Imagine a standard number line. All the numbers we know are located on it: natural numbers (1, 2, 3…), integers (…, −2, −1, 0, 1, 2…), fractions, and irrational numbers. They all lie on a single line.
Now, let's try to solve a simple equation:
$x^2 = -1$
To find $x$, we need to take the square root of $-1$. However, there is no number on the number line that produces a negative result when multiplied by itself. It is well known that the product of two positive numbers is positive, and the product of two negative numbers is also positive. Therefore, this equation has no solution on the familiar number line. This means we need to expand our understanding of numbers.
To solve this problem, mathematicians introduced a new concept — the imaginary unit. It is denoted by the letter $i$. By definition:
$i^2 = -1$
This number allows us to take square roots of negative numbers and build a new number system. Now the equation $x^2 = -1$ has a solution: it is the number $i$ or $-i$.
To visualize these new numbers, a single line is not enough. Mathematicians added a second number axis that passes vertically through zero.
The plane on which complex numbers are represented is called the complex plane. Now, a number is represented not as a point on a line, but as a point on a plane.
A complex number is written in general form as:
$z = x + iy$
Where:
It is important to note that any ordinary (real) number is a special case of a complex number where the imaginary part is zero ($y = 0$).
Every complex number is represented by a point on the plane. Each such point defines a vector directed from the origin $(0,0)$ to that point.
A complex number can be defined not only by the Cartesian coordinates $x$ and $y$, but also by using polar coordinates:
Modulus ($r$):
The distance from the origin to the point representing the number. The modulus is always a positive number or zero. To calculate it, we use the Pythagorean theorem:
$r = |z| = \sqrt{x^2 + y^2}$
Argument ($\varphi$):
The angle between the positive direction of the real axis and the vector of the number. The angle is measured counter-clockwise. We use trigonometric functions to find it:
$\cos(\varphi) = \frac{x}{r}, \quad \sin(\varphi) = \frac{y}{r}$
Interactive Figure 1: A draggable complex number on the complex plane
This graph demonstrates how a complex number is defined by its Cartesian coordinates ($x$ and $y$) as well as its polar coordinates ($r$ and $\varphi$). You can drag the blue control point to see how these values relate to each other in real time.
In this chapter, we will explore the rules for interacting with complex numbers. It is important to remember that for complex numbers, the concepts of "greater than" or "less than" are meaningless, as points on a plane cannot be arranged in a single chain like numbers on a line. However, the moduli (magnitudes) of complex numbers are standard real numbers, and therefore, they can be compared with one another.
Two complex numbers are equal if and only if their real and imaginary parts match:
$z_1 = z_2 \iff x_1 = x_2$ and $y_1 = y_2$
The algebraic form of a complex number ($z = x + iy$) is convenient for addition and subtraction operations. These operations are performed component-wise:
Addition: To add two complex numbers, their real and imaginary parts are added separately:
$z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2)$
Subtraction: Similarly, for subtraction, the corresponding parts are subtracted:
$z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2)$
Geometric Interpretation: The addition and subtraction of complex numbers are analogous to vectors. Furthermore, the modulus of the difference $|z_1 - z_2|$ is equal to the distance between points $z_1$ and $z_2$ on the complex plane.
When multiplying complex numbers, we use the FOIL method (distributive property) as in standard algebra, keeping in mind that $i^2 = -1$.
For the general case:
$z_1 \cdot z_2 = (x_1 + iy_1)(x_2 + iy_2) = x_1x_2 + ix_1y_2 + iy_1x_2 + i^2y_1y_2$
Replacing $i^2$ with $-1$, we get:
$z_1 \cdot z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + y_1x_2)$
Special Case: Multiplying the purely imaginary number $i$ by itself confirms the fundamental definition:
$i^2 = (0 + i)(0 + i) = 0 \cdot 0 + 0 \cdot i + i \cdot 0 + i^2 = -1$
The Visual Meaning of Multiplying by $i$:
Imagine the number 1 on the real axis. If we multiply it by $i$, we get $i$ — a point on the imaginary axis. Geometrically, this is a 90° counter-clockwise rotation. If we multiply $i$ by $i$ again, we rotate the vector by another 90° and land on $-1$. Thus, two 90° rotations result in a 180° turn, which explains why $i^2 = -1$.
For a complex number $z = x + iy$, we introduce the concept of the complex conjugate $\bar{z}$, which differs only by the sign of the imaginary part:
$\bar{z} = x - iy$
The product of a complex number and its conjugate always results in a real number. This property is critically important for the division operation:
$(x + iy)(x - iy) = x^2 - (iy)^2 = x^2 - i^2y^2 = x^2 + y^2$
Note that $x^2 + y^2 = |z|^2 = r^2$.
The division operation $z_1/z_2$ is performed by multiplying both the numerator and the denominator by the conjugate of the denominator. This allows us to eliminate the imaginary unit from the denominator of the fraction:
$\frac{z_1}{z_2} = \frac{x_1 + iy_1}{x_2 + iy_2} \cdot \frac{x_2 - iy_2}{x_2 - iy_2} = \frac{(x_1x_2 + y_1y_2) + i(y_1x_2 - x_1y_2)}{x_2^2 + y_2^2}$
The algebraic form $z = x + iy$ is very convenient for performing addition and subtraction or for checking the equality of numbers. However, for operations such as multiplication and division, as well as for raising numbers to a power or extracting roots, other forms of notation — trigonometric and exponential — are much more practical.
From the geometric representation of a complex number, we know the relationships between Cartesian and polar coordinates:
$x = r\cos\varphi, \quad y = r\sin\varphi$
Let us substitute these expressions into the algebraic form $z = x + iy$:
$z = r\cos\varphi + ir\sin\varphi = r(\cos\varphi + i\sin\varphi)$
This is known as the trigonometric form of a complex number. It demonstrates that any complex number is uniquely defined by its modulus $r$ and its argument $\varphi$.
For real numbers, the function $e^x$ (the exponential function) is well known: it is always positive and possesses a remarkable property: $e^{a+b} = e^a \cdot e^b$. Mathematicians discovered an amazing way to extend this concept to complex numbers. The key result was Euler's formula, discovered by Leonhard Euler in the 18th century:
$e^{i\varphi} = \cos\varphi + i\sin\varphi$
This formula establishes a deep connection between the exponential function, trigonometric functions, and the imaginary unit. Using this formula, the trigonometric form can be easily converted into the exponential form:
$z = r(\cos\varphi + i\sin\varphi) = re^{i\varphi}$
The expression $e^{i\varphi}$ represents a complex number with a modulus of 1 and an argument of $\varphi$. Let us consider some important special cases:
A more general relationship is $e^{i \cdot 2\pi k} = 1$ for any integer $k$. This reflects the periodicity of trigonometric functions: after making a full rotation of $2\pi$, we return to the starting point on the complex plane.
The symbol $e^{i\varphi}$ possesses all the familiar properties of the exponential function. This makes it an incredibly convenient tool for calculations.
Using these properties, we can easily perform operations with complex numbers in exponential form. Let two numbers be $z_1 = r_1e^{i\varphi_1}$ and $z_2 = r_2e^{i\varphi_2}$. Then:
Multiplication: When multiplying complex numbers, their moduli are multiplied, and their arguments are added:
$z_1 \cdot z_2 = (r_1e^{i\varphi_1}) \cdot (r_2e^{i\varphi_2}) = (r_1r_2)e^{i(\varphi_1 + \varphi_2)}$
Division: When dividing complex numbers, their moduli are divided, and their arguments are subtracted:
$\frac{z_1}{z_2} = \frac{r_1e^{i\varphi_1}}{r_2e^{i\varphi_2}} = \left(\frac{r_1}{r_2}\right)e^{i(\varphi_1 - \varphi_2)}$
These rules are much simpler than the multiplication and division of polynomials in the algebraic form.
The property of exponentiation in the exponential form directly leads to an important relationship known as De Moivre's formula. If a complex number is represented in trigonometric form $z = r(\cos\varphi + i\sin\varphi)$, then raising it to a natural power $n$ results in:
$z^n = [r(\cos\varphi + i\sin\varphi)]^n = r^n(\cos(n\varphi) + i\sin(n\varphi))$
In the exponential form, this looks even more concise:
$(re^{i\varphi})^n = r^ne^{in\varphi}$
De Moivre's formula is a powerful tool for calculating the powers of complex numbers and for deriving trigonometric identities.
Euler's formula allows us to express trigonometric functions in terms of the exponential function. This is extremely useful in various mathematical transformations.
First, let us observe what happens if we replace $\varphi$ with $-\varphi$ in Euler's formula. Considering that cosine is an even function ($\cos(-\varphi) = \cos\varphi$) and sine is an odd function ($\sin(-\varphi) = -\sin\varphi$), we get:
$e^{-i\varphi} = \cos(-\varphi) + i\sin(-\varphi) = \cos\varphi - i\sin\varphi$
Notice that $e^{-i\varphi}$ is the complex conjugate of $e^{i\varphi}$. If we denote $e^{i\varphi} = \cos\varphi + i\sin\varphi$, then its conjugate is $\overline{e^{i\varphi}} = \cos\varphi - i\sin\varphi = e^{-i\varphi}$. Geometrically, this corresponds to a reflection across the real axis: the argument changes sign ($\varphi \to -\varphi$), while the modulus remains equal to 1.
Now we have a system of two equations:
$\begin{cases} e^{i\varphi} = \cos\varphi + i\sin\varphi \\ e^{-i\varphi} = \cos\varphi - i\sin\varphi \end{cases}$
By adding these two equations, we can express the cosine function:
$e^{i\varphi} + e^{-i\varphi} = 2\cos\varphi \Longrightarrow \cos\varphi = \frac{e^{i\varphi} + e^{-i\varphi}}{2}$
By subtracting the second equation from the first, we obtain the expression for the sine function:
$e^{i\varphi} - e^{-i\varphi} = 2i\sin\varphi \Longrightarrow \sin\varphi = \frac{e^{i\varphi} - e^{-i\varphi}}{2i}$
These relationships are called Euler's formulas for sine and cosine. They demonstrate a deep internal unity between these seemingly different functions.
The $n$-th root of a complex number $w$ is a complex number $z$ such that, when raised to the power of $n$, it equals $w$:
$z^n = w$
One of the most important features of complex numbers is that the operation of extracting a root of degree $n$ always results in exactly $n$ distinct solutions.
This occurs due to the periodicity of trigonometric functions. Since adding a full circle ($2\pi$, $4\pi$, etc.) to the argument does not change the number itself, dividing these different representations by $n$ leads to different results on the complex plane. To perform this operation, it is most convenient to use the exponential form of a complex number, which we introduced in the previous chapter.
Suppose we are given a number $w$ from which we want to extract the $n$-th root. Let us write it in exponential form:
$w = re^{i\varphi}$
We will also represent the sought-after number $z$ (or rather, each of its values) in exponential form:
$z = \rho e^{i\theta}$
where $\rho$ is its modulus and $\theta$ is its argument (both currently unknown).
Then the equation $z^n = w$ becomes:
$(\rho e^{i\theta})^n = \rho^n e^{in\theta} = re^{i\varphi}$
Two complex numbers in exponential form are equal if their moduli are equal and their arguments differ by a multiple of $2\pi$ (a full rotation). This gives us two conditions:
For the moduli: $\rho^n = r$. Since moduli are non-negative real numbers, we get:
$\rho = \sqrt[n]{r}$
where $\sqrt[n]{r}$ is the standard arithmetic $n$-th root of the positive number $r$.
For the arguments: $n\theta = \varphi + 2\pi k$, where $k$ is any integer. From this:
$\theta_k = \frac{\varphi + 2\pi k}{n}, \quad k = 0, 1, 2, \ldots, n-1$
We only need to take values of $k$ from $0$ to $n-1$. If we take $k = n$, we get the argument $\frac{\varphi + 2\pi n}{n} = \frac{\varphi}{n} + 2\pi$, which corresponds to the same number as $k = 0$ (since adding a full $2\pi$ rotation does not change the position of the point on the plane). Therefore, there are exactly $n$ distinct roots.
Thus, all $n$-th roots of the number $w$ are calculated using the formula:
$z_k = \sqrt[n]{r} \cdot e^{i\frac{\varphi + 2\pi k}{n}}, \quad k = 0, 1, 2, \ldots, n-1$
In trigonometric form, the same formula looks like this:
$z_k = \sqrt[n]{r}\left(\cos\frac{\varphi + 2\pi k}{n} + i\sin\frac{\varphi + 2\pi k}{n}\right), \quad k = 0, 1, \ldots, n-1$
The geometric interpretation of extracting a root is very clear. All $n$ distinct roots of the number $w$ are arranged on the complex plane in a consistent pattern: