A Number system or numeral system is defined as an elementary system to express numbers and figures. It is the unique way of representing numbers in arithmetic and algebraic structures.
The four common types of Number systems are:
Types of Number System
Decimal Number System
In the decimal number system, each digit in a number occupies a specific position, and its value depends on its place value, which is given by a power of 10. Starting from the right, the first position is called the units place with place value 10. The second position is the tens place with place value 101. The third position is the hundreds place with place value 102, followed by the thousands place with place value 103, and so on towards the left.
A number System with a base value of 2 is termed a Binary number system. It uses 2 digits i.e. 0 and 1 for the creation of numbers. The numbers formed using these two digits are termed Binary Numbers. The binary number system is very useful in electronic devices and computer systems because it can be easily performed using just two states ON and OFF i.e. 0 and 1.
Decimal Numbers 0-9 are represented in binary as 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001
For example, 14 can be written as 1110, 19 can be written as 10011, and 50 can be written as 110010.
Example of 14 in the binary system
14 as Binary
Here 14 can be written as 1110
Octal Number System
Octal Number System is one in which the base value is 8. It uses 8 digits i.e. 0-7 for the creation of Octal Numbers. Octal Numbers can be converted to Decimal values by multiplying each digit with the place value and then adding the result. Here the place values are 80, 81, and 82. Octal Numbers are useful for the representation of UTF8 Numbers.
(81)10 can be written as (121)8
(125)10 can be written as (175)8
Hexadecimal Number System
A number System with a base value of 16 is known as Hexadecimal Number System. It uses 16 digits for the creation of its numbers. Digits from 0-9 are taken like the digits in the decimal number system but the digits from 10-15 are represented as A-F i.e. 10 is represented as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. Hexadecimal Numbers are useful for handling memory address locations.
(185)10 can be written as (B9)16
(5440)10 can be written as (1540)16
(4265)10 can be written as (10A9)16
Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Other than these, one ancient number system which precedes the decimal number system is the Roman number system. Let's learn about it in detail as follows:
Roman Number System
The Roman numeral system is an ancient numerical system that was used in ancient Rome and throughout the Roman Empire. It is based on a combination of letters from the Latin alphabet to represent numbers. Here are the basic symbols used in the Roman numeral system along with their corresponding values:
Symbol
Value
Symbol
Value
Symbol
Value
I
1
X
10
C
100
II
2
XX
20
CC
200
III
3
XXX
30
CCC
300
IV
4
XL
40
CD
400
V
5
L
50
D
500
VI
6
LX
60
DC
600
VII
7
LXX
70
DCC
700
VIII
8
LXXX
80
DCCC
800
IX
9
XC
90
CM
900
X
10
C
100
M
1000
Rule of Roman Numeral
To write numbers in Roman numerals, we can use the following rules:
The symbols I, X, C, and M can be repeated up to three times in a row.
When a smaller value symbol appears before a larger value symbol, the smaller value is subtracted. For example, IV represents 4 (5 - 1) and IX represents 9 (10 - 1).
When a smaller value symbol appears after a larger value symbol, the smaller value is added. For example, VI represents 6 (5 + 1) and XI represents 11 (10 + 1).
Sample Problems on Types of Number System
Problem 1: Convert (4525)8 into a decimal.
Solution:
(4525)8 = 4 × 83 + 5 × 82 + 2 × 81 + 5 × 80
⇒ 45258 = 4 × 512 + 5 × 64 + 2 × 8 + 5 × 1
⇒ 45258 = 2048 + 320 + 16 + 5
⇒ 45258 = 238910
Problem 2: Convert (17)10 as a binary number.
Solution:
17 as Binary
Therefore (17)10 = (10001)2
Problem 3: Convert (1011110)2 into an octal number.
Solution:
Given (1011110)2 a binary number, to convert it into octal number
Octal Number
Binary Number
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111
Using this table we can write give number as
001 011 110 i .e 001 = 1 011 = 3 110 = 6
So (1011110)2 in octal number is (136)8
Problem 4: Convert the Roman numeral XLVIII into its decimal equivalent.
Solution:
As we know, all numbers before the greatest symbol are subtracted from it and all the numbers after the greatest symbol are added,
XLVIII = 50 - 10 + 5 + 1 + 1 + 1 = 48
Thus, XLVIII is 48 in decimal representation.
Problem 5: Convert the Roman numeral MCCXLV into its decimal equivalent.
Solution:
As we know, all numbers before the greatest symbol are subtracted from it and all the numbers after the greatest symbol are added,
