# Types of Number

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In studying mathematics, we see various types of numbers in maths such as 1, 2, 3, 4, 5, 6, 7……., or 1.1, 1.72, 4.3, 2.4323 or  5/6, 7/3, 6/4 or  5, -4, -3, -2, -1, etc.

We have seen these types of numbers during our studies but ever you have thought that what are they and why are they different from each other? What is the name of these types of numbers?

That’s the thing we are going to know in this guide. We will learn about the types of numbers. We are going to know what is Natural Number, Whole number, Real Number, Rational Number, Irrational Numbers, Integers, Counting numbers, etc.

There are different types of number but we shall discuss the important ones. We are going to discuss these topics in detail, about their definition, examples, properties, and many more things.

## HISTORY

(i) The Pythagoreans in Greece introduce,’ irrationals’ in 400 B.C.

(ii) German mathematicians Cantor and Dedekind showed in 1870 that corresponding to every real number there is a point on the real number line and corresponding to every point on the number line, there exists a unique real number.

(iii) The Greek genius Archimedes (476-550 AD) computed digits in the decimal expansion of n. He showed 3.140845 < n < 3.142857.

(iv) Ramanujan (1887-1920) born in Chennai (India) was able to calculate the value of π(Pi) correct to millions of decimal places.

## What are Counting Numbers

The set of numbers that can be used for counting are known as Counting Numbers. Example 1, 2, 3, 4, 5, 6,7…….. Counting number does not include 0, negative number, fraction, and decimals. Counting numbers are also called Natural Numbers

## What is Natural Number

Counting numbers are called natural numbers i.e all the positive integers from 1 till infinity are known as natural numbers. Thus, 1, 2, 3, 4, 5, 6, 7, …, etc., are all natural numbers.

Natural numbers do not include 0 hence 0 is not a natural number.

## What is Whole Number

All-natural numbers together with 0 form the collection of all whole Numbers. Thus, 0, 1, 2, 3, 4, 5, 6, 7, …, etc., are all whole numbers.

REMARKS

• Every natural number is a whole number.
• 0 is a whole number which is not a natural number.

## What is Real Number

A number whose square is non-negative is called a Real Number.
In fact, all rational and all irrational numbers form the collection of all real numbers.
Every real number is either rational or irrational.

Consider a Real Number
(i). If it is an integer it has a terminating or repeating decimal representation then it is rational.
(ii). If it has non-terminating and non-repeating decimal representation then it is irrational.

The totality of rationals and irrationals forms the collection of all Real Numbers.

COMPLETENESS PROPERTY On the number line, each point corresponds to a unique real number. And, every real number can be represented by a unique point on the real line.

DENSITY PROPERTY Between any two real numbers, there exist infinitely many real numbers.

## Additional Property of Real Numbers

(i) CLOSURE PROPERTY: The sum of two real numbers is always a number.

(ii) ASSOCIATIVE LAW: (a b) c = a + (b c) for all real numbers a ,b , c,

(iii) COW COMMUTATIVE LAW: a + b = b + a for all real numbers a and b.

(iv) EXISTENCE OF ADDITIVE IDENTITY: Clearly, 0 is a real number such that 0 + a = a + 0 = a for every real number a. 0 is called the additive identity for real numbers.

(v) EXISTENCE OF ADDITIVE INVERSE: For each real number a, there exists a real number (-a) such that a + (-a) = (-a) + a = 0. a and (-a) are called the additive inverse (or negative) of each other.

## Multiplication Properties of Real Number

(i) CLOSURE PROPERTY The product of two real numbers is always a real number.

(ii) ASSOCIATIVE LAW (ab)c = a(bc) = ba for all real numbers a, b , c.

(iii) COMMUTATIVE LAW ab = ba for all real numbers a and b.

(iv) EXISTENCE OF MULTIPLICATIVE IDENTITY Clearly, I is a real number such that 1 .a=a• 1 = a for every real number a. 1 is called the multiplicative identity for real numbers.

(v) EXISTENCE OF MULTIPLICATIVE INVERSE For each nonzero real number a, there exists a real number
(1/a) such that a • 1/a =1/a.a =1.
a and 1/a are called the multiplicative inverse (or reciprocal) of each other.

(vi) DISTRIBUTIVE LAWS OF MULTIPLICATION OVER ADDITION We have: a(b + c) = ab + ac and (a + b)c = ac + bc for all real numbers a, b, c.

## What are Integers in Math

All natural numbers, 0 and negatives of natural numbers form the collection of all integers. Thus, …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, etc., are all integers.

REMARKS

• (i) Every natural number is an integer.
• (ii) Every whole number is an integer.

## What are Rational Numbers

The number of the form p/q where p and q are integers and q 0 is known as Rational Number. Example: 4/5, 6/9, 3/5, 2/8, 9/3.

Remarks

• 0 is a rational number since we can write, 0=0/1.
• Every natural number is a rational number since we can write, 1=1/1, 2=2/1, 3=3/1, etc.
• Every integer is a rational number since an integer a can be written as a/1.

## Equivalent Rational Numbers

We know that,
1/2=2/4=3/6=……=15/30 =16/32= ….144/288=…..
These are known as Equivalent Rational Numbers

Example: Write four rational numbers equivalent to 4/7

Solution: we have,
7/4 = 4×2/7×2 = 4×3/7×4 = 4×4/7×4 = 4×5/7×5
4/7 = 8/14 = 12/21 = 16/2 8= 20/35
Thus four rational numbers equivalent to 4/7 are
8/14, 12/21, 16/28 and 20/35.

## Simplest Form of A Rational Number

A Rational Number p/q is said to be the simplest form if p and q are integers having no common factor other than 1 and q ≠ 0.

Thus, the simplest form of each of 2/4, 3/6, 4/8, 5/10, etc., is 1/2.

Similarly, the simplest form of 6/9 is 2/3 and that of 95/133 is 5/7.

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## Special Characteristics of Rational Number

• Every Rational Number is expressible either as a terminating decimal or as a repeating decimal.
• Every terminating decimal is a Rational Number.
• Every repeating decimal is a Rational Number.

## What are Terminating Decimals

Every fraction p/q can be expressed as a decimal. If the decimal expression of p/q terminates, i.e., comes to an end then the decimal so obtained is called terminating decimal.

Examples: We have, (i) 1/4=0.25, (ii) 5/8=0.625, (iii) 13/5=2.6
Thus, each of the number 1/4, 5/8, and 13/5 can be expressed in the form of terminating decimal

An Important Observation A fraction p/q is a terminating decimal only when prime factors of q are 2 and 5 only.

Examples:
Each one of the fraction1/2, 3/4, 7/20, 13/25 is terminating decimal since the denominator of each has no prime factor other than 2 and 5.

## What are Repeating (or Recurring) Decimals

A decimal in which a decimal or a set of digits repeats periodically is called a repeating or recurring decimal.

In a recurring decimal, we place a bar over the first block of the repeating part and omit the other repeating blocks.

## What are Irrational Numbers

A number that can neither be expressed as a terminating decimal nor as a repeating decimal, is called an irrational number. Thus, non-terminating, non-repeating decimals are an irrational number.

## Examples of Irrational Numbers

Type 1
(i) Clearly, 0.01001000100001… is a non-terminating and non-repeating decimal, and therefore, it is irrational.

Similarly,
(ii) 0.02002000200002… is irrational
0.03003000300003… is irrational, and so on.

(iii) 0.12112111211112… is irrational
0.13113111311113… is irrational, and so on.

(iv) 0.54554555455554… is an irrational number,
0.64664666466664… is irrational, and so on.

Type 2
If m is a positive integer which is not a perfect square, then √m is irrational
Thus, √2, √3, √3, √5, √6, √7, √8, √10, √11, etc are all irrational numbers

Type 3
If m is a positive integer which is not a perfect cube, then 3√m is irrational.
Thus, 3√2, 3√3, 3√4, 3√5, 3√6, 3√7, 3√8, 3√9, etc., are all irrational numbers.

The number ‘π’
π is a number whose exact value is 22/7.
In fact, π has a valve that is non-terminating and non-repeating.
So, π is irrational, while 22/7 is rational.

## Conclusion

You have learned about the types of numbers i.e., what is Natural Number, Whole number, Real Number, Rational Number, Irrational Numbers, Integers, Counting numbers, etc.

I hope you liked the way I guide you through this article and solved your queries regarding the types of numbers in math. If you still have any queries, then ask in the comment session.