Numbers
Numbers are an integral part of our everyday lives, right from the number of hours we sleep at night to the number of rounds we run around the racing track and much more. In math, numbers can be even and odd numbers, prime and composite numbers, decimals, fractions, rational and irrational numbers, natural numbers, integers, real numbers, rational numbers, irrational numbers, and whole numbers. In this chapter, we'll get an introduction to the different types of numbers and to all the concepts related to it.
Introduction to Numbers
Numbers form the basis of mathematics. We should befriend numbers in order to understand maths. Numbers are of various kinds. We have a long list that includes ordinal numbers, consecutive numbers, odd numbers, even numbers, natural numbers, whole numbers, integers, real numbers, rational numbers, irrational numbers, and complex numbers.
Along with numbers, we come across the interesting world of factors and multiples. This world includes prime numbers, composite numbers, coprime numbers, perfect numbers (yes, numbers could be perfect!) HCF, LCM, and prime factorization.
Let’s get started on our journey of numbers. You can go ahead and explore all important topics in Numbers by selecting the topics from this list below:
Prenumber Math
Building prenumber math skills is a prerequisite to understanding numbers. Prenumber skills like matching, sorting, classifying, ordering, and comparing will set the stage to build a strong number sense. Prenumber Math skills are builtin preschool years. Kids learn how to stand before they start taking small steps. In the same way, the pre number concept is very important for them to start understanding Mathematics. In this section, we will cover the different prenumber concepts like Matching and Sorting, Comparing and Ordering, Classification, and Shapes and patterns.
Example:
In the above example, the left column displays the numbers 1 to 4. The right column displays rows of items. The numbers are matched to the quantities they represent. This is an essential skill for children aged 3 to 4 years.
Number Names
Number names are used to represent numbers in an alphabetical format. A specific word is used to refer to each number. To write a number in words in English, you should know the place value of each digit in the number.
Example:
PEMDAS
The rules of PEMDAS outline the order of the operations and give structure to nested operations. In mathematics, PEMDAS is an acronym that stands for P Parentheses, E Exponents, M Multiplication, D Division, A Addition, and S Subtraction.
Let’s learn in detail the different concepts of PEMDAS such as Addition, Subtraction, Multiplication, and Division.
Number Systems
The decimal number system is the most commonly used number system. The digits 0 to 9 used to represent numbers. A digit in any given number has a place value. The decimal number system is the standard system for denoting integers and nonintegers. We will be using the decimal number system for representation of Numbers up to 2Digits, Numbers up to 3Digits, Numbers up to 4Digits, Numbers up to 5Digits, Numbers up to 6Digits, Numbers up to 7Digits, Numbers up to 8Digits, Numbers up to 9Digits and Numbers up to 10Digits.
Example:
Cardinal Numbers and Ordinal Numbers
A cardinal number is a number that denotes the count of any object. Any natural number such as 1, 2, 3, etc., is referred to as a cardinal number, whereas, an ordinal number is a number that denotes the position or place of an object. For example, 1^{st}, 2^{nd}, 3^{rd}, 4^{th}, 5^{th}, etc. It indicates the order of things or objects, such as first, second, third, fourth, and so on.
Example:
In the example given above, ordinal numbers will help define the position of the children. Such as, Jim is the fourth child from the left.
Consecutive Numbers
Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number. They usually have a difference of 1 between every two numbers.
Example:
Integers
Integers are numbers that are whole numbers and negative numbers. All integers are represented by the alphabet Z. A number line is full of integers. On the left side, you can find negative integers while on the right side you have the positive ones. Don’t forget the zero in between! Integers do sound interesting right? We will now learn about the Addition and Subtraction of Integers, Multiplication, and Division of Integers, Euclid's Division Lemma, and Euclid's Division Algorithm.
Z = { ...., 4, 3, 2, 1, 0 , 1, 2, 3, 4,....}
Natural Numbers and Whole Numbers
A natural number is a nonnegative integer and is always greater than zero. It is represented by the symbol N. We will learn about the various properties of natural numbers as part of understanding this concept. Now, the whole number does not contain any decimal or fractional part. It means that it represents the whole thing without pieces. It is represented by the symbol W.
We will learn about the various properties of whole numbers as part of understanding this concept.
Even Numbers and Odd Numbers
Even numbers are those numbers that can be divided into two equal groups or pairs and are exactly divisible by 2. For example, 2, 4, 6, 8, 10, and so on. In other words, these are whole numbers that are exactly divisible by 2.
Example:
Odd numbers are whole numbers that cannot be exactly divided by 2. These numbers cannot be arranged in pairs. Interestingly, all the whole numbers except the multiples of 2 are odd numbers.
Example:
Prime Numbers and Composite Numbers
A prime number is a number that has exactly two factors, 1 and the number itself whereas a composite number is a number that has more than two factors, which means it can be divided by number 1 and itself, and at least one more integer. Any whole number greater than 1 that has exactly two factors, 1 and itself is defined to be a prime number.
Examples: 2, 5, 7, 11 , etc
Well, now we know that a prime number has just two factors. One and the number itself. But how do we know that a number is prime? You will learn How to find out if a Number is a Prime Number in this section. along with the concepts of Prime Numbers and Euclid’s Proof 3.
What about Composite Numbers? Any number greater than 1 that is not a Prime number, is defined to be a composite number. Thus composite numbers will always have more than 2 factors.
Examples: 6, 8, 9, 12, etc
 Factors of 6 = 1, 2, 3, 6 (factors other than 1 and 6)
 Factors of 8 = 1, 2, 4, 8 (factors other than 1 and 8)
 Factors of 9 = 1, 3, 9 (factors other than 1 and 9)
 Factors of 12 = 1, 2, 3, 4, 6, 12 (factors other than 1 and 12)
Composite numbers are positive integers and you already know that they have more than one factor. Let us now learn about the Fundamental Theorem of Arithmetic.
Coprime Numbers
If a pair of numbers has no common factor apart from 1, then they are called coprime numbers. In other words, a set of numbers or integers which have only 1 as their common factor, which means their highest common factor (HCF) will be 1, are coprimes. These are also known as mutually prime numbers or relatively prime numbers. Also, there should be two numbers in order so to form coprimes.
Example:
Perfect Numbers
Perfect numbers are the positive integers which are equal to the sum of its factors except the number itself. In other words, perfect numbers are the positive integers which are the sum of its proper divisors. The smallest perfect number is 6, which is the sum of its proper divisors: 1, 2 and 3
Example:
Fractions and Decimals
Fractions are a part of a whole. They are represented by numbers that have two parts to them. There is a number at the top, which is called the numerator, and the number at the bottom is called the denominator.
Example:
Now that we already know about fractions and how it is represented, let us explore some more fraction related topics like Equivalent Fractions, Improper Fractions and Mixed Fractions, Addition and Subtraction of Fractions, Multiplication of Fractions and Division of Fractions.
What about Decimals? A decimal number has a whole number part and a fractional part. These parts are separated by a decimal point.
Example:
Decimals are really interesting. They have a whole number part and they can also be represented as fractions. Let’s dive deeper and find out how! In this section, we will cover decimals related concepts such as the Relationship between Fractions and Decimals, Addition and Subtraction of Decimals, Multiplication of Decimals, and Division of Decimals.
Rational Numbers and Irrational Numbers
A rational number, denoted by Q, is represented in the form p/q, where q is not equal to zero. Integers, Fractions, Decimals, Whole numbers, and Natural numbers are all Rational numbers.
Example:
In order to get a better understanding of Rational numbers, we will cover topics like Decimal Representation of Rational Numbers, and Operations on Rational Numbers.
Now, Irrational Numbers are the numbers that cannot be represented using integers in the p/q form. The set of irrational numbers is denoted by Q'.
Example: √5, √2, etc
Irrational numbers cannot be represented as a simple fraction. Their decimal expansion neither terminates nor becomes periodic. You must be wondering how! We will find out once we study some more topics related to irrational numbers such as Square Root of Two is Irrational, Decimal Representation of Irrational Numbers, The exactness of Decimal Representation, Rationalize the Denominator, Surds, and Conjugates and Rationalization.
Real Numbers
Any number that can be found in the real world is a real number. Any number that we can think of, except complex numbers, is a real number. The set of real numbers is the union of the set of Rationals (Q) and Irrationals (Q'). It is denoted by R. The set of real numbers, R = Q ∪ Q' .
Example:
Complex Numbers
A complex number is a number that can be expressed in the form (a + bi) where a and b are real numbers, and i is a solution of the equation x^{2} = −1. Because no real number satisfies this equation, i is called an imaginary number. Complex numbers have a real part and an imaginary part. Wait, do you think Complex numbers are really complex? Well, let us study them in detail to find out. In this section, we cover different topics like Complex Numbers – Points in the Plane, A Complex Number as a Point in the Plane, What is i? Magnitude and Argument, Powers of iota, Addition, and Subtraction of Complex Numbers, Multiplication of Complex Numbers, Conjugate of a Complex Number, Division of Complex Numbers, Addition, Subtraction, and Interpretation of z1z2.
Example:
Factors and Multiples
The factors and multiples are the two key concepts studied together. Factors are the numbers that divide the given number completely without leaving any remainder, whereas the multiples are the numbers that are multiplied by the other number to get specific numbers.
Factors of a given number are numbers that can perfectly divide that given number.
Examples:
 Factors of 6: 1, 2, 3, 6
 Factors of 8: 1, 2, 4, 8
 Factors of 14: 1, 2, 7, 14
 Factors of 36: 1, 2, 3, 4, 6, 9, 18, 36
A multiple of a number is a number obtained by multiplying the given number by another whole number.
Examples:
 Multiples of 3: 3, 6, 9, 12, 15, ......
 Multiples of 5: 5, 10, 15, 20, 25, .....
 Multiples of 10: 10, 20, 30, 40, 50,...
 Multiples of 12: 12, 24, 36, 48, 60, ....
Highest Common Factor (HCF)
The highest common factor (HCF) of two numbers is the largest whole number which is a factor of both. It is also called the Greatest Common Factor(GCF).
Example:
Least Common Multiple (LCM)
When we consider two numbers, each will have its own set of multiples. Some multiples will be common to both numbers. The smallest of these common multiples is called the least common multiple (LCM) of the two numbers.
Example:
Prime Factorization
Prime factorization allows us to write any number as a product of prime factors. It is a way of expressing a number as a product of its prime factors. To do prime factorization, you need to break a number down to its prime factors. In this section, we will learn about concepts such as Divisibility, GCD, and LCM. We will also have a look at the various applications of prime factorization.
Example:
Important Notes

The numbers which start from 1 and go up to infinity are called natural numbers.

The numbers which start from zero are called whole numbers.

Integers consist of positive and negative numbers along with zero.

Rational numbers consist of integers, fractions, and ratios.

Irrational numbers are not expressed in the form of fractions or ratios.

Real numbers consist of natural numbers, whole numbers, rational numbers, and irrational numbers.
Solved Examples on Numbers

Example 1: Classify the following numbers as are even numbers and odd numbers: 1, 3, 4, 7, 12, 21, 29, 32
Solution:
Even numbers are the numbers that are exactly divisible by 2. Therefore, 4, 12, and 32 are even numbers, whereas, odd numbers are the numbers that are not divisible by 2. Therefore, 1, 3, 7, 21, and 29 are odd numbers.

Example 2: William has a collection of number cards with the following numbers written on them. −1, √22, 11, 44, and −11. Help William pick out the natural numbers from this!
Solution:
Natural numbers are positive numbers, not fractions, and begin from 1. Therefore, William can choose 11 and 44.
FAQs on Numbers
What is the Smallest Whole Number?
0 is the smallest whole number.
What are the First 10 Numbers?
If someone asks you this question, it means that they want to know the first ten natural numbers. These are 1,2,3,4,5,6,7,8,9 and 10.
What is Not a Real Number?
If you look at the diagram above, all basic numbers, including irrational numbers fall under the subset of real numbers. There is NO number that isn't a real number unless it's an imaginary number.
How Numbers are Formed?
A number is an arranged group of digits. Numbers can be formed with or without the repetition of digits. For example, the largest number which can be formed using 8 and 9 is 98.
Why Numbers are Important in our Life?
Numbers are a part of our everyday life. These are used in an unlimited range of ways, from mathematical calculations, mobile numbers, and phone calls, identification of bank accounts, exchange of money to cooking, etc.