Types of Numbers: Definition, Properties, Examples

Numbers are everywhere. From the date on which a person was born to the time that the clock needles are ticking and much more, we are surrounded by the numbers. Moreover, numbers are the basic and integral aspect of mathematical concepts. For mathematical operations, there are types of numbers such as odd and even numbers, natural and whole numbers, prime and composite numbers, fractions, decimals, real numbers, rational and irrational numbers, integers, and so on. Let's have a look at all the types of numbers, their properties, and examples of their usage in mathematics. 

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Types of numbers in Mathematics

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One can classify the numbers in various categories or types based on their properties along with the representation of these numbers on the number line. There is a logic behind every type of number which makes it different from other types of numbers. A list of all the types of numbers along with their properties and examples is provided below.

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Natural Numbers

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Natural numbers which are also known as “counting numbers” are the set of positive numbers beginning from 1 till infinity. The letter "N" is used to represent the set of natural numbers.   The natural number set is as follows:

N = {1, 2, 3, 4, 5, ……….till infinity}

Examples: 53,95,210 and any positive number (except zero).

What are the Properties of Natural Numbers?

1. Closed, associative, and commutative 

Closed, associative, and commutative are the properties of natural numbers during addition and multiplication.

2. Identify element is equal to zero under addition

Natural numbers' identity element under addition is zero. That means n+0 = n.

3. Identify element is equal to one under multiplication

Natural numbers' identity element under multiplication is one. That means n x 1 = n.

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Whole Numbers

The group of Whole numbers as types of numbers is very similar to natural numbers. The only difference there is that zero is also included in the group this time. The set has all the positive numbers along with zero. But, decimals and fractions are not a part of this set. Letter "W" is used for the representation of the set of whole numbers. The whole number set is provided below:

W = {0,1, 2, 3, 4, 5, ……….till infinity}

Examples: 39, 0, 58,304.

What are the Properties of Whole Numbers?

Whole numbers are closed under addition and multiplication.

1. Closed under addition and multiplication

Whole numbers are found as closed under addition and repetitive addition (multiplication). In other words, whether you add or you multiply two whole numbers, the result would always be a whole number only.

2. Identify element is equal to zero under addition

The whole numbers' identity element under addition is zero. That means W+0 = W.

3. Identify element is equal to one under multiplication

The whole numbers' identity element under multiplication is one. That means W x 1 = W.

4. Commutative and associative property

Associative and commutative are the properties of whole numbers during addition and multiplication.

5. Distributive property of multiplication

Whole Numbers work in line with the distributive property of multiplication over addition. Moreover, vice versa is also the case here.

Read more:

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Integers

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The set of all whole numbers with all the natural numbers in negative mode, i.e., -1,-2,-3, and so on in the number line is known as Integers. This set is represented with the help of the letter "Z". Here is the set of Integers:

Z =  {-3, -2, -1, 0, 1, 2, 3...till infinity}

Examples: -25, 0, -178, 61, 96, etc.

What are the Properties of Integers?

1. Closed under addition, multiplication, and subtraction

Integers are observed as closed under addition, subtraction, and multiplication. It indicates that when two or more integers are added, subtracted, or multiplied; the result would always be an integer only.

2. Distributive property of multiplication

Integers satisfy the commutative, associative, and distributive property of integers. 

3. Identify element is equal to zero under addition

Since Integers include whole and natural numbers, therefore, Integers' identity element under addition is also zero. That means Z+0 = Z.

4. Identify element is equal to one under multiplication

Integers' identity element under multiplication is also one. That means Z x 1 = Z.

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Real Numbers

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Positive integers, negative integers as well as zero all were there. But, fractional numbers and decimals are not included yet. Real numbers are all the numbers that are found in the number line. From 1 to ½ to 1.111111 and so on. Real numbers are represented by the letter “R”. Anything that is not in the number line is not a part of real numbers. 

Examples: ¾, 0.989, √5, 0, -100, 29, etc.

What are the Properties of Real Numbers?

1. Distributive, commutative and associative properties of multiplication

Real Numbers are distributive, commutative, and, associative under addition and multiplication.

2. Multiplication Identity Element equal to One

Real numbers' identity element under multiplication is also one. That means R x 1 = R.

3. Additive Identity Element equal to zero

Real numbers' identity element under multiplication is also one. That means Z x 0 = Z.

4. Inverse property

Inverse is the property followed by Real numbers.

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Rational Numbers

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Rational numbers are the numbers that one can write in the form of p/q. In other words, a ratio of one number over the other number. Rational numbers as types of numbers are represented by the symbol "Q". 

Examples: 7/9, 2/5, 1/1, 0/1, etc.

What are the Properties of Rational Numbers?

1.  Closed under addition, subtraction, multiplication, and division

Rational numbers are closed under arithmetic operations of subtraction, addition, multiplication as well as division.

2. Commutative and associative property under addition and multiplication

Rational Numbers are commutative and associative under addition and multiplication.

3. Distributive property under addition and subtraction

Rational numbers follow the distributive property.

Also read:

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Irrational Numbers

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Those types of numbers that one cannot express in the form of ratio or p/q are expressed as Irrational numbers. Irrational numbers are represented by the symbol ”P”.

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

What are the Properties of Irrational Numbers?

1. Not Closed
2. Commutative and associative properties under addition and multiplication
3. Distributive property under addition and subtraction

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Complex Numbers

Complex numbers are the numbers that are written in the form of a+b. "a and b” are the real numbers there. Whereas, “i” is an imaginary number.

Examples: 8 + 4i, -2 + 6i, 3+√2i, etc

What are the Properties of Complex Numbers?

The following properties hold for the complex numbers:

1. Associative and Commutative property under addition and multiplication.
2. Distributive property of multiplication over addition.

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Complex Numbers

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Imaginary numbers, as types of numbers, are a category under complex numbers only. The product of real numbers added with the imaginary unit “i” can be known as imaginary numbers. The imaginary aspect of the complex numbers is denoted by the symbol Im (Z).

Examples: √2, i5, 6i, etc.

What are the Properties of Imaginary Numbers?

Imaginary Numbers have an interesting property. 

1. A cycle of four different values under multiplication

This is an interesting property of imaginary numbers. Imaginary number, when performed in multiplication operation, goes through four different values just by changing the place of value.

For example:

1 × i = i

i × i = -1

-1 × i = -i

-i × i = 1

Therefore, imaginary numbers can be written as...

i = √1

i2 = -1

i3 = -i

i4 = +1

i4n = 1

i4n-1= -i

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Things to Remember 

• Natural numbers don't include zero with them. The symbol of natural numbers is "N".
• Zero is a whole number and not a natural number. The symbol of the Whole number is "W".
• Numbers that can be written in a ratio form are rational numbers. "Q" is the symbol for rational numbers.
• Irrational Numbers are numbers that cannot be written in ratio form. "P" is for irrational numbers.
• Imaginary Numbers are a part of Complex numbers.
• Imaginary Numbers are represented as "i" in the formula.

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