be 0 through 7. The number of digits used in this system is 8, so base of the octal number system is 8.
The digits use for the octal system would be 0 through 7. The number of digits used in this system is 8, so base of the octal number system is 8.
The number 8 of the decimal number system is
represented by 10 , 9 by 11 and so on.
For example , a number represented as 123 in the octal
number system would have a value of 1 x 8^{2}+2 x 8^{1}+3 x
8^{0} which amounts to 64 + 16 +
3=83 as represented in the decimal system.
Hexadecimal Number
System:
The base of the hexadecimal number system is 16. The 10 digits from 0 to 9 and letters from A to F are used to represent numbers in this system.
The base of the hexadecimal number system is 16. The 10 digits from 0 to 9 and letters from A to F are used to represent numbers in this system.
The digits are 0 to 9 same as those for the decimal
number system and digits from 10 to 15 are represented by alphabets A to F
respectively.
For example, a number represented as 6A9 in the hexadecimal number system would
have a value of 6 x 16^{2} + A^{ }x^{ }16^{1 }+
9 x 16^{0 }which amounts to 6 x 256 + 10 x 16 + 9 = 1705 as represented
in the decimal number system.
Conversion of Base 10
to Base 2 :
Conversion of Decimal Number to Binary Number:
Conversion of Decimal Number to Binary Number:
In case a number ABC (with A, B and C as the digits)
represented in the base 10 system, is to be converted to a binary system, then
it has to be repeatedly divided by 2. The remainders obtained on each division
provide the digits that would appear in the binary system.
Thus a number 25 can be repeatedly divided by 2 as
follows.
Number

/

Base

Quotient

Remainder

25

/

2

12

1

12

/

2

6

0

6

/

2

3

0

3

/

2

1

1

1

/

2

0

1

Repeated division until we get the remainder 0, gives the
remainders .1,0,0,1 and 1 in that order. On writing these in the reverse order,
we get the binary number 11001.
Example 2: Convert 125_{(10)} to equivalent binary number.
2  125
2  62  1
2  31 0
2  15  1
2 
7  1
2  3
 1
2  1
 1
0  1
The binary
equivalent of 125_{(10)} is 1111101_{(2)}
Conversion
of a Decimal Fraction to Binary Fraction:
The conversion of a
decimal fraction to a binary fraction can be done using the successive multiplication
by 2 technique. Following example will explain the procedure to convert a decimal fraction to a binary fraction
Example :Convert 0.25 to equivalent binary number.
Example :Convert 0.25 to equivalent binary number.
Fraction

Fraction x 2

Remainder
Fraction

Integer

0.25

0.5

0.5

0

0.5

1

0

1

0.25_{(10)}
= 0.01
Conversion of A Binary Number System To Decimal Number System
To
convert binary number to decimal number, follow the procedure :
Example : Convert 11001_{(2)}
to equivalent decimal number
Solution
: 11001 = 1 x 2^{4} + 1 x 2^{3} + 0 x 2^{2} + 0 x2^{1} + 1
=16 + 8 + 0 + 0 + 1
=25
Example 2: Convert 10.101
to equivalent decimal number
Solution
:
10.101_{(2)} = 1 x 2 + 0 x 2 + 1 x 2^{1} + 0 x 2^{2} + 1^{ }x 2^{3}
10.101_{(2)} = 1 x 2 + 0 x 2 + 1 x 2^{1} + 0 x 2^{2} + 1^{ }x 2^{3}
^{ }= 1 x 2 + 0 x 2+ $ \frac{1}{2}$
+ $ \frac{0}{2^{2}}
$
+ $\frac{1}{2^{3}}
$
= 2 + 0 +0.5 +0
+0.125
= 3.625
Conversion
of a decimal number to octal number :
How
to convert a number from base 10 to base 8 :
In case a number
ABC (with A,B and C as the digits) represented
in the base 10 system, is to be converted to an octal system, then it
has to be divided by 8 repeatedly. The remainders obtained on each division
provide the digits that would appear in the octal number.
Number

/

Base

Quotient

Remainder

205

/

8

25

5

25

/

8

3

1

3

/

2

1

1

We have to repeat the division until we get the
remainder zero. The remainders obtained are 5,1 and 1. On writing these in the
reverse order, we get the octal number 115.
Example 2: Convert
463_{(10)} to equivalent octal number.
Solution :
8  463
8  463
8  57 7
8  7 1
0  7
The octal equivalent
of 463_{(10)} is 717_{(8)}
Conversion of octal number to decimal number :
Example 1:Convert 716_{(8)} to equivalent decimal number
Solution: 716_{(8)} = 7 x 8^{2} + 1 x 8^{1}
+ 6 x 8^{0}
= 7 x 64 + 1 x 8 + 6
= 448 + 8 + 6 = 462
Example 2 : Convert 65.24 to equivalent decimal number
Solution : 65.24_{(8)} = 6 x 8 + 5 + $\frac{2}{8}$
+ $\frac{4}{8^{2}}$
=48 + 5 +0.25+0.0625
= 53.3125
Conversion of a decimal
number to Hexadecimal Number:
In case a number
ABC (with A,B and C as the digits) represented
in the base 10 system, is to be converted to an octal system, then it
has to be divided by 16 repeatedly. The remainders obtained on each division
provide the digits that would appear in the octal number.
Number

/

Base

Quotient

Remainder

410

/

16

25

10

25

/

16

1

9

9

/

16

0

9

We have to repeat the division until we get the
remainder zero. The remainders obtained are 10, 9 and 9.
In hexadecimal number system, the equivalent number for
10 is A.
On writing these in the reverse order, we get the octal
number 99A.
Example 2: Convert 8655_{(10)}
to equivalent decimal number.
16  8655
16  540 15
16  33 12
16  2  1
0  2
The hexadecimal
equivalents to 12 and 15 are C, F
The
octal equivalent of 8655_{(10)} is 21CF_{(8)}
Conversion
of Hexadecimal number to Decimal Number:
To convert a
hexadecimal number to a decimal number, multiple each digit of the given
hexadecimal number by powers of 16 as follows :
Example : Convert 90BA(16) to equivalent decimal number
Solution :
90BA(16) = 9 x 16^{3} + 0 x 16^{2} + B x 16^{1} + A x 16^{0}
90BA(16) = 9 x 16^{3} + 0 x 16^{2} + B x 16^{1} + A x 16^{0}
= 9 x 4096 + 0 x
256 + 11 x 16 +10 x 1
= 36864 + 0 + 176
+10 =37050