Base of Number System

In the case of a positional numeral system, a base refers to the total number of unique digits that we use to represent all the numbers. It also includes the digit zero. Now, for instance, for a decimal system, the base number is ten. It is because it utilises the ten digits ranging from 0 to 9.

In this article, we will take a look at the Base of Number System according to the GATE Syllabus for CSE (Computer Science Engineering). Read ahead to learn more.

Table of Contents

What is the Base of the Number System?
- Examples
  - Number System with Base 10
  - Number System with Base 2
- Important Note
Types of Number System
FAQs

What is the Base of the Number System?

The base of a number system refers to the total number of digits that are actually used in the given number system. The number system that has the base ‘b’ consists of its digits in the [0, b-1] range. This base of the number system is also known as the radix of a number system.

Let us take a look at a few examples to understand this better.

Examples

We will now consider the examples given below-

Number System with Base 10

Let us consider a base 10 number system, normally known as a decimal number system, here:

The total number of digits that are used in this number system is 10 since it has a base of 10.
Thus, these digits would be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The digits here lie clearly in the [0, base-1] range = [0, 9].

Number System with Base 2

Let us consider a base 2 number system normally known as a binary number system, here:

The total number of digits that are used in this number system is 2 since it has a base of 2.
Thus, these digits would be 0 and 1.
The digits here lie clearly in the [0, base-1] range = [0, 1].

Important Note

It is critical to note that:

The digits of a number system that has the base ‘b’ will always be less than ‘b’.
It is clear from the [0, base-1] range in which all the digits of a number system lie.

Types of Number System

The four most generally used number systems are these:

1. Decimal

2. Binary

3. Octal

4. Hexadecimal

The table given below shows what bases and digits are used in these mentioned number systems:

Number System	Digits Used	Base
Decimal Number System	(Total 10 digits) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9	10
Binary Number System	(Total 2 digits) 0, 1	2
Octal Number System	(Total 8 digits) 0, 1, 2, 3, 4, 5, 6, 7	8
Hexadecimal Number System	(Total 16 digits) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F	16

Frequently Asked Questions

Which number system has a base 2?

The binary number system that has only two independent digits is referred to as a base-2 number system. All the larger types of binary numbers get represented in terms of ‘1’ and ‘0’.

How many types of number systems are there?

There are two primary types of number systems: Positional as well as Non-positional. The positional system utilises digits to represent the numbers. On the other hand, a non-positional number system utilises specific symbols to represent the numbers. For instance, octal is a positional type of number system that has the base 8.

What is the base of the octal number system?

The base of the octal number system is 8. Thus, the maximum value of any given single-digit is 7. Here, it takes digits ranging from 0 to 7. Now, in a number system, every position of a digit would represent some specific power of the given base.

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The Base of Number System