In Mathematics, a number is an arithmetic value which is used to represent the quantity of an object. We are using numbers in our day-to-day life, such as counting money, time, things, and so on. We have different types of numbers in the number system. In this article, we are going to discuss the types of numbers in Maths, properties and examples.

Types of Numbers in Maths

According to the properties and how they are represented in the number line, the numbers are classified into different types. Each classification of number is provided herewith description, properties, and examples to understand it in a better way. The different types of numbers are as follows:

Natural Numbers

Natural numbers are also called “counting numbers” which contains the set of positive integers from 1 to infinity. The set of natural numbers is represented by the letter “N”. The natural number set is defined by:

N = {1, 2, 3, 4, 5, ……….}

Examples: 35, 59, 110, etc.

Properties of Natural Numbers:

• Addition of natural numbers is closed, associative, and commutative.
• Natural Number multiplication is closed, associative, and commutative.
• The identity element of a natural number under addition is zero.
• The identity element of a natural number under Multiplication is one.

Whole Numbers

Whole numbers are also known as natural numbers with zero. The set consists of non-negative integers where it does not contain any decimal or fractional part. The whole number set is represented by the letter “W”. The natural number set is defined by:

W = {0,1, 2, 3, 4, 5, ……….}

Examples: 67, 0, 49, 52, etc.

Properties of Whole Numbers:

• Whole numbers are closed under addition and multiplication.
• Zero is the additive identity element of the whole numbers.
• 1 is the multiplicative identity element.
• It obeys the commutative and associative property of addition and multiplication.
• It satisfies the distributive property of multiplication over addition and vice versa.

Learn more about whole numbers here.

Integers

Integers are defined as the set of all whole numbers with a negative set of natural numbers. The integer set is represented by the symbol “Z”. The set of integers is defined as:

Z =  {-3, -2, -1, 0, 1, 2, 3}

Examples: -52, 0, -1, 16, 82, etc.

Properties of Integers:

• Integers are closed under addition, subtraction, and multiplication.
• The commutative property is satisfied for addition and multiplication of integers.
• It obeys the associative property of addition and multiplication.
• It obeys the distributive property for addition and multiplication.
• Additive identity of integers is 0.
• Multiplicative identity of integers is 1.

Real Numbers

Any number such as positive integers, negative integers, fractional numbers or decimal numbers without imaginary numbers are called the real numbers. It is represented by the letter “R”.

Examples: ¾, 0.333, √2, 0, -10, 20, etc.

Properties of Real Numbers:

• Real Numbers are commutative, associate, and distributive under addition and multiplication.
• Real numbers obey the inverse property.
• Additive and multiplicative identity elements of real numbers are 0 and 1, respectively.

Rational Numbers

Any number that can be written in the form of p/q, i.e., a ratio of one number over another number is known as rational numbers. A rational number can be represented by the letter “Q”.

Examples: 7/1, 10/2, 1/1, 0/1, etc.

Properties of Rational Numbers:

• Rational numbers are closed under addition, subtraction, multiplication, and division.
• It satisfies commutative and associative property under addition and multiplication.
• It obeys distributive property for addition and subtraction.

Irrational Numbers

The number that cannot be expressed in the form of p/q. It means a number that cannot be written as the ratio of one over another is known as irrational numbers. It is represented by the letter ”P”.

Examples: √2, π, Euler’s constant, etc

Properties of Irrational Numbers:

• Irrational numbers do not satisfy the closure property.
• It obeys commutative and associative property under addition and multiplication.
• Irrational Numbers are distributive under addition and subtraction.

Complex Numbers

A number that is in the form of a+bi is called complex numbers, where “a and b” should be a real number and “i” is an imaginary number.

Examples: 4 + 4i, -2 + 3i, 1 +√2i, etc

Properties of Complex Numbers:

The following properties hold for the complex numbers:

• Associative property of addition and multiplication.
• Commutative property of addition and multiplication.
• Distributive property of multiplication over addition.

Imaginary Numbers

The imaginary numbers are categorized under complex numbers. It is the product of real numbers with the imaginary unit “i”. The imaginary part of the complex numbers is defined by Im (Z).

Examples: √2, i², 3i, etc.

Properties of Imaginary Numbers:

Imaginary Numbers has an interesting property. It cycles through 4 different values each time when it is under multiplication operation.

• 1 × i = i
• i × i = -1
• -1 × i = -i
• -i × i = 1

So, we can write the imaginary numbers as:

• i = √1
• i² = -1
• i³ = -i
• i⁴ = +1
• i⁴ⁿ = 1
• i^4n-1= -i

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