A number system is defined as a system of writing to express numbers.
It is the mathematical
notation for representing numbers of a given set by using digits or other symbols in a consistent
manner.
Types of Number Systems
1. Decimal number system (Base 10)
2. Binary number system (Base 2)
3. Octal number system (Base 8)
4. Hexadecimal number system (Base 16)
DECIMAL NUMBER SYSTEM (Base 10)
The decimal number system has a base of 10 because it uses ten digits from 0 to 9. In the
decimal number system, the positions successive to the left of the decimal point represent units,
tens, hundreds, thousands and so on. This system is expressed in decimal numbers. Every position
shows a particular power of the base (10).
Example:
The decimal number 1457 consists of the digit 7 in the units position, 5 in the tens place, 4 in the
hundreds position, and 1 in the thousands place whose value can be written as:
(1×103) + (4×102) + (5×101) + (7×100)
(1×1000) + (4×100) + (5×10) + (7×1)
1000 + 400 + 50 + 7
1457
BINARY NUMBER SYSTEM (Base 2)
The base 2 number system is also known as the Binary number system wherein, only two binary
digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under
this system are known as binary numbers which are the combination of 0 and 1.
Example:
Write (14)10 as a binary number.
∴ (14)10 = 11102
OCTAL NUMBER SYSTEM (Base 8)
In the octal number system, the base is 8 and it uses numbers from 0 to 7 to represent numbers.
Octal numbers are commonly used in computer applications. Converting an octal number to
decimal is the same as decimal conversion and is explained below using an example.
Example:
Convert 2158 into decimal.
Solution:
2158 = 2 × 82 + 1 × 81 + 5 × 80
= 2 × 64 + 1 × 8 + 5 × 1
= 128 + 8 + 5
= 14110
HEXADECIMAL NUMBER SYSTEM (Base 16)
In the hexadecimal system, numbers are written or represented with base 16. In the hexadecimal
system, the numbers are first represented just like in the decimal system, i.e. from 0 to 9. Then, the
numbers are represented using the alphabet from A to F. The below-given table shows the
representation of numbers in the hexadecimal number system.
Number System Chart
In the number system chart, the base values and the digits of different number systems can be
found. Below is the chart of the numeral system.
Number System Conversion
Numbers can be represented in any of the number system categories like binary, decimal,
hexadecimal, etc. Also, any number which is represented in any of the number system types can
be easily converted to another. Check the detailed lesson on the conversions of number
systems to learn how to convert numbers in decimal to binary and vice versa, hexadecimal to
binary and vice versa, and octal to binary and vice versa using various examples.
With the help of the different conversion procedures explained above, now let us discuss in brief
about the conversion of one number system to the other number system by taking a random
number.
Assume the number 349. Thus, the number 349 in different number systems is as follows:
The number 349 in the binary number system is 101011101
The number 349 in the decimal number system is 349.
The number 349 in the octal number system is 535.
The number 349 in the hexadecimal number system is 15D
Number System Solved Examples
Example 1:
Convert (1056)16 to an octal number.
Solution:
Given, 105616 is a hex number.
First, we need to convert the given hexadecimal number into decimal number
(1056)16
= 1 × 163 + 0 × 162 + 5 × 161 + 6 × 160
= 4096 + 0 + 80 + 6
= (4182)10
Now we will convert this decimal number to the required octal number by repetitively dividing by
8.
Therefore, taking the
value of the remainder
from bottom to top, we
get;
(4182)10 = (10126)8
Therefore,
(1056)16 = (10126)8
Example 2:
Convert (1001001100)2 to a decimal number.
Solution:
(1001001100)2
= 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 0 × 20
= 512 + 64 + 8 + 4
= (588)10
Example 3:
Convert 101012 into an octal number.
Solution:
Given,
101012 is the binary number
We can write the given binary number as,
010 101
Now as we know, in the octal number system,
010 → 2
101 → 5
Therefore, the required octal number is (25)8
Example 4:
Convert hexadecimal 2C to decimal number.
Solution:
We need to convert 2C16 into binary numbers first.
2C → 00101100
Now convert 001011002 into a decimal number.
101100 = 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 0 × 20
= 32 + 8 + 4
= 44
