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◉ Science·2 min·Updated Mar 12, 2026

What is Complex Number?

Complex Number

Quick Answer

A complex number is a type of number that includes a real part and an imaginary part. It is expressed in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part, with 'i' representing the square root of -1.

Overview

A complex number combines a real number and an imaginary number, making it useful in various fields of mathematics and engineering. The real part represents a value on the horizontal axis of a graph, while the imaginary part represents a value on the vertical axis. This duality allows complex numbers to be represented as points or vectors in a two-dimensional space, known as the complex plane. Complex numbers work by following arithmetic rules similar to real numbers, but with the inclusion of the imaginary unit 'i'. For example, when you add or multiply complex numbers, you treat the real and imaginary parts separately. This approach helps solve equations that have no real solutions, such as the equation x^2 + 1 = 0, which leads to solutions of x = i and x = -i. In real-world applications, complex numbers are essential in fields like electrical engineering and quantum physics. For instance, they are used to analyze alternating current circuits, where voltages and currents can be represented as complex numbers to simplify calculations. This mathematical tool not only enhances our understanding of theoretical concepts but also aids in practical problem-solving.

Frequently Asked Questions

'i' is defined as the square root of -1, which does not exist among real numbers. It allows for the extension of the number system to include solutions to equations that would otherwise be impossible to solve using only real numbers.
To add or subtract complex numbers, you combine their real parts and their imaginary parts separately. For example, (3 + 4i) + (2 + 3i) results in (3 + 2) + (4i + 3i) = 5 + 7i.
Complex numbers are widely used in engineering, particularly in electrical engineering to analyze AC circuits. They also appear in physics, computer science, and applied mathematics, making them a crucial tool in various scientific and technical fields.