Complex Numbers as Vectors in the Complex Plane

Complex Numbers as Vectors in the Complex Plane

Complex numbers become much more useful in physics when they are interpreted geometrically rather than treated only as algebraic symbols. A number such as

(1)

contains two real numbers, a and b. These may be displayed as Cartesian coordinates in a plane: a along a horizontal real axis and b along a vertical imaginary axis. The same complex number can then be viewed as a point or as a directed line segment from the origin to that point.

This geometric viewpoint is the foundation for rotating complex vectors, phasors, sinusoidal steady-state analysis, Fourier methods, and many wave calculations. It also prepares the key result developed in the next topic: complex multiplication simultaneously changes magnitude and angle.

Throughout this article, j denotes the imaginary unit,

(2)

Engineering texts commonly use j because i is often reserved for electric current. Mathematics and much of theoretical physics usually use i. The algebra is identical.

1 Cartesian form of a complex number

A complex number in Cartesian form is written

(3)

where a and b are real numbers.

The real part of z is

(4)

and the imaginary part is

(5)

Notice that the imaginary part is the real coefficient b, not the quantity jb.

The horizontal axis of the complex plane contains purely real numbers. The vertical axis contains purely imaginary numbers. The complex number z = a + jb is represented by the point (a,b).

2 Why it is reasonable to call z a vector

As a geometric object in the complex plane,

(6)

can be associated with the real two-dimensional vector

(7)

Complex addition is exactly ordinary two-dimensional vector addition. If

(8)

and

(9)

then

z1 + z2

= (a + c) + j(b + d),

(10)

which corresponds to

(11)

There is, however, an important distinction:

• complex numbers form a two-dimensional vector space over the real numbers;
• complex numbers also possess their own multiplication operation;
• that multiplication is not the ordinary dot product or cross product of two-dimensional vectors.

The extra multiplication rule is precisely what makes complex numbers especially powerful for oscillations and rotations.

3 Length: the modulus of a complex number

The vector from the origin to (a,b) forms a right triangle with legs a and b. By the Pythagorean theorem, its length is

(12)

The modulus or magnitude of z is therefore defined as

(13)

Because a length is nonnegative,

(14)

Furthermore,

(15)

only for z = 0.

Example: z = 3 + 4j

For

(16)

the modulus is

|z|	=	(17)
	=	(18)
	= 5.	(19)

Thus the familiar 3-4-5 right triangle appears directly in the complex plane.

4 Direction: the argument of a complex number

Let 𝜃 denote the angle measured counterclockwise from the positive real axis to the vector z. This angle is called an argument of the complex number.

For a≠0, the right-triangle geometry gives

(20)

A tempting formula is therefore

(21)

That formula alone is not sufficient because the tangent function cannot distinguish opposite quadrants. For example, (1, 1) and (−1,−1) have the same value of b∕a but their directions differ by π.

For actual calculations, the angle should be determined from both coordinates. In numerical software this is usually the two-argument arctangent,

(22)

The argument is not unique because adding a full revolution leaves the same direction:

(23)

A commonly used principal argument chooses one representative angle, often

(24)

Other branch choices are possible, so a source should state its convention when the distinction matters.

5 Cartesian and polar descriptions of the same number

The geometry gives

(25)

Substituting these into z = a + jb gives

z	= r cos 𝜃 + jr sin 𝜃	(26)
	= r(cos 𝜃 + j sin 𝜃).	(27)

Thus the polar form of a nonzero complex number is

(28)

where

(29)

This is not a new complex number. Cartesian and polar forms are two coordinate descriptions of the same geometric object.

Example: convert 3 + 4j to polar form

The magnitude was already found to be

(30)

Because the point lies in the first quadrant,

𝜃	= tan −1	(31)
	≈ 0.9273 rad	(32)
	≈ 53.13∘.	(33)

Therefore

(34)

6 Quadrants matter

Consider

(35)

Its magnitude is still

(36)

but the vector lies in quadrant II. The ratio b∕a is

(37)

A simple one-argument inverse tangent returns approximately −53.13^∘, which points into quadrant IV and is therefore the wrong geometric direction for z.

The correct principal argument is

(38)

This is why complex calculations should use quadrant-aware angle extraction.

7 The complex conjugate as a geometric reflection

For

(39)

the complex conjugate is defined as

(40)

The real coordinate is unchanged while the imaginary coordinate changes sign. Geometrically, conjugation reflects the complex vector across the real axis.

Consequently,

(41)

and, away from branch-cut subtleties,

(42)

Multiplying z by its conjugate gives

zz∗	= (a + jb)(a − jb)	(43)
	= a2 + b2.	(44)

Hence an important identity is

(45)

This identity becomes extremely useful in wave amplitudes, impedance calculations, and quantum mechanics.

8 Euler’s formula and exponential form

Euler’s formula connects the complex exponential to ordinary trigonometric functions:

(46)

Using this in the polar form gives the complex exponential form

(47)

For the earlier example,

(48)

The exponential notation is compact, but its meaning should remain geometric:

• r gives the vector length;
• 𝜃 gives the vector direction;
• e^j𝜃 represents a unit-length complex number at angle 𝜃.

9 Special directions in the complex plane

Euler’s formula immediately identifies several important points:

ej0	= 1,	(49)
ejπ∕2	= j,	(50)
ejπ	= −1,	(51)
ej3π∕2	= −j,	(52)
ej2π	= 1.	(53)

Thus the numbers

(54)

correspond to directions separated by 90^∘ around the unit circle.

This geometry foreshadows why multiplication by j produces a quarter-turn in the complex plane. That result is developed carefully in the follow-up article on the geometric meaning of complex multiplication.

10 Complex numbers and physical quantities

A complex number is usually a mathematical representation rather than a directly measured two-component physical vector. In oscillation and wave problems, one often writes a complex quantity such as

(55)

and obtains the physical scalar displacement from its real projection,

(56)

The complex-plane vector is then a computational and geometric device. Its rotating direction stores phase while its length stores amplitude.

It is useful to keep three different uses of the word “vector” distinct:

• a complex number may be drawn as a two-dimensional vector in the complex plane;
• a physical vector such as force or velocity has components in physical space; and
• a phasor may associate a complex amplitude with each component of a physical vector field.

These structures can interact, but they are not automatically the same object.

11 Worked example: complete Cartesian-to-polar conversion

Let

(57)

The real and imaginary parts are

(58)

The modulus is

|z|	=	(59)
	=	(60)
	= 10.	(61)

The point lies in quadrant III. A quadrant-aware evaluation gives the principal argument

(62)

which is equivalent to

(63)

Thus

(64)

Its conjugate is

(65)

which has the same modulus and the reflected principal angle

(66)

12 What this article establishes

The geometry of the complex plane can be summarized as follows:

• z = a + jb corresponds to the point (a,b);
• Re(z) = a and Im(z) = b are Cartesian coordinates;
• |z| = is the length of the complex-plane vector;
• arg z gives its direction modulo 2π;
• z = r(cos 𝜃 + j sin 𝜃) = re^j𝜃 is its polar/exponential representation;
• z^∗ = a − jb reflects the vector across the real axis; and
• |z|² = zz^∗ connects geometric length to complex algebra.

The next natural result is deeper: when two complex numbers are multiplied, their lengths multiply and their angles add. That is the mathematical step that turns the complex plane from a static coordinate picture into a language for scaling, rotation, phasors, and oscillations.

References