Types of Numbers: Definitions, Properties and Examples
Types of Numbers classify numbers according to their values and properties. The main groups are natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers. Aptitude questions also use prime, composite, even, odd, co-prime and perfect numbers. Their definitions, relationships and divisibility rules help solve number system problems accurately.
What Are the Types of Numbers?
Natural numbers are counting numbers: 1, 2, 3, and so on. Whole numbers include zero along with natural numbers. Integers include negative numbers, zero and positive numbers. Rational numbers can be written as p/q, where p and q are integers and q ≠ 0. Irrational numbers cannot be written in this form. Both rational and irrational numbers are real numbers. The usual aptitude convention is N = {1, 2, 3, ...} and W = {0, 1, 2, 3, ...}.
Types of Numbers Video Lessons
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Types of Numbers Classification
Understand the classification of numbers, including natural, whole, integer, rational, irrational, real and complex numbers, and see how these categories are related in the classification tree.
Types of Numbers Formula & Tricks
Important Formulas
A number is rational if it can be expressed as the quotient of two integers with a non-zero denominator.
Every rational or irrational number belongs to the set of real numbers.
An integer is even when it is exactly divisible by 2.
An integer is odd when division by 2 leaves remainder 1.
Quick Tricks
For positive counting numbers, use N ⊂ W ⊂ Z ⊂ Q ⊂ R. This quickly identifies the smallest set containing a number.
A terminating or recurring decimal is rational. A non-terminating, non-recurring decimal is irrational.
To check whether a number n is prime, test divisibility only by prime numbers less than or equal to √n.
Types of Numbers Concepts
Natural, Whole and Integer Numbers
Natural numbers are N = {1, 2, 3, ...}. Whole numbers are W = {0, 1, 2, 3, ...}. Integers are Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}. Therefore, N ⊂ W ⊂ Z.
Rational and Irrational Numbers
Integers, fractions with non-zero integer denominators, terminating decimals and recurring decimals are rational. Numbers such as √2, √3 and π are irrational. The square root of a positive integer is rational only when that integer is a perfect square.
Real Numbers and Their Subsets
The set of real numbers is R = Q ∪ irrational numbers. Every natural number, whole number, integer, rational number and irrational number is real. Numbers involving √(-1), such as i, are not real numbers.
Prime, Composite and Co-prime Numbers
The number 1 is neither prime nor composite because it has only one positive factor. The number 2 is the only even prime number. Two numbers are co-prime if their greatest common divisor is 1; they need not both be prime.
Even, Odd and Perfect Numbers
Even numbers have the form 2k and odd numbers have the form 2k + 1, where k is an integer. A perfect number equals the sum of its proper positive divisors. For example, the proper divisors of 6 are 1, 2 and 3, and their sum is 6.
Types of Numbers Data Table
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Classification of Common Number Types
Definitions and examples of frequently used number types
| Number type | Definition or property | Example |
|---|---|---|
| Natural | Positive counting numbers | 1, 2, 3 |
| Whole | Natural numbers including zero | 0, 1, 2 |
| Integer | Negative numbers, zero and positive numbers | -3, 0, 5 |
| Rational | Can be written as p/q, q ≠ 0 | 3/4, -2, 0.5 |
| Irrational | Cannot be written as p/q | √2, π |
| Real | All rational and irrational numbers | -7, 2/3, √5 |
Note: Under the usual aptitude convention, 0 is a whole number but not a natural number.
Prime and Composite Classification
Examples based on the number of positive factors
| Number | Positive factors | Classification |
|---|---|---|
| 1 | 1 | Neither prime nor composite |
| 2 | 1, 2 | Prime and even |
| 9 | 1, 3, 9 | Composite and odd |
| 17 | 1, 17 | Prime and odd |
| 20 | 1, 2, 4, 5, 10, 20 | Composite and even |
Note: A prime number has exactly two positive factors; a composite number has more than two.
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Types of Numbers: Quick Revision
Key definitions, relationships and special cases for rapid revision.
- N ⊂ W ⊂ Z ⊂ Q ⊂ R under the convention N = {1, 2, 3, ...}.
- Zero is whole, integer, rational and real, but is neither positive nor negative.
- A rational number has the form p/q, where p and q are integers and q ≠ 0.
- Terminating and recurring decimals are rational; non-terminating, non-recurring decimals are irrational.
- One is neither prime nor composite.
- Two is the only even prime number.
- Co-prime numbers have HCF 1; both numbers do not have to be prime.
- Even integers have the form 2k, and odd integers have the form 2k + 1.
Types of Numbers FAQs
Is 0 a natural number?
In the usual aptitude convention, 0 is not a natural number. It is a whole number, integer, rational number and real number.
Is every integer a rational number?
Yes. Every integer n can be written as n/1, so every integer is rational.
Is 0.272727... rational?
Yes. It is a recurring decimal, so it is rational. In fact, 0.272727... = 3/11.
Is the square root of every number irrational?
No. The square root of a perfect square is rational, such as √64 = 8. The square root of a positive non-perfect square, such as √10, is irrational.
Why is 1 neither prime nor composite?
The number 1 has only one positive factor, itself. A prime number must have exactly two positive factors, and a composite number must have more than two.
Are 14 and 25 co-prime even though both are composite or not prime?
Yes. Their only common factor is 1, so HCF(14, 25) = 1. Co-primality depends on the HCF, not on whether each number is prime.