Types of numbers

Following is the list of the most commonly used types of numbers in maths with examples:

  1. Natural
  2. Whole
  3. Integers
  4. Rational
  5. Irrational
  6. Real
  7. Imaginary
  8. Complex

1. Natural

Any number that matches any of 1, 2, 3, 4, …. and so on is called a natural or counting number.

Note

Natural numbers start from 1.

Example of natural

4, 9 , 11, 35, 99, 500 or 9876

They are commonly used to count and order a group of numbers which further divides them into the two types cardinal and ordinal numbers.

Note

Set of natural numbers is denoted by N.

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a) Cardinal

These are used for counting of objects. i.e. how many objects are there 1, 2, 3. etc.

Example of cardinal

Example 1: There are 7 apples in a box
So, the count of apples in the box is 7, so 7 is the cardinal number.

Example 2: There are 365 days in a year.
The count of days in a year is 365, so 365 is the cardinal number here.

b) Ordinal

These are used to order the positions of objects or in other words they are used to tell the position of an object like 1st, 2nd, 3rd, 4th or 5th … etc.

Example of ordinal

Example 1: Girls of the class bagged 1st position in the school race.
1st is the ordinal number here.

Example 2: Canada is the 2nd largest country in the world.
2nd is the ordinal number here.

2. Whole

Natural numbers including 0 are called whole numbers.

The only difference between whole and natural numbers is that the whole start from 0 unlike natural which start from 1.

Example of whole

0, 2, 14, 23, 48, 172, 623 or 245

Note

Set of whole numbers is denoted by W.

3. Integers

All natural numbers including their negatives and zero are called Integers.
It includes 0, all negative and positive natural numbers.
That means 1, 2, 3, 4, 5, 6, … so on and -1, -2, -3, -4, -5, -6, … so on and 0 are integers

Example of integers

-65, -12, -5, 0, 56, 89 or 354

Note

Set of integers is denoted by Z.

4. Rational numbers

The numbers which can be expressed in the form of pq , where p and q are integers and q is not equal to zero, are called Rational Numbers.
These are non terminating and repeating.

Non terminating numbers never end and are represented with the repeating dots “…………….” at the end.

Repeating numbers repeats the same digits again and again.

Example of rational

Example 1: 25 , 13 , 89 and 310

Example 2: 0.1111………, 0.3333……… and 1.27272727……..
In 0.1111………, 1 is non terminating and repeating
In 0.3333………, 3 is non terminating and repeating
In 1.27272727………, 27 is non terminating and repeating

Note

Set of rational numbers is denoted by Q.

5. Irrational numbers

The numbers which are not rational are called irrational numbers.

Irrational are non terminating and non repeating decimal numbers.

Example of irrational

Example 1: 1.10526315789……………. and 1.21052631579…………….
because 1.10526315789……………. and 1.21052631579……………. have non terminating and non repeating digits after the decimal

Example 2: √2 and √5

Further, irrationals can be classified into transcendental and algebraic numbers.

a) Transcendental

A number that is not the root of a non-zero polynomial with integer as coefficient is called as transcendental number. It is a real or complex number.

Example of transcendental

π (pi) and e (euler’s number).

b) Algebraic

A number which is root of a non-zero polynomial in one variable with integer as coefficient is called as algebraic number.

Example of algebraic

√2 is the root of x² – 2.
So, √2 is an algebraic number.

The golden ratio 1 + √5 2 is an algebraic number as it is a root of x² – x – 1

Note

A transcendental is not an algebraic number.

6. Real

Set of rational and irrational are called Real Numbers.

All real numbers can be represented on a number line which is also called a real line.

Example of real

√7 , 18 and 310

Note

Set of real numbers is denoted by R.

7. Imaginary

Square root of a negative number is called as imaginary number.

Square root of negative 1 is called the imaginary unit. It is denoted by symbol i. It is pronounced as iota.
The value of i, i², i³ and i4 can be calculated as following:

Value of i
√-1

Value of i²
i² = i . i
√-1 . √-1 = -1

Value of i³
i³ = i² . i
= (-1) . i = -i

Value of i4
i4 = i² . i²
=(-1) . (-1) = 1

The value of i-1 and i-2 are calculated as following:
Value of i-1
i-1 = 1i
= 1i × ii
=ii²
i-1 = -i

Value of i-2
i-2 = 1i²
= 1-1 = -1

Example of imaginary

2i, -5i and √6i

8. Complex

A number which can be expressed in the form of a + ib is called a complex number. In the complex number a + ib, a and b are real numbers and i is an imaginary unit. It has two parts real part and imaginary part. The real part is a and the imaginary part is ib.

Example of complex

3 + 5i and 6 – 7i

Note

Set of complex numbers is denoted by C.

Other than the above types, other more commonly known numbers are even, odd, prime, composite and polygonal. But these are not considered as the types of the numbers.

More types of numbers

Other than the basic types listed above, there are some more different kind of numbers used in maths.

1. Even

Any integer which is multiple of 2 is called an even number.

Example of even

2, 4, 6, 8, 20 and 42

2. Odd

Any integer which is not multiple of 2 is called an odd number.

Example of odd

1, 3, 7, 9, 11, 13 and 15

3. Prime

A positive integer which is divisible by itself and 1 only is called a prime number.

Example of prime

2, 3, 5, 7, 11, 13 and 17

4. Composite

Any number which can be expressed as product of smaller positive integers is called a composite number.

Example of composite

4, 28, 102, 112 and 700

5. Polygonal

The numbers which can be expressed as dots and can be arranged in the shape of regular polygon are called polygonal numbers.
These are further classified into the following types:

  1. Triangular numbers
    1, 3, 6, 10, 15, 21, 28, 36, 45, 55 and so on
  2. Square numbers
    1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on
  3. Pentagonal numbers
    1, 5, 12, 22, 35, 51 and so on
  4. Hexagonal numbers
    1, 6, 15, 28, 45, 66, 91, 120, 153, 190 and so on
  5. Heptagonal numbers
    0, 1, 7, 18, 34, 55, 81, 112, 148, 189 and so on
  6. Octagonal numbers
    1, 8, 21, 40, 65 and so on
  7. Nonagonal numbers
    0, 1, 9, 24, 46, 75, 111, 154, 204, 261 and so on
  8. Decagonal numbers
    0, 1, 9, 24, 46, 75 and so on
  9. Hexadecagonal numbers
    1, 7, 19, 37, 61, 91, 127, 169, 217, 271 and so on
  10. Dodecagonal numbers
    0, 1, 12, 33, 64, 105 and so on

What is additive inverse of a number?

If a positive number and its corresponding negative number are added up, their sum results to 0.
The negative numbers are the additive inverse of the positive corresponding numbers.

Example of additive inverse

2 is the additive inverse of -2.
Because -2 + 2 = 0.

Zero is neither a negative number nor positive number. It is a neutral number.