Number System: A Guide To Learn About

This article is completely dedicated to Number Systems. Here we will be providing information about types of number systems, conversions, number systems in general as well as in computer language and a lot more.

It is definitely a very vast topic and has a lot of information on it. We will try to compress the information for you so that you get to know all the necessary terms regarding number systems in this article itself.

Let us first begin with the number of systems in general terms.

Definition

The way representing or naming the numbers is known as the numeral or number system. In mathematics, there are various types of number systems like binary, decimal, etc. These types are discussed later in the article.

But first, let us understand what do number systems mean in maths and what role do they play in it?

As we have already mentioned above that number systems are the collection of numbers. They are of different types such as whole number, natural number, etc.

The table below makes it clear regarding different types of numbers.

NAMESYMBOLEXAMPLE
Natural NumbersN1,2,3,4,…
Whole NumbersW0,1,2,3,4,5,…
IntegersZ…,-4,-3,-2,-1,0,1,2,3,…
Rational NumbersQp/q form where p and q are integers and q is not a zero 1/2, 4/5, 26/8, etc.
Irrational NumbersWhich can’t be represented as rational numbers √3, √5, √11, etc.
Real NumbersAll the rational and irrational numbers are collectively called as real numbers.

Characteristics of Number Systems

Rational Numbers

Every rational number can be expressed in the decimal form either as terminating or recurring decimal.

Terminating Decimal: If the decimal p/q terminates or comes to an end after reducing it to the last then it is called as terminating decimal.

Example: 1/5=0.20

Recurring Decimal: If the decimal p/q keeps on repeating the digits in the reduced form then it can be called as recurring decimal.

Example: 1/3=0.3333…

Irrational Numbers

The numbers that are non-repeating and non-terminating after their reduction are known as irrational numbers.

Example: 1/7=0.142…

Real numbers

A number whose square is non-negative is known as real number.

Example: 4^2=16 (real number)

This was a pretty basic and general classification of number system. Next we will be discussing the detailed classification of number systems.

Classification of Number Systems

A number system can be defined as a system of writing to express numbers. It is the mathematical expression for representing numbers of a given set by using digits or other symbols in a consistent manner.

It gives a unique identity to every digit and helps in using those digits for equations and algorithmic operations.

It also allows us to operate arithmetic operations like addition, subtraction and division.

The value of any digit in a number is represented by:

  • The digit itself
  • The position of the digit in the number
  • The base value of the number system

Different categories of Number System

There are various categories in number systems. Some of the basic types are already discussed above. Now here we will see further enhanced categories of number systems.

The four most common types of number system are:

Decimal number system (Base- 10)

Binary number system (Base- 2)

Octal number system (Base-8)

Hexadecimal number system (Base- 16)

Decimal Number Systems

Decimal number systems are the numbers that use 10 as their base and that is because they use 10 digits from 0 to 9.

Let us make this a bit more clear to you. In this number system the position of each digit represents certain power to the base 10. For example if a number if 526 then the digit 6 are at unit’s place, 2 is at ten’s place and 5 is at hundred’s place.

Now with the base 10 the number can be expressed as:

5×10^2=5×100=500

2×10^1=2×10=20

6×10^0=6×10=60

Adding the resultant we get 500+20+6=526

The decimal number systems are the most common and widely used number systems all over the world. This number system is accepted almost everywhere in all the countries.

Binary Number Systems

The binary number system has 2 as the base. In this number system only two binary digits exist i.e. 0 & 1.

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All the numbers are represented in the form of these two binary digits.

It might look and sound a bit complicated to you but we have illustrated this number system very thoroughly in the below given example.

Example: Write ‘14’ as a binary number

Answer:

First of all you need to keep in mind that here the base is digit 2. So all the numbers and its answers will be only divided by 2.

Dividing 14 by 2 gives 0 as remainder (14 is divisible by 2) therefore we will place ‘0’

Dividing Quotient i.e. 7 by 2 gives 3 as remainder (7 is not divisible by 2) therefore we will place ‘1’

Dividing quotient i.e. 3 by 2 gives 1 as remainder (3 is not divisible by 2) therefore we will place ‘1’

The remainder of the above division is to considered as well, so we write ‘1’

Now to write the sequence of 0s and 1s always start from bottom to top. This means that our answer will be:

1110

Specifically to be mentioned here is that the base 2 is a radix of 2. Therefore the figures in this number system are called as binary numbers.

Octal Number System

The octal number system has base as 8 and it uses 0 to 7 digits to represent the answers. We know you might be a little bit confused here. But you will understand it once we share an illustration with you.

The octal numbers are mostly used in computer applications. Converting an octal number to decimal is same as decimal conversion.

Example: Convert Decimal number 348 to an octal number.

Dividing 348 by 8 gives 43 as quotient and 4 as remainder

Dividing 43 by 8 gives 5 as quotient and 3 as remainder

5 cannot be divided by 8 (incomplete equation) therefore taking 5 as it is

Writing the answer from bottom to top

Answer: (534) base 8

Hexadecimal Number System

In the hexadecimal number system the base of the numbers is taken as 16. This number system is similar to decimal numbers.

From 0 to 9 they are represented identically as the decimal numbers are written. After that numbers are replaced by alphabets. This means that the number ‘10’ is represented by ‘A’, ‘11’ by ‘B’ and so on till the number ‘15’ being represented by ‘F’.

DECIMAL0123456789101112131415
HEXA-DECIMAL0123456789ABCDEF

Converting a decimal number into hexadecimal is similar to the conversions mentioned above. The only difference is that the decimal number is divided by 16 to get its hexadecimal form.

Again you might be confused on how to use them in general. Well the confusion will be gone once you read the example given below.

Example: Write 8402 in hexadecimal

Answer:

Divide 8402 by 16 and then separate what is on the left and right side of the decimal point.

8402/16 = 525.125

Multiply the right side of the decimal point by 16 (if the answer is more than 9 then use the decimal to hexadecimal conversion table)

0.125×16 = 2

Again divide the left side of the decimal point by 16 and separate the left and right of the decimal point in the answer

525/16 = 32.8125

0.8125×16 = 13 = ‘D’

32/16 = 2

0×16 = 0

2/16 = 0.125

0.125×16 = 2

Keep repeating the above steps until the answer is 0

Then enter the digits from bottom to top or in reverse order of the solving.

Answer: 20D2

The hexadecimal number systems are also used in computer applications. They are one of the main languages that computers can understand.

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Number Systems in Computer

The language we use to communicate comprises of alphabets and characters. But that is not the case with computers.

Computers do not understand words, sentences, phrases instead they only understand numbers. Thus, when we type any letter or word, the computer translates them into numbers since computers can understand only numbers.

When we enter the data into a computer it recognizes it and converts it into an electronic pulse. Each pulse has its own code and this code is converted into numbers by ASCII.

ASCII provides each character and symbol with a numeric value that can be understood by the computer.

So to understand the computer language, you must be aware of all the number systems we have discussed above.  

In general, the binary number system is used in computers. However, the octal, decimal, and hexadecimal systems are also used sometimes.

Why do computers use numbers only?

This is the most commonly asked question and we are sure that a lot of our readers might be having a similar query that why computers only understand numbers.

Well, there is a very rational and strategic explanation for it.

Computers only understand numbers because the binary number system is the most simplified number system and it helps them keep the calculations very simple.

Another reason is that the amount of necessary circuitry and other equipment reduces significantly. This ultimately results in less space coverage, which means more compact units, and also reduces the cost of production and consumption simultaneously.

So, it is a very clever move that computer language can only understand numbers.

Conclusion

So in this article, you learned about Number systems, their types, their uses, and also different types that are used in computer applications.

We tried our best to explain each and every term in the simplest way possible with all the necessary examples.

We hope this article was helpful in solving most of the queries of our readers.

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