We are all familiar with the concept of numbers. A number is a mathematical entity employed for counting objects, serving as the foundation for arithmetic calculations. Numbers manifest in diverse types, encompassing:

- Natural Numbers
- Whole Numbers
- Rational Numbers
- Irrational Numbers and more

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## What, then, is a number system?

A number System is a method of representing Numbers on the Number Line with the help of a set of Symbols and consistent rules. These symbols range from 0-9 and are termed as digits. Number System comes in handy to perform mathematical computations. That said, Number Systems can be of different types. They include:

- Decimal number system (Base- 10)
- Binary number system (Base- 2)
- Octal number system (Base-8)
- Hexadecimal number system (Base- 16)

Let us look at each of these systems in some detail.

## Decimal Number System

This is occasionally denoted as the Base 10 Number System, deriving its name from the utilization of ten digits ranging from 0 to 9. Given its widespread application, you may observe that the digits to the left of the decimal point signify distinct positions, including Units, Tens, Hundreds, and beyond.

#### Let us take an example.

The number 1254 has 4 in the unit’s place, 5 in the tens place, 2 in the hundreds place, and 1 in the thousands place. If we were to express its value, it would be:

(1×103) + (2×102) + (5×101) + (4×100)

(1×1000) + (2×100) + (5×10) + (4×1)

1000 + 200 + 50 + 4 = 1254

To sum up, here are some important points of the decimal number system:

- It uses ten digits.
- The place value of each digit is determined from right to left in the number.
- The decimal number system is the most common number system in daily life.

## Binary Number System

This is known as the Binary Number System, operating on the recognition of two binary digits: 0 and 1. The base of the Binary Number System is also referred to as the radix. Consequently, in a binary number system, a number is conventionally represented as (87065) ₂.

Converting a binary number into a decimal number involves multiplying each digit of the binary number by the corresponding power of 2, either 1 or 0. Conversely, transforming a decimal number into a binary number requires continuous division of the given decimal number by 2 until the quotient becomes 1. The resulting binary number is then written from bottom to top.

### Steps for Conversion of Binary to Decimal Number System

The steps involved in converting a number from the binary to the decimal system are as follows:

- You need to multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
- The exponents should start with 0 and increase by one every time we move from right to left.
- Simplify each of the above products and add them.

To sum up, here are some important aspects of the Binary number system:

- It uses two digits: 0 and 1.
- The digits 0 and 1 are called bits.
- The binary number system is also used in computer systems.

## Octal Number System

The octal numeral system is the base-8 number system. It uses 8 digits from 0 to 7. With fewer digits than the decimal or the hexadecimal number system, it is prone to fewer errors. A number in the octal number system is represented with the number 8 at the base.

Let us take an example where we convert an Octal Number into a decimal.

Example: (16)_{8} to decimal

= 1 x 8^{1} + 6 x 8^{0}

= 8 + 6

= (14)_{10}

Here are some salient features of the Octal Number System:

- It uses 8 digits: 0-7
- This number system has lesser digits as compared to other systems.

## Hexadecimal Number System

In the hexadecimal system, numbers are written or represented with base 16. The exciting thing about this system is that in the hexadecimal system, the numbers are first represented like in the decimal system, i.e., from 0 to 9. Then, the numbers are defined using the alphabet from A to F. The base of the hexadecimal number system is 16 because it has 16 alphanumeric values. Here, A is 10, B is 11, C is 12, D is 13, E is 14, and F is 15.

### Number System Conversion

As we saw in several examples above, a number can be represented in any number system and converted from one to the other.

The number 349, for instance, can be written in different ways in different number systems. The decimal number 349 can thus be represented as 15D regarding the hexadecimal number system.

### Computer Numeral System

Besides the above number systems, there is also the Computer Numeral System. Essentially, when we type any letter or word, the computer, in turn, translates it into numbers. The computer primarily makes use of the binary number system.

How about ending the blog with some trivia on the Number System?

#### Trivia

- The Egyptians had a base 10 system of hieroglyphs for numerals
- The decimal number system is also called the Hindu–Arabic numeral system
- The decimal system is the most used number system, possibly because humans have five fingers on each hand and ten in both.
- The Greeks used a numeral system based on the 27 letters in the Greek alphabet.
- The Romans used 7 symbols to express their number system
- The Indian numerals form the basis of the European number systems, which are now widely used.

Keep watching this space for more such informational and fun blogs. At Pragyanam, one of the best CBSE schools in Gurgaon, we aim to pique students’ curiosity and ensure that students remain learners for life.

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