Numbers: Definition, Types, and Properties

Numbers are part and parcel of our daily lives. Whenever you count how many chocolates you have, mention the time, or write the score of a cricket match, you use numbers.

Can you think of a day without numbers? When you are calculating the grocery bills, scientists somewhere are busy with complex scientific measurements, all involving numbers.

They are important building blocks in math, and they help us understand the world around us. In this article, we will explore different types of numbers in maths, learn how to define numbers, and understand the different categories under types of numbers.

What Are Numbers?

Let us start by learning how to define numbers. 

Numbers have evolved from the earlier concept of tally marks or counting with fingers to form a fundamental element in mathematics. We can’t think of mathematics without numbers, which are frequently used to represent quantity, time, weight, age, etc.

Numbers form the basis for counting, measurement, ordering, indexing, and many other activities and are used for basic arithmetic operations to find new numbers. Numbers are given names, which are expressed as numerals as well as words.

For example, "5" shows you have five apples. Numbers are used to count, measure, and compare things. This is why mathematics numbers are very helpful in school and in life.

Definition of Numbers

The number is an arithmetic value that quantifies entities and is used for counting and calculations. It is typically denoted by a symbol or digits known as numerals. For example, the symbols “5” and ‘’ 24’’ represent two different numbers.

Number Names

Number names are the representation of numerals in words. A specific word is used to express a particular number. The first twenty numbers have unique names, and to write a number after that, you should know the place value of each digit in the number.

For example, the number 12 is written as 'twelve’ and 17 as 'seventeen’ whereas the number 45 is written as ‘forty-five’ in the number name form. 

Numbers are used to count, measure, and compare things. This is why mathematics numbers are very helpful in school and in life.

Basic Numbers

We use the digits from 0 to 9 to represent the first 10 numbers. It is interesting to note that we can create infinite numbers using these basic numbers. For example: 21, 748, 5036, etc.

The Number Zero

In mathematics numbers, zero is very useful in arithmetic and place value. The concept of the number zero (0) plays an important role in mathematics, although it doesn’t quantify anything and denotes the absence of anything. The number 0 becomes relevant when expressing numbers as multiples of ten, such as 50, 920, 4400, etc.

It also acts as an additive identity for counting numbers, which means by adding zero to some quantity, the quantity remains the same.

On the other hand, if we remove all quantities from a given quantity, the result becomes zero. For example, if you have 6 chocolates and have eaten all of them, you have zero chocolates with you. This concept is frequently used in arithmetic calculations.

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Types of Numbers

In real life, we use different numbers, but for mathematical calculation purposes, they are classified into different categories, as explained below.

Let us now look at the different types of numbers. These help us solve many problems in math and real life. We will now explore the most common types of numbers in maths.

Natural Numbers

Natural numbers are used for counting. It contains all the positive numbers from 1 to infinity. The set of natural numbers is denoted as N = {1, 2, 3, 4, 5,...}.}

Whole Numbers

Whole numbers refer to all natural numbers and also the number zero. The set of whole numbers is denoted as W = {0, 1, 2, 3, 4, 5, ……….}

Integers

All whole numbers and their negative numbers form the set of integers. Therefore, the set of integers is represented by Z = { -3, -2, -1, 0, 1, 2, 3}

Real Numbers

All the positive and negative integers, along with fractional and decimal numbers, are called real numbers. The set of real numbers is represented by "R.”.

Rational Numbers

A rational number is a number that can be expressed as a ratio of two numbers. This means that any number that can be written in the form of a/b represents a rational number. It is typically denoted by Q. For example, 3/4, -12/25, etc. are rational numbers.

Irrational Numbers

The number that cannot be expressed as the ratio of two numbers is known as an irrational number. The symbol ‘P’ typically denotes irrational numbers. For example, √2, √3, etc. are irrational numbers.

Complex Numbers

The numbers that can be written in the form of a+bi, where “a and b” are the real numbers and “i” is an imaginary number, are known as complex numbers.

Even Numbers

The numbers that are divisible by 2 without leaving any remainder are called even numbers . Even numbers can be positive or negative, such as 4, 10, -22, 38, -542, 766, and so on.

Odd Numbers

These numbers are not exactly divisible by 2 and leave a remainder of 1. There can be positive and negative odd numbers such as 3, -9, 35, 17, -43, 251, and so on.

Prime Numbers

Prime numbers are the numbers that are divisible by only two numbers: 1 and the number itself. For example, 2, 3, 11, 29, 47, etc. are prime numbers.

Composite Numbers

A composite number is a number that can be written as the product of two or more prime numbers. For example, 14 is a composite number, which is a product of 2 and 7. Other examples of composite numbers are 8, 9, 15, 28, and so on.

Why Are There So Many Types of Numbers?

You might wonder why we have so many types of numbers. That’s because each type helps us with different kinds of problems. Some help with measuring, others with sharing equally. Knowing the types of numbers in maths makes solving math questions easier.

Fractions and Decimals

Fractions are numbers that represent the part of a whole number. It is denoted by the numeral as a ratio of two numbers. It has two parts: the top number is called the numerator, and the bottom number is called the denominator.

For example , the fraction 5/8 represents 5 parts out of the whole 8 parts, where 5 is the numerator and 8 is the denominator.

Decimal numbers contain a whole part followed by a fractional part. These two parts are separated by a decimal point. A decimal number can be represented as a fraction and vice versa.

For example, the decimal number for the fraction 7/2 is written as 3.5.

Factors and Multiples

Factors and multiples are important concepts related to numbers. Factors of a number are a set of numbers each of which divides the given number without leaving a remainder.

For example , factors of 24 are 2, 3, 4, 6, 8, 12, and 24.

Multiples of a number are the numbers obtained by multiplying the given number by another whole number.

For example, multiples of 5 are 5, 10, 15, 20, 25...

Some Special Numbers

• Cardinal Numbers : A Cardinal number defines the quantity of, such as one, six, fifteen, etc.
• Ordinal Numbers : Ordinal numbers indicate the position of an entity when placed in an order such as first, third, tenth, etc.
• Pi (π): Pi is a constant used in mathematical calculations, and its approximate value is 3.14159.
• Euler’s Number (e): Euler’s number is another important number used in many mathematical calculations. Its value is approximately equal to 2.7182818.
• Golden Ratio (φ): It is a special number with an approximate value of 1.618.

Properties of Numbers

The common properties related to real numbers are as follows:

Commutative Property

The commutative property states that if p and q are two real numbers, then

p + q = q + p and p x q = q x p

For example:

5 + 6 = 6 + 5

and 5 × 6 = 6 × 5

Associative Property

According to this property, for three real numbers a, b, and c,

(a + b) + c = a + (b + c)

(a x b) x c = a x (b x c)

For example:

(3 + 8) + 4 = 3 + (8 + 4)

(3 x 8) x 4 = 3 x (8 x 4)

Distributive Property

If x, y, and z are three real numbers, then according to distributive property,

x × (y + z) = x × y + x × z

For example:

4 × (7 + 8) = 4 ×7 + 4 × 8

Or, 4 × 15 = 28 + 32

Or, 60 = 60

Identity Property

If we add zero to a number or multiply by 1, the number will remain unchanged.

For any number m and n, m + 0 = m and n x 1 = n

Example: 16 + 0 = 16 and 36 x 1 = 36

Additive Inverse

If a number is added to its negative number, then the sum becomes zero.

For example , 45+(-45) = 45-45 = 0

Multiplicative Inverse

If a non-zero number is multiplied by its reciprocal, then the result is 1.

For example , 34 x (1/34) = 1

Zero Product Property

If the product of two real numbers is zero, then either of these numbers is zero.

It means for two numbers a and b, if a x b = 0, then either a = 0 or b = 0.

For example , 9 x 0 = 0 or 0 x 8 = 0

This article provides useful insights about numbers that form the basis of mathematics. Numbers also play a significant role in everyday life, allowing us to quantify, measure, compare, and analyze the entities around us.

Cardinal and Ordinal Numbers

Let’s learn about two more types of numbers used daily:

• Cardinal numbers tell us how many: one, two, five.

• Ordinal numbers tell position: first, second, third.

These are simple ways to define numbers used in real life.

Why Learning About Numbers Matters

Understanding what is number in maths helps you solve real problems. Whether you are sharing cookies, calculating scores, or playing games, maths numbers are useful.

If you define numbers and learn their types, math becomes super easy and fun.

Let’s Be Number Ninjas

You now know what numbers are, how to define numbers, and many types of numbers in maths! From counting candies to solving puzzles, mathematics numbers make everything more interesting. So keep practicing and exploring.