# The 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.

## 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 â€˜bâ€™ 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 â€˜bâ€™ will always be less than â€˜bâ€™.
• 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

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

Q1

### 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 â€˜1â€™ and â€˜0â€™.

Q2

### 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.

Q3

### 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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