# Number Systems in Digital Electronics – Decimal, Binary, Hexadecimal, Octal

## Number systems in digital electronics :

Number systems in digital electronics are the techniques of defining any quantity. It is basically a mathematical system model which is used to express digital quantity.

In ancient time human being was started counting with the help of fingers, stones, sticks, pebbles etc. So at that time, there was no any robust system to count. Mostly they counted with the help of fingers.With the development of civilisation, these basic counting techniques were founded to be not up to date. So, the complexity of the civilisation started demanding a more robust and complex mathematical system. So to overcome the problem of counting method, they started thinking of a more robust system. There are various number systems were developed, if we categorise these advanced counting systems then we get two types of number systems.

## Positional Number System :

In positional number systems there is a fixed place or position for each and every discrete element. These elements always represent a discrete value. The value of a certain element depends on the specific position of that element, base of the number systems and also on the element itself. Digital computers only understand positional number systems. We can classify the positional number systems into 4 different ways.

## Non Positional Number System :

In non positional number systems, the position of a number is not fixed at any instant. All the elements in non positional number systems represent value regardless of its position.

Roman number system is an example of non positional number system.

In digital number systems the most important thing that we have to understand is the radix or base of a number system. So the question is what is the base of a number system?

## What is the base of a number system :

The Base of a number system is the number of discrete elements that is used to represent a number system.

For example when we use binary number system to represent any number then we only used two numbers 0(zero) and 1(one). So, there are two types of discrete elements to represent any number in a binary number system. That’s the reason that base of a binary number system is 2.

## Decimal number system

#### Definition of decimal number system :

Decimal number system can be defined as method of representing a number using the integers from 0 to 9. It is the most commonly used number system.

Any number can be represented in decimal number system using 10 types of discrete elements. We use 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to represent a number in decimal format. Since here we are using 10 discrete elements from 0 to 9 that is why base or radix of decimal number system is only and only 10.

(16)10 is an example of a decimal number.

### Advantages of decimal number system

• Representing a number in decimal format is shorter than binary.
• It is easy to read and write since this number system is most commonly used in normal life.
• It is more efficient.

### Disadvantages of decimal number system

• Since digital computers only understand ON and OFF, we can’t represent these 2 states with 10 different states using decimal number system.
• To perform arithmetical operations by a CPU using decimal number system, it requires a very complex ALU design which is costly, time consuming and also a space eater.
• The speed of operations performed using decimal number system is slow because of complex hardware design.

## Binary number system

### Definition of binary number system :

Binary number system can be defined as the technique or way of representing any numbers by using only 0’s (zeros) and 1’s (onesA). Since only two elements are used in binary number system therefore the base of the binary number system is 2. Each digit in binary number system is referred as a bit. In digital world 0 represents OFF and 1 represents ON.

Digital computers recognize only these two States ON and OFF. Using only these two States computer’s CPU performs operations like processing, memory management, controlling different input output devices etc.

(10101)2 is an example of a binary number.

### Advantages of binary number system

• It has only two elements that are 0 and 1, so to deal with binary number system is quite easier as compared to the other number system.
• Using binary number system we can minimize a circuit design which helps in saving the space, energy as well as the cost.
• Conversion from other number systems like decimal, octal and hexadecimal to binary number system is quite easy since it has the lowest base of 2.
• In transmission of data, binary number system provides a robust way. Since, any noise can be easily detected and also easy to reject.
• Binary number system is the universal alphabet in mathematics which is used to communicate with any kind of extraterrestrial life in the universe.

### Disadvantages of binary number system

• It is difficult to read those 0’s (zeros) and 1’s (ones) for a large number. So, its difficult to represent a large number like 99 millions in binary.
• Its not an easy task to perform multiplications, divisions between binary numbers.

## Octal number system

### Octal number system definition :

In octal number system we use 8 discrete elements. We use 0, 1, 2, 3, 4, 5, 6, and 7 to represent any number in octal number system. So each and every octal number system should be represented using these 8 discrete elements. That’s why base of octal number system is 8.

(132)8 is an example of a octal number.

### Advantages of octal number system

• Each octal digit is equal to 3 bits. So it is a great way to represent a bit pattern.
• It is easy to write a number in octal format rather than a large binary number of many bits. Since conversation process from binary to octal is easy.
• Octal number systems are used to express large numbers which are normally used in 16 bits and 32 bits computer to express memory instructions, data, CPU status etc.
• Multiplications, divisions operations are easy to perform between octal numbers as compared to binary and hexadecimal number system.

### Disadvantages of octal number system

• Digital computer does not understand octal number system. Using octal number system, we can’t represent ON and OFF state using 8 different States.
• As like decimal number system we have to use a complex ALU design to deal with octal number system in digital computers.
• It is not efficient due to the need of complex hardware.

### Definition of hexadecimal number system :

The word ‘Hexa’ means 6 and the decimal consists of 10 discrete elements from 0 to 9 as we already read in the decimal number system section. So hexadecimal has the base or radix of 16. It consists of numbers 0 to 9 and specially 10 to 15 but they are represented as A, B, C, D, E and F respectively.

(AF32)16 is an example of a hexadecimal number.

• Each hexadecimal digit is equal to 4 bit which is helpful to generate a bit pattern.
• Hexadecimal number system provides an easy way to implement a large binary number.
• It is the most commonly used number system to represent computer memory addresses.