Number System In Digital Electronics

In this lecture, we will learn about the most basic and most important topic of digital electronics i.e. Number System In Digital Electronics. This topic is the foundation of digital electronics. So let’s start by knowing about the Number system and then we will learn the types of number systems and then see them in detail.

Number System In Digital Electronics

• The number system in digital electronics defines a set of values used to represent quantity. We talk about the number of people attending class, and the number of modules taken per student and also use numbers to represent grades achieved by students in tests.

• The study of number system in digital electronics is not just limited to computers. We apply numbers every day and knowing how numbers work will give us an insight into how a computer manipulates and stores numbers.

Before getting to know about the types of number system in digital electronics, let’s know some important points to be remembered about number systems.

Important Definitions Related to Number system

All the number systems have a few common elements as follows:

• The number of values that a digital (one character) can assume is equal to the base of the system. It is also called as Redix of the system.
  • For Example for a decimal number system, the base is ’10’ hence every digital can assume 10 values (0, 1, 2, …, 9).

• The largest value of a digit is always one less than the base.
  • For example, the largest digit in a decimal system is 9. (one less than the base 10).

• Each digit position (place) represents a different multiple of the base. i.e. the numbers have positional importance.
  • For example, consider the decimal number (349.25)₁₀

Also Read: What Do You Mean By Twos Complement?

Types of Number System in Digital Electronics

Number systems can be divided into four main types according to the basis or radix:

Types of Number System in Digital Electronics
Name of Number System	Base or Redix
Binary Number System	2
Octal Number System	8
Decimal Number System	10
Duodecimal Number System	12
Hexadecimal Number System	16

Now we will learn each number system in digital electronics in detail.

Decimal Number System

The number system we have been taught to use since school is called the decimal number system. We are familiar with counting and math that uses this decimal number system. Using this number system it will help us to understand other number systems.

Characteristics of Decimal Number System

Some of the important characteristics of a decimal number system are:

1. It uses the base or Redix of 10.
2. The largest value of a digit is 9.
3. Each place (column number) represents a different multiple of 10. These multipliers are also called weighted values.

Let a decimal number 5245 which has a total of four digits, this number can also be written as follows:

(5×103)+(2×102)+(4×101)+(5×100)

(5×1000)+(2×100)+(4×10)+(5×1)

5000+200+40+5=5245

From the above example, we can see that the value of the different digits depends on the position of the number.

Also Read: Classification of Logic Families |Characteristics of Logic Families

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Binary Number System

A computer cannot operate in the decimal number system which we have seen in the above section. Most modern computer systems operate using the binary number system only.

A binary number system uses only two digits namely 0 and 1.

The binary number system works like the decimal number system except for one change. It uses base 2.

Hence the largest value of a digit is 1 and the number values a digit can assume is two i.e. 0 and 1.

The binary digits (o and 1) are also called bits. Thus the binary system is a two-bit system.

The leftmost bit in a given binary number with the highest weight is called the Most Significant Bit (MSB) whereas the rightmost bit in a given number with the lowest weight is called the Least Significant Bit (LSB).

Let a binary number 1011 and we have to convert it into an equivalent decimal number, then:

(1×23)+(0×22)+(1×21)+(1×20)

8+0+2+1=11

∴ (1011)2=(11)10

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Binary Number system formats

• We typically write binary numbers as a sequence of bits (bits are short for binary digits). We have defined boundaries for these bits. These boundaries are:

Binary Number Formats
Name	Size (bits)	Example
Bit	1	1
Nibble	4	0101
Byte	8	0000 0101
Word	16	0000 0000 0000 0101
Double Word	32	0000 0000 0000 0000 0000 0000 0000 0101

In any number base, we may add many leading zeros as we wish without changing their values. However, we normally add leading zeros to adjust the binary number to a desired size boundary. For example, we can represent the number Five (5) as:

Nibble	0101
Byte	0000 0101
Word	0000 0000 0000 0101

Bit: The smallest unit of data is defined as a single bit. With a single bit, we can represent any two distinct items like true or false, on or off, male or female, right or wrong, etc.

The Nibble: A nibble is a combination of four bits. It would not be a particularly interesting data structure except for two items: BCD (binary coded decimal) numbers and hexadecimal (base 16) numbers because it takes four bits to represent a single BCD or hexadecimal digit. With a nibble, we can represent up to 16 distinct values.

The Byte: A byte is a combination of 8 binary bits. The number of distinct values represented by a byte is 256, ranging from 00000000 to 11111111. That means a number of distinct values = 2^N = 2⁴ = 16, where N = a number of bits used to represent a number.

The Word: A word is a combination of 16 bits. Hence it consists of two bytes.

The Double Word: A double word is exactly what its name implies, two words. Therefore, a double-word quantity is 32 bits. Naturally, this double word can be divided into a higher order word and a low order word, four bytes, or eight nibbles.

Disadvantages of the Binary Number system

• The binary number system has an important drawback. It requires a very long string of 1’s and 0’s to represent a decimal number. For example (128)₁₀ = (1 0 0 0 0 0 0 0 )₂
• So we require 8 bits to represent (128)₁₀. Imagine what will happen if (26275)₁₀ is to be converted.
• To overcome this, the other numbering system was developed.

Also Read: Comparison of Logic Families

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Octal Number System

The important features of the octal number systems are as follows:

Base: The Base used for the octal number system is 8.

The number of values assumed by each digit: Each digit in the octal system will assume different values from 0 to 7 ( 0, 1, 2,…,6, 7).

The largest value of a digit: The largest value of a digit in the octal system will be 7. That means the octal numbers higher than 7 will not be 8, instead of that it will be 10. The blow table gives you a clear idea about this. The largest two-digit number in the octal system is 77. The next higher number after 77 is 100.

Each digit has a different multiple of base, This is shown in the figure below.

Let an octal number 345 and we need to find its equivalent in decimal, then

(3×82)+(4×81)+(5×80)

192+32+5=229

∴ (345)8=(345)10

Application of Octal Number System

• In order to represent large numbers, we will need a huge string of 0’s and 1’s of the binary system I used. So the octal system can be used for reducing the size of the large binary number.
• But it is important to note that the digital circuits and computers internally work strictly with the binary structure, and not with the octal structure.
• The octal structure is used only for the convenience of the user.

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Hexadecimal Number System

The important feature of the Hexadecimal Number System is as follows:

Base: The base of the Hexadecimal Number System is 16.

The number of values assumed by each digit: The number of values assumed by each digit is 16. The values include digits 0 through 9 and letters A, B, C, D, E, and F. Hence the sixteen possible values are 0 1 2 3 4 5 6 7 8 9 A B C D E F.

0 represents the least significant digit whereas F represents the most significant digit.

The base of 16 needs 16 digits. hence it borrows 0 through 9 but needs another 6. For these, the first six letters, A, B, C, D, E, and F are used. Hence A represents 10, B represents 11, and so on.

When dealing with large values, binary numbers quickly become too unwise. the hexadecimal number system solves this problem. Hexadecimal numbers offer two features:

1. Hex numbers are very compact
2. It is easy to convert from hex to binary and binary to hex.

The largest value of a digit: The largest value of a digit in the hexadecimal number system is 15 and it is represented by F. The hexadecimal number higher than F will be 10.

The largest two-digit hexadecimal number is FF which corresponds to 255 decimal. The next higher number after FF is 100.

Positional weights: The positional weights for a hexadecimal number to the left and right decimal point are shown in the figure below:

Let 1AB be a hexadecimal number, we can convert it into the equivalent decimal number as follows:

(1×162)+(A×161)+(B×160)

(1×162)+(10×161)+(11×160)

256+160+11=427

∴ (1AB)16=(427)10

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Relation between Binary, Decimal, Octal, and Hexadecimal Numbers

The following table shows the decimal numbers from 0 to 15 and their equivalent binary, octal, and hexadecimal numbers:

Important Books: List Of Best Books For Digital Electronics

FAQs on Number System in Digital Electronics