In mathematics, numbers are categorized into various types based on their properties and uses. From natural numbers to complex numbers, each category serves a unique purpose. This guide covers all the essential types of numbers that you may encounter in aptitude tests, competitive exams, and everyday math.
Natural numbers are the most basic type of numbers that are used for counting and ordering. They are denoted by ‘N’ and consist of positive integers starting from 1. The set of natural numbers is:
N = {1, 2, 3, 4, 5, …}
Whole numbers are similar to natural numbers but include zero. When you include 0 in the set of natural numbers, you get whole numbers, denoted by ‘W’. The set of whole numbers is:
W = {0, 1, 2, 3, 4, 5, …}
Integers consist of both positive and negative whole numbers, as well as zero. They are represented by ‘I’ and include all natural numbers, their negatives, and zero:
I = {…, -3, -2, -1, 0, 1, 2, 3, …}
Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., in the form p/q, where p and q are integers, and q ≠ 0. The set of rational numbers is denoted by ‘Q’.
Example:
Rational numbers can be both terminating (like 0.5) and recurring decimals (like 0.333…).
Irrational numbers cannot be expressed as the ratio of two integers. They are non-terminating and non-recurring decimals. Examples include numbers like √2 and π.
Irrational numbers are denoted by ‘RQ’.
Real numbers include both rational and irrational numbers. They represent any quantity along a continuous number line and can be expressed in infinite decimal form. Real numbers include values like -5, 3/4, √2, and π.
Real numbers are denoted by ‘R’.
Complex numbers are numbers that have both a real and an imaginary part. They are written in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i² = -1.
Example:
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 7 is a prime number.
Note: 1 is not considered a prime number.
To determine if a number is prime:
Example:
To check if 131 is a prime number, find the square root (√131 ≈ 11.45). Check divisibility by prime numbers up to 11 (i.e., 2, 3, 5, 7, 11). Since 131 is not divisible by any of these, it is a prime number.
A perfect number is a positive integer that is equal to the sum of its divisors, excluding the number itself.
Example:
A composite number is a number that has more than two distinct divisors. In other words, it is not a prime number.
Example:
Note: 2 is the only even prime number.
A recurring number is a decimal in which one or more digits repeat indefinitely. For example, 0.3333… or 7.353535….
To convert a recurring decimal to a fraction:
Thus, 0.¯78 = 78/99.
Example 1:
Example 2:
Understanding the different types of numbers is essential for solving various mathematical problems, especially in aptitude tests. By familiarizing yourself with these concepts, you can improve your problem-solving skills and approach questions with greater ease.