Following is the list of the most commonly used types of numbers in maths with examples:
- Natural numbers
- Whole numbers
- Integers numbers
- Rational numbers
- Irrational numbers
- Real numbers
- Imaginary numbers
- Complex numbers
1. Natural numbers
Any number that matches to a number from 1, 2, 3, 4, …. and so on is called a natural number or counting number.
4, 9 , 11, 35, 99, 500 or 9876 are natural numbers.
These are commonly used for counting and ordering and due to that, they can be further divided into cardinal and ordinal numbers.
a) Cardinal numbers
Cardinal numbers are used for counting of objects. i.e. how many objects are there 1, 2, 3. etc.
There are 7 apples in a box
So, the count of apples in the box is 7, so 7 is a cardinal number here.
There are 365 days in a year.
So, the count of days in a year is 365, so 365 is a cardinal number here.
b) Ordinal numbers
Ordinal numbers are used to order the positions of objects or in other words, we can tell the position of an object like 1st, 2nd, 3rd, 4th or 5th … etc.
Girls of the class bagged 1st position in the school race.
1st is the ordinal number here.
Canada is the 2nd largest country in the world.
2nd is the ordinal number here.
Set of natural numbers is denoted by N.
2. Whole numbers
Natural numbers including 0 are called as whole numbers.
The only difference between whole numbers and natural numbers is that whole numbers start from 0 unlike natural numbers which start from 1.
0, 2, 14, 23, 48, 172, 623 or 245
Set of whole numbers is denoted by W.
3. Integers
All natural numbers and negative of natural numbers including zero are called Integers.
It includes 0 and all negative numbers 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
-65, -12, -5, 0, 56, 89 or 354
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.
The rational numbers are non terminating and repeating numbers.
25 , 13 , 89 and 310
0.1111………, 0.3333……… and 1.27272727……..
because 1 is non terminating and repeating in 0.1111………
3 is non terminating and repeating in 0.3333………
27 is non terminating and repeating in 1.27272727……..
Set of rational numbers is denoted by Q.
5. Irrational numbers
The numbers which are not rational are called irrational numbers.
Irrational numbers are non terminating and non repeating decimal numbers.
√2 and √5
1.10526315789……………. and 1.21052631579…………….
because 1.10526315789……………. and 1.21052631579……………. have non terminating and non repeating
digits after the decimal
Further, irrational numbers can be classified into transcendental numbers and algebraic numbers.
a) Transcendental number
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.
π (pi) and e (euler’s number) are transcendental numbers.
b) Algebraic number
A number which is root of a non-zero polynomial in one variable with integer as coefficient is called as algebraic number.
√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
A transcendental number is not an algebraic number.
6. Real numbers
Set of rational numbers and irrational numbers are called Real Numbers.
All real numbers can be represented on a number line which is also called a real line.
√7 , 18 and 310
Set of real numbers is denoted by R.
7. Imaginary numbers
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
2i, -5i and √6i
8. Complex numbers
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.
3 + 5i and 6 – 7i
Set of complex numbers is denoted by C.
Other than these types of numbers, there are other more commonly known numbers like even numbers, odd numbers, prime numbers, composite numbers and polygonal numbers, but these are not considered as the types of the numbers. So let’s learn these too one by one.
What are even numbers?
Any integer which is multiple of 2 is called an even number.
2, 4, 6, 8, 20 and 42
What are odd numbers?
Any integer which is not multiple of 2 is called an odd number.
1, 3, 7, 9, 11, 13 and 15
What are prime numbers?
A positive integer which is divisible by itself and 1 only is called a prime number.
2, 3, 5, 7, 11, 13 and 17
What are composite numbers?
Any number which can be expressed as product of smaller positive integers is called a composite number.
4, 28, 102, 112 and 700
What are polygonal numbers?
The numbers which can be expressed as dots and can be arranged in the shape of regular polygon are called
polygon numbers.
Such numbers are classified into the following number types:
-
Triangular numbers
1, 3, 6, 10, 15, 21, 28, 36, 45, 55 and so on -
Square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on -
Pentagonal numbers
1, 5, 12, 22, 35, 51 and so on -
Hexagonal numbers
1, 6, 15, 28, 45, 66, 91, 120, 153, 190 and so on -
Heptagonal numbers
0, 1, 7, 18, 34, 55, 81, 112, 148, 189 and so on -
Octagonal numbers
1, 8, 21, 40, 65 and so on -
Nonagonal numbers
0, 1, 9, 24, 46, 75, 111, 154, 204, 261 and so on -
Decagonal numbers
0, 1, 9, 24, 46, 75 and so on -
Hexadecagonal numbers
1, 7, 19, 37, 61, 91, 127, 169, 217, 271 and so on -
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 positive corresponding numbers.
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.
