NUMBERSYSTEM
Number systems are the technique to represent numbers in the computer system architecture,
every value that you are saving or getting into/from computer memory has a defined number
system.
Computer architecture supports following number systems.
Binary number system Octal number system Decimal number system
Hexadecimal (hex) number system
BINARY NUMBER SYSTEM
A Binary number system has only two digits that are 0 and 1. Every number (value)
represents with 0 and 1 in this number system. The base of binary number system is 2,
because it has only two digits.
OCTAL NUMBER SYSTEM
Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents
with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because
it has only 8 digits.
DECIMAL NUMBER SYSTEM
Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents
with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10,
because it has only 10 digits.
HEXADECIMAL NUMBER SYSTEM
A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F.
Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number
system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values.
Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.
Number system Base(Radix) Used digits Example
Binary 2 0,1 (11110000)2
Octal 8 0,1,2,3,4,5,6,7 (360)8
Decimal 10 0,1,2,3,4,5,6,7,8,9 (240)10
Hexadecimal 16 0,1,2,3,4,5,6,7,8,9, (F0)16
A,B,C,D,E,F
CONVERSIONS DECIMALTO OTHER
1. DECIMAL TO BINARY
Decimal Number System to Other Base
To convert Number system from Decimal Number System to Any Other Base is quite easy;
you have to follow just two steps:
A) Divide the Number (Decimal Number) by the base of target base system (in which you
want to convert the number: Binary (2), octal (8) and Hexadecimal (16)).
B) Write the remainder from step 1 as a Least Signification Bit (LSB) to Step last as a
Most Significant Bit (MSB).
Decimal to Binary Conversion Result
Binary Number is
(11000000111001)2
Decimal Number is : (12345)10
2. DECIMAL TO OCTAL
Decimal to Octal Conversion Result
Octal Number is
(30071)8
Decimal Number is : (12345)10
3. DECIMAL TO HEXADECIMAL
Decimal to Hexadecimal Conversion Result
Example 1 Hexadecimal Number is
(3039)16
Decimal Number is : (12345)10
Example 2
Hexadecimal Number is
(2D5)16 Convert
10, 11, 12, 13, 14,
15 to its
equivalent...
A, B, C, D, E, F
Decimal Number is : (725)10
BINARYTOOTHER
A) Multiply the digit with 2(with place value exponent). Eventually add all the multiplication
becomes the Decimal number.
1. BINARY TO DECIMAL
2. BINARY TO OCTAL
An easy way to convert from binary to octal is to group binary digits into sets of three, starting
with the least significant (rightmost) digits.
Binary: 11100101 11 100 101
=
011 100 101 Pad the most significant digits with zeros if
necessary to complete a group of three.
Then, look up each group in a table:
Binary: 000 001 010 011 100 101 110 111
Octal: 0 1 2 3 4 5 6 7
Binary 01 10 10
= 1 0 1
Octal = 3 4 5 = 345
oct
3. BINARY TO HEXADECIMAL
An equally easy way to convert from binary to hexadecimal is to group binary digits into sets
of four, starting with the least significant (rightmost) digits.
Binary: 11100101 = 1110 0101
Then, look up each group in a table:
Binary: 0000 0001 0010 0011 0100 0101 0110 0111
Hexadecimal: 0 1 2 3 4 5 6 7
Binary: 1000 1001 1010 1011 1100 1101 1110 1111
Hexadecimal: 8 9 A B C D E F
Binary = 1110 0101
Hexadecimal E 5 = E5
= hex
OCTAL TO OTHER
1. OCTAL TO BINARY
Converting from octal to binary is as easy as converting from binary to octal. Simply look up
each octal digit to obtain the equivalent group of three binary digits.
Octal: 0 1 2 3 4 5 6 7
Binary 00 00 010 01 10 10 11 11
: 0 1 1 0 1 0 1
Octal = 3 4 5
Binary 01 10 10 = 011100101
= 1 0 1 binary
2. OCTAL TO HEXADECIMAL
When converting from octal to hexadecimal, it is often easier to first convert the octal number
into binary and then from binary into hexadecimal. For example, to convert 345 octal into
hex:
(from the previous example)
Octal = 3 4 5
Binary 011 100 101 = 011100101
= binary
Drop any leading zeros or pad with leading zeros to get groups of four binary digits (bits):
Binary 011100101 = 1110 0101
Then, look up the groups in a table to convert to hexadecimal digits.
Binary: 000 000 001 001 010 010 011 011
0 1 0 1 0 1 0 1
Hexadecimal 0 1 2 3 4 5 6 7
:
Binary: 100 100 101 101 110 110 111 111
0 1 0 1 0 1 0 1
Hexadecimal 8 9 A B C D E F
:
Binary = 111 010
0 1
Hexadecimal E 5 = E5
= hex
Therefore, through a two-step conversion process, octal 345 equals binary 011100101 equals
hexadecimal E5.
3. OCTAL TO DECIMAL
The conversion can also be performed in the conventional mathematical way, by showing
each digit place as an increasing power of 8.
345 octal = (3 * 82) + (4 * 81) + (5 * 80) = (3 * 64) + (4 * 8) + (5 * 1) = 229 decimal
OR
Converting octal to decimal can be done with repeated division.
1. Start the decimal result at 0.
2. Remove the most significant octal digit (leftmost) and add it to the result.
3. If all
4. Otherwise, multiply the result by 8.
5. Go to step 2.
Octal Digits Operation Decimal Result Operation Decimal Result
345 +3 3 ×8 24
45 +4 28 ×8 224
5 +5 229 done.
(345)8 =(229)10
HEXADECIMALTOOTHER
1. HEXADECIMAL TO BINARY
Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal.
Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.
Hexadecimal 0 1 2 3 4 5 6 7
:
Binary: 000 000 001 001 010 010 011 011
0 1 0 1 0 1 0 1
Hexadecimal 8 9 A B C D E F
:
Binary: 100 100 101 101 110 110 111 111
0 1 0 1 0 1 0 1
Hexadecimal A 2 D E
=
Binary = 101 001 110 111 =
0 0 1 0 1010001011011110
binary
2. HEXADECIMAL TO OCTAL
