Classification of Numbers

Based on characteristics, numbers can be classified into different types, such as:

• Natural Numbers
• Whole Numbers 
• Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
• Complex Numbers

Natural Numbers

Natural Numbers, which are thought to be the first numbers to exist, are the most fundamental and obvious types of numbers. Natural numbers are defined as numbers that are used for counting. The Natural Numbers are therefore 1, 2, 3, 4, . . . and go on forever. The general representation of Natural numbers is N, and it is widely used in textbooks for the same. The image added below shows natural or counting numbers.

Whole Numbers

After the discovery of 0 Whole Numbers became the natural continuation of Natural Numbers. As Whole Numbers are defined as the collection of Natural Numbers including 0 i.e., 0, 1, 2, 3, 4, . . . and goes on forever. Whole Numbers are represented by the letter W. The image added below shows Whole Numbers.

Integers

When the use of negative numbers was popularized, they were very useful for many real-life use cases, such as debt-oriented calculations. Integers came into existence, as these are collections of whole numbers as well as the negative of each natural number, i.e., . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .,  and these go forever on both sides. Integers are represented by Z.

Rational Numbers

There was a problem in ancient Egypt with how to represent half or one-third of something in the records, so they came up with the solution known as fractions, and these fractions further evolved into Rational Numbers as we know them today. For a definition, Rational Numbers are those numbers that can be represented in the p/q form, where p and q are both integers and q can never be 0. For example, 1/2, 3/5, 17/41, 13/7, etc. (As we can’t list all rational numbers as a list of natural numbers or integers).

The image added below shows the rational and irrational number

Irrational Numbers

Irrational Numbers came into existence due to geometry, as Pythagoras discovered a very elegant solution for a right-angled triangle known as the Pythagoras Theorem. If there is a right-angled triangle with its base and height both being 1 unit, then using Pythagoras’ theorem, its hypotenuse comes to be √2, which back then wasn’t known as anything.

Also there was a dark story about it that goes like one of the Pythagoras’s disciple named Hippasus of Metapontum proved the existence of irrational numbers representing √2 as fraction and proofing that it is a contradiction but Pythagoras believed in the absoluteness of numbers and couldn’t accept the existence of irrational number but he also didn’t able to disproof logically that irrational numbers doesn’t exist. So, he sentenced Hippasus’ death by drowning to impede the spread of such things which were against the philosophies of Pythagoras.

Irrational numbers are defined as such numbers that can’t be represented as the ratio of two integers and are represented by P. Irrational Numbers are non-terminating and non-repeating in nature i.e. they don’t have decimal value limited to finite places and also the preparation of digits in decimal expansion is not periodic. Some examples of Irrational Numbers include √2, √3, √11, 2√2,  π(pi), etc. (As we can’t list all rational numbers as a list of natural numbers or integers).

Real Numbers

The collection of rational and irrational numbers is known as Real Numbers but the name comes from the fact that they can be represented on the real number line. All the examples of rational and irrational numbers are examples of Real Numbers as well. All the numbers except imaginary numbers are included under Real Numbers. Real Numbers are represented by R.

Imaginary Numbers

For a long period of time, people thought that the number system was incomplete and needed some new sort of numbers to complete it, as there was no solution to the equation x²+a=0(where a>0) in real numbers, but we now know by the fundamental theorem of algebra that every polynomial of degree n needs to have n roots. So there must be a new sort of number needed to find the solution to the above equation. 

The solution of the equation x² + a = 0 is simply x = ±√-a, which in ancient times was not accepted as the solution because they didn’t know any such number whose square was a negative number, but eventually, some mathematicians started using such a number and saw that this made sense for a lot of other calculations as well. Some things that mathematicians saw as impossible before the use of the square root of negative numbers now seem graspable. One of the first mathematicians to use this notion was Rafael Bombelli, an Italian mathematician. Eventually, this concept of using the square root of negative numbers is becoming a useful tool for many fields of mathematics as well as physics.

A new symbol, “i(iota)” was used by Euler first for -1 so he could easily represent an imaginary number without writing √-1 repetitively, and it spread across the world and became second nature to use “i” for √-1. Numbers that give a negative value when squared are generally called Imaginary Numbers. Some examples of these numbers are √-21(which can be written as √-1×√21 or i√21), i, 2i, etc.  

Complex Numbers

Complex numbers are the result of the endeavor of hundreds of mathematicians to complete the number system and are defined in the form of a+ib, where a and b are real numbers and “i” is the iota, which represents √-1. Complex numbers are represented by C and are the most useful in the different fields of modern physics, such as quantum mechanics and electromagnetic waves. Some examples of Complex Numbers are 1+i, √2-3i, 2-i√5, etc. The image added below represents the general structure of complex numbers.

The set of numbers is discussed in the image added below which explains that all the numbers known to humans are the subset of complex numbers.

Numbers in math are the most fundamental thing invented by mankind to serve its vast variety of endeavors in science and technology. From sending rockets to Mars to calculating bills for groceries, numbers are used everywhere. Nowadays, we can’t think of mathematics without Understanding numbers.

There are different types of numbers like natural numbers, integers, rational numbers, irrational numbers, real numbers, complex numbers, prime numbers, composite numbers, algebraic numbers, transcendental numbers, even and odd numbers, and many, many more.

In this article, we will discuss what are numbers with examples, definition, types, history, operations on numbers, and practice problems.