Table of Contents

  1. The Complex Plane
  2. Modulus: The Length of a Vector
  3. Argument: The Direction of a Vector
  4. Trigonometric Form of a Complex Number
  5. Adding Complex Numbers as Vector Addition
  6. Difference and Distance Between Points
  7. Multiplication: Scaling and Rotation
  8. Division: Scaling Down and Reverse Rotation
  9. Summary

Complex numbers are useful not only for calculations, but also for representing points on a plane. This perspective turns formulas into visual operations: addition becomes vector addition, the modulus becomes distance, and multiplication becomes rotation and scaling.

Main idea: a complex number \(z=a+bi\) can simultaneously be viewed as a number, a point \((a,b)\), and a vector drawn from the origin to that point.

The Complex Plane

Let a complex number be given:

\[ z=a+bi. \]
Complex plane

The number \(a\) is called the real part of \(z\), while \(b\) is called its imaginary part:

\[ \operatorname{Re}z=a,\qquad \operatorname{Im}z=b. \]

On the complex plane, the real part is plotted along the horizontal axis and the imaginary part along the vertical axis. Thus, the number \(z=a+bi\) corresponds to the point with coordinates \((a,b)\).

Example

The number \(z=3+2i\) corresponds to the point \((3,2)\): starting at the origin, move 3 units to the right and 2 units up.

The same point can also be described as the endpoint of a vector starting at \((0,0)\). This is why complex numbers are naturally connected with vectors on a plane.

Modulus: The Length of a Vector

The modulus of a complex number is the distance from its corresponding point to the origin. It is denoted by \(|z|\).

\[ |z|=\sqrt{a^2+b^2}. \]

This formula follows from the Pythagorean theorem: the real and imaginary parts form the legs of a right triangle.

Example

For the number \(z=-3+4i\):

\[ |z|=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=5. \]

Although the point lies to the left of the origin, its distance from the origin is still 5.

The modulus cannot be negative. Moreover, \(|z|=0\) only for \(z=0\).

Argument: The Direction of a Vector

Length alone is not enough to specify a vector uniquely: its direction is also needed. The argument of a complex number is the angle between the positive direction of the real axis and the number's vector.

\[ \varphi=\arg z. \]

The counterclockwise direction is considered positive. The same ray is obtained if a full turn is added to the angle:

\[ \arg z=\varphi+2\pi k,\qquad k\in\mathbb{Z}. \]

The principal argument is often used: one selected angle in the interval \((-\pi,\pi]\). It is denoted by \(\operatorname{Arg}z\).

Important: the argument of zero is undefined. The point \((0,0)\) has no direction.

Examples of Arguments

  • for \(z=1+i\): \(\operatorname{Arg}z=\frac{\pi}{4}\);
  • for \(z=-1+i\): \(\operatorname{Arg}z=\frac{3\pi}{4}\);
  • for \(z=-1-i\): \(\operatorname{Arg}z=-\frac{3\pi}{4}\).

When calculating an argument, it is important to consider the quadrant in which the point lies. The formula \(\tan\varphi=\frac{b}{a}\) gives only a ratio and does not always immediately determine the correct angle.

Trigonometric Form of a Complex Number

If the modulus \(\rho=|z|\) and the argument \(\varphi\) are known, the number can be written in trigonometric form:

\[ z=\rho(\cos\varphi+i\sin\varphi). \]

This form is especially convenient for multiplying, dividing, and raising complex numbers to powers.

Example: The Number \(1+i\)

Its modulus equals:

\[ |1+i|=\sqrt{1^2+1^2}=\sqrt{2}. \]

Its argument is \(\frac{\pi}{4}\), so:

\[ 1+i=\sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right). \]

Using Euler's formula \(e^{i\varphi}=\cos\varphi+i\sin\varphi\), the same representation can be written more compactly:

\[ z=\rho e^{i\varphi}. \]

Adding Complex Numbers as Vector Addition

Adding complex numbers as vectors
A geometric interpretation of the sum of complex numbers.

The sum of complex numbers is found using the usual rule:

\[ (a+bi)+(c+di)=(a+c)+(b+d)i. \]

Geometrically, this is vector addition. The second vector can be imagined as moved to the endpoint of the first, while the sum vector is drawn from the origin to the final point.

Example

Let \(z_1=2+i\) and \(z_2=1+3i\). Then:

\[ z_1+z_2=(2+i)+(1+3i)=3+4i. \]

On the plane, the sum vector ends at the point \((3,4)\).

The triangle inequality follows immediately from geometry:

\[ |z_1+z_2|\leq |z_1|+|z_2|. \]

The length of the resulting vector cannot exceed the sum of the lengths of the two original vectors.

Difference and Distance Between Points

The modulus of the difference between two complex numbers has a simple geometric meaning:

\[ |z_1-z_2| \]

It is the distance between the points corresponding to \(z_1\) and \(z_2\).

Example

Let \(z_1=1+2i\) and \(z_2=4+6i\). Then:

\[ |z_1-z_2|=|(1+2i)-(4+6i)|=|-3-4i|=5. \]

Therefore, the distance between the points \((1,2)\) and \((4,6)\) is 5.

Another useful inequality is:

\[ |z_1-z_2|\geq \bigl||z_1|-|z_2|\bigr|. \]

It states that the distance between the endpoints of two vectors cannot be less than the difference between their lengths.

Multiplication: Scaling and Rotation

In algebraic form, complex numbers are multiplied by expanding brackets and using \(i^2=-1\). However, the geometric meaning of multiplication is particularly clear in trigonometric form.

If:

\[ z_1=\rho_1(\cos\varphi_1+i\sin\varphi_1), \qquad z_2=\rho_2(\cos\varphi_2+i\sin\varphi_2), \]

then:

\[ z_1z_2= \rho_1\rho_2 \left( \cos(\varphi_1+\varphi_2) +i\sin(\varphi_1+\varphi_2) \right). \]

Thus, during multiplication:

  • the moduli are multiplied;
  • the arguments are added.

In other words, multiplication by a complex number changes a vector's length and rotates it at the same time.

Special Case: Multiplication by \(i\)

The number \(i\) has modulus 1 and argument \(\frac{\pi}{2}\). Therefore, multiplying any complex number by \(i\) rotates its vector by \(90^\circ\) counterclockwise.

\[ 1\cdot i=i,\qquad i\cdot i=-1. \]

Division: Scaling Down and Reverse Rotation

A similar rule applies to division. If \(z_2\neq 0\), then:

\[ \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2} \left( \cos(\varphi_1-\varphi_2) +i\sin(\varphi_1-\varphi_2) \right). \]

During division, the moduli are divided and the argument of the divisor is subtracted. Geometrically, this changes the scale and rotates the vector in the opposite direction.

Summary

The geometric interpretation makes complex numbers easier to visualize. The real and imaginary parts define the coordinates of a point, the modulus gives its distance from the origin, and the argument gives the vector's direction.

This approach helps reveal simple geometric actions behind formulas: vector addition, distance measurement, rotation, and scaling. It becomes especially useful when studying De Moivre's formula, complex roots, and transformations on the plane.