In this article, we will learn about the most fundamental concept in digital electronics namely, Number Systems. A number system can be defined as a system of representing or expressing numbers. In the field of digital electronics, number systems are critical elements because all the digital systems work by using information represented in numbers.
What is a Number System?
A number system is nothing but a method of expressing numbers using some specific symbols, called digits. Each of the digits or symbols has a certain value assigned to it depending on the number system rules.
In a number system, there are following three things that determine the value of a given number –
- Digit itself
- Base/radix of the number system
- Position of the digital within the number
We will understand this concept in more detail in the upcoming sections of this article.
A number system has two important components namely,
- Radix or Base
- Symbols or Digits
The base of a number system is defined as the total number of symbols or digits used in that number system. For example, the decimal number system (we use in everyday life calculations) has ten digits, from 0 to 9, thus the base of the decimal number system is 10.
Types of Number System
In digital electronics, the following four major types of number systems are used –
- Decimal Number System
- Binary Number System
- Octal Number System
- Hexadecimal Number System
Let’s discuss each type of number system in detail.
Decimal Number System
The decimal number system is also known as the natural number system and we use this number system in our everyday life for basic calculations. The decimal number system comprises of ten unique digits, which are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Hence, the base of the decimal number system is 10, and each position in a decimal number represents a power of 10.
For example, consider a decimal number (2215)10, the value of each digit of this decimal number will be given as follows –
(2 × 103) + (2 × 102) + (1 × 101) + (5 × 100) = 2000 + 200 + 10 + 5 = (2215)10
Binary Number System
The binary number system is the foundation of digital electronics. It is a number system that uses only two digits i.e., 0 and 1. Hence, the base of the binary number system is 2. It is used in digital electronics to represent two distinct states of electronic signals, which are High (represented by 1) and Low (represented by 0).
Each digit of the binary number system is called a Binary Digit (Bit). The following are two common groups of bits and their names –
- A group of 4 bits is called a Nibble. For example, 0101, 1001, etc.
- A group of 8 bits is called a Byte. For example, 11101110, 11001111, etc.
In a binary number, each position represents a power of two. Consider the following example to understand this.
A binary number, let (1110)2, is expressed as follows –
(1 × 23) + (1 × 22) + (1 × 21) + (0 × 20) = 8 + 4 + 2 + 0 = 14
Octal Number System
The octal number system is another widely used numeral system in digital electronics. The octal number system comprises eight unique digits or symbols, which are 0, 1, 2, 3, 4, 5, 6, and 7. These digits are used to express octal numbers. Since the octal number system uses 8 digits, hence its radix or base is 8. Therefore, each position in an octal number represents a power of 8.
Additionally, each octal digit can be represented using a group of 3 binary digits. Thus, the octal number system provides a more efficient way of representing binary codes. For this reason, it is widely used in computing and computer architecture for expressing machine-level instructions and memory addresses.
For example, consider an octal number (572)8, this number be expressed in decimal as follows –
(5 × 82) + (7 × 81) + (2 × 80) = 320 + 56 + 2 = (378)10
In the binary number system, it can be expressed as,
5 = 101
7 = 111
2 = 010
Thus,
(572)8 = (101 111 010)2
Hexadecimal Number System
The hexadecimal number system is a base 16 number system. Thus, it uses 16 unique digits or symbols, which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Here, the alphabetic letters represent the following numbers –
A = 10; B = 11; C = 12; D = 13; E = 14; F = 15
Since the hexadecimal number system has a radix or base of 16 (24), thus it uses 4 binary digits to represent a hexadecimal digit.
The most significant advantages of hexadecimal numbers include compact size and the requirement of less memory. Due to easy conversion between hexadecimal and binary, this number system is widely used in computer programming and microcontroller’s input-output handling.
For example, consider a hexadecimal number (AD5)16, which can be expressed in decimal as follows –
(A ×162) + (D × 161) + (5 × 160) = (10 ×162) + (13 × 161) + (5 × 160) = 2560 + 208 + 5 = (2773)10
In the binary number system,
A = 1010
D = 1101
5 = 0101
Thus,
(AD5)16 = (1010 1101 0101)2
Summary of Number Systems
The following table summarizes the number systems discussed above –
|
Number System |
Radix/Base |
Digits Used |
| Decimal Number System |
10 |
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Binary Number System |
2 |
0, 1 |
| Octal Number System |
8 |
0, 1, 2, 3, 4, 5, 6, 7 |
| Hexadecimal Number System |
16 |
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (10), B (11), C (12), D (13), E (14), F (15) |
Conclusion
Hence, this is all about different types of number systems in digital electronics. Number systems are ways of representing numbers in a systematic manner. In digital electronics, there are four main types of number systems used to design and implement digital systems, and these number systems are described above in detail.