The number system or the numeral system is the system of naming or representing numbers. There are various types of number systems in maths like binary, decimal, etc. This lesson covers the entire concepts of the numeral system with their types, conversions and questions.
What is Number System in Maths?
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. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction and division.
The value of any digit in a number can be determined by:
- The digit
- Its position in the number
- The base of the number system
What is Number?
A number is a mathematical value used for counting or measuring or labeling objects. Numbers are used to performing arithmetic calculations. The examples of numbers are natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents null value.
A number has many other variations such as even and odd numbers, prime and composite numbers. Even and odd terms are used when a number is divisible by 2 or not, whereas prime and composite differentiate between the numbers that have only two factors and more than two factors, respectively.
In a number system, these numbers are used as digits. 0 and 1 are the most common digits in the number system, that are used to represent binary numbers. On the other hand, 0 to 9 digits are also used for other number systems. Let us learn here the types of number systems.
Types of Number System
There are various types of number system in mathematics. The four most common number system types are:
- Decimal number system (Base- 10)
- Binary number system (Base- 2)
- Octal number system (Base-8)
- Hexadecimal number system (Base- 16)
Decimal Number System (Base 10 Number System)
Decimal number system has base 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). For 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 Number System)
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. For example, 110101 is a binary number.
We can convert any system into binary and vice versa.
Example
Write (14)10 as a binary number.
Solution:
Base 2 Number System Example
∴ (14)10 = 11102
Octal Number System (Base 8 Number System)
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 Number System)
In the hexadecimal system, numbers are written or represented with base 16. In the hex system, the numbers are first represented just like in decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabets from A to F. The below-given table shows the representation of numbers in the hexadecimal number system.
| 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 |
Number System Chart
In the number system chart, the base values and the digits of different number system can be found. Below is the chart of the numeral system.
Number System Chart
Number System Conversion
Numbers can be represented in any of the number system categories like binary, decimal, hex, etc. Also, any number which is represented in any of the number system types can be easily converted to other. 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.
Solved Examples
Q.1: Convert (1056)16 to octal number.
Solution: Given, 105616 is an hex number.
First we need to convert the given hexadecimal number into decimal number
(1056)16
= 1 x 163 + 0 x 162 + 5 x 161 + 6 x 160
= 4096 + 0 + 80 + 6
= (4182)10
Now we will convert this decimal number to required octal number by repetitively dividing by 8.
| 8 | 4182 | Remainder |
| 8 | 522 | 6 |
| 8 | 65 | 2 |
| 8 | 8 | 1 |
| 8 | 1 | 0 |
| 0 | 1 |
Therefore, taking the value of remainder from bottom to top, we get;
(4182)10 = (10126)8
Therefore,
(1056)16 = (10126)8
Q.2: Convert (1001001100)2 to decimal number.
Solution: (1001001100)2
= 1 x 29 + 0 x 28 + 0 x 27 + 1 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 0 x 20
= 512 + 64 + 8 + 4
= (588)10
Q.3: Convert 101012 into octal number.
Solution: Given,
101012 is the binary number
We can write the given binary number as:
010 101
Now as we know, in octal number system,
010 → 2
101 → 5
Therefore, the required octal number is 258
Q.4: Convert hexadecimal 2C to decimal number.
Solution: We need to convert 2C16 into binary number first.
2C → 00101100
Now convert 001011002 into a decimal number.
101100 = 1×25+1×23+1×22
=32+8+4
=44
Computer Numeral System (Number System in Computers)
When we type any letter or word, the computer translates them into numbers since computers can understand only numbers. A computer can understand only a few symbols called digits and these symbols describe different values depending on the position they hold in the number. In general, the binary number system is used in computers. However, the octal, decimal and hexadecimal systems are also used sometimes.
Frequently Asked Questions
What is Number System and its Types?
The number system is simply a system to represent or express numbers. There are various types of number systems and the most commonly used ones are decimal number system, binary number system, octal number system, and hexadecimal number system.
Why is the Number System Important?
Number system helps to represent numbers in a small symbol set. Computers, in general, use binary numbers 0 and 1 to keep the calculations simple and to keep the amount of necessary circuitry less, which results in the least amount of space, energy consumption and cost.
What is Base 1 Number System Called?
Base 1 number system is called the unary numeral system and is the simplest numeral system to represent natural numbers.
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