The term for 复数 in English is complex number. This might seem straightforward, but understanding complex numbers requires delving into their structure, properties, and applications. This article aims to provide a comprehensive overview of complex numbers, covering everything from their basic definition to their various representations and uses in different fields.

The foundation of complex numbers lies in the concept of the imaginary unit, denoted by i. This i is defined as the square root of -1 (i.e., i = √-1). This allows us to handle the square roots of negative numbers, which are not possible within the realm of real numbers.

A complex number is formally defined as a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit. In this expression, a is referred to as the real part of the complex number, and b is called the imaginary part. For instance, in the complex number 3 + 4i, the real part is 3 and the imaginary part is 4. It's important to note that both the real and imaginary parts are real numbers themselves.

When b is equal to 0, the complex number a + bi simply reduces to the real number a. This demonstrates that the set of real numbers is a subset of the set of complex numbers. In other words, every real number can be considered a complex number with an imaginary part of 0. Conversely, when a is equal to 0, the complex number a + bi becomes bi, which is called a purely imaginary number. Examples of purely imaginary numbers include 5i, -2i, and i√2.

Complex numbers can be manipulated using basic arithmetic operations such as addition, subtraction, multiplication, and division. Addition and subtraction are performed component-wise. For example, to add two complex numbers (a + bi) and (c + di), you simply add the real parts and the imaginary parts separately: (a + c) + (b + d)i. Subtraction follows the same principle: (a + bi) - (c + di) = (a - c) + (b - d)i.

Multiplication of complex numbers involves using the distributive property and the fact that i² = -1. So, (a + bi) (c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i. The result is another complex number.

Division of complex numbers is a bit more involved. It requires multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its complex conjugate results in a real number: (a + bi)(a - bi) = a² + b². This allows us to eliminate the imaginary part from the denominator and express the quotient as a complex number in the standard form a + bi.

Besides the algebraic form (a + bi), complex numbers can also be represented geometrically in the complex plane (also known as the Argand diagram). The complex plane is a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. A complex number a + bi is plotted as a point with coordinates (a, b) in the complex plane.

Another way to represent complex numbers is in polar form. In polar form, a complex number is described by its magnitude (or modulus) r and its argument (or angle) θ. The magnitude r is the distance from the origin to the point representing the complex number in the complex plane, and it can be calculated as r = √(a² + b²). The argument θ is the angle between the positive real axis and the line connecting the origin to the point representing the complex number in the complex plane. It can be calculated using trigonometric functions (θ = arctan(b/a)), taking into account the quadrant in which the complex number lies. The polar form of a complex number is then expressed as r(cos θ + i sin θ), which can be further simplified using Euler's formula as re^(iθ).

The polar form is particularly useful for performing multiplication and division of complex numbers. When multiplying two complex numbers in polar form, you simply multiply their magnitudes and add their arguments. When dividing two complex numbers in polar form, you divide their magnitudes and subtract their arguments. This simplifies the calculations significantly compared to using the algebraic form.

Complex numbers have numerous applications in various fields of science and engineering. In electrical engineering, they are used to represent alternating currents and voltages, as well as impedances in electrical circuits. In physics, they are used in quantum mechanics to describe wave functions and in signal processing to analyze and manipulate signals. In mathematics, they are fundamental in complex analysis, a branch of mathematics dealing with functions of complex variables. They also play a crucial role in fractals, such as the Mandelbrot set, which are generated using iterative calculations involving complex numbers.

Furthermore, complex numbers find applications in control systems, fluid dynamics, and computer graphics. Their ability to represent two-dimensional quantities elegantly and their powerful mathematical properties make them an indispensable tool in these fields. The study of complex numbers and their applications continues to be an active area of research, with new discoveries and applications constantly emerging. Understanding complex numbers is therefore essential for anyone pursuing advanced studies in science, engineering, or mathematics. Their seemingly abstract nature belies their practical significance and the power they bring to problem-solving in diverse domains.