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Complex numbers are constituents of a number system that extend the real numbers with a specific element denoted i, called the imaginary unit.

This imaginary unit is defined by

where i2 = -1.

From our basic grasp of multiplication of real numbers, the square of any number will always be positive, thus it is a natural consequence that the square root of a negative number should not exist. However, this imaginary unit is incredibly significant in many fields of modern mathematics and engineering. Indeed, both complex numbers and vectors are important areas of study in mathematics. Physical problems that might be difficult to solve in the real domain can be more conveniently solved when transformed into complex variables due to the special properties of the complex domain.

Vectors are also applied to geometry, kinematics and vector calculus in three dimensions and are therefore indispensable tools and concepts in mechanics. The two areas of study, complex numbers and vectors, overlap with respect to vector representation of the additive property of complex numbers in the complex plane (complex vectors). Complex numbers behave much like two dimensional vectors. Indeed real numbers are one dimensional vectors (with one real component, on a line) and complex numbers are two dimensional vectors (consisting of two components, in a plane).

Definition of a Complex Number

If a and b are real numbers, then the number

is a complex number in rectangular form, where a is the real part and bi is the imaginary part of the number. The set of real numbers is a subset of the set of complex numbers. Every real number a can be written as a complex number with b = 0. That is, for every real number a,

The real and imaginary parts are unique characteristics of a complex number. We can only say that two complex numbers are equal if and only if their real and imaginary parts are equal. That is, if there are two complex numbers written in rectangular form, then

if and only if a = c and b = d.

For complex numbers, the rectangular form is the standard form.

The Complex Plane

Since a complex number is uniquely determined by its real and imaginary parts, it is natural to associate the number a + bi with the ordered pair (a, b). With this association, we can graphically represent complex numbers as points in a coordinate plane that we call the complex plane or the Argand plane. This plane is an adaptation of the rectangular (Cartesian) coordinate plane. Specifically, we call the horizontal axis the real axis and the vertical axis the imaginary axis. For instance, Figure 1 shows the graph of two complex numbers 2 + 3i and -2 - i. The number 2 + 3i is associated with the point (2,3) and the number -2 - i is associated with the point (-2,-1).

Complex number imaginary real axis
Figure 1. Imaginary and Real Axis of a Complex Number

Another way to represent the complex number a + bi is as a vector whose horizontal component is a and vertical component is b. (Letter i is only used to represent the imaginary unit and is unrelated to î which is used to represent a unit vector.)

Complex number vector representation
Figure 2. Vector representation of a complex number

Absolute Value and Conjugate

The absolute value of a real number is equal to the positive value of the number itself, represented by modulus | x |. Thought of as a number’s distance from zero, the modulus always gives a positive value, such that; |5| = |-5| = 5.

In the case of complex numbers, finding the modulus is not so straightforward.

Suppose z = x + iy is a complex number. Then, mod of z will be:

The mod of complex number z is dependent on its real and imaginary parts. These values form a right triangle, where the hypotenuse is the distance from 0 to z, hence mod of z. Applying the Pythagorean theorem in a complex plane yields this expression.

Every complex number has an associated complex conjugate. The complex conjugate of a complex number has the same real part and an imaginary part that is equal in magnitude but with the opposite sign.

The conjugate of z = x + iy is denoted by :

Complex conjugates are useful in rationalizing complex numbers and simplifying expressions.

Algebraic Operations on Complex Numbers

Addition

When we have two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2, we obtain the sum by separately adding the real and imaginary components:

z1 + z2 = ( x1 + x2 ) + i ( y1 + y2 )

Complex plane complex numbers addition
Figure 3. Addition of complex numbers on the complex plane

Subtraction

Much like with addition, for subtraction, you simply subtract the real and imaginary components separately. Visually, we can think of this as getting the negative of the number to be subtracted, and then ‘adding’ that vector on the first number from which we are subtracting.

z1 - z2 = ( x1 + iy1 ) - ( x2 + iy2 ) = ( x1 -x2 ) + i( y1 - y2 )

Complex plane complex numbers subtraction
Figure 4. Subtraction of complex numbers on the complex plane

This relationship is sometimes called the parallelogram rule for the addition/subtraction of complex numbers, as it is based on the idea of regarding complex numbers as vectors in the complex plane. If you have two complex numbers z1 and z2 , z1 ± z2 is the fourth vertex on a parallelogram where z1, z2 and the origin are the other vertices.

Multiplication

az = a ( x + iy ) = ax + aiy, when a is a real number

Complex plane complex number multiplication scalar
Figure 5. Multiplication of a complex number by a scalar on the complex plane

When two complex numbers in rectangular form are multiplied, the process of obtaining the product is similar to the multiplication of two binomials. The FOIL method (Distributive multiplication process) is used.

( x1 + iy1 ) ( x2 + iy2 ) = ( x1x2 - y1y2 ) + i ( x1y2 + y1x2 )

Multiplying a complex number by i,

iz = ix + i2y = - y + ix

i2z = (-1)( z ) = - x - iy

i3z = ( i2 . i ) z = - iz = y - ix

i4z = ( i2 . i2 ) z = ( 1 ) z = z

For complex numbers,

When working with products involving square roots of complex numbers, we must convert to a multiple of i before multiplying. For instance, consider the following:

Depending on the power of i, it can take the following values: (i, -1, -i, 1)

i4k+1 = i,

i4k + 2 = - 1,

i4k+3 = - i

i4k = 1

where k is a positive or negative integer.

Complex plane multiplying complex number i effects
Figure 6. The effect of multiplying a complex number by i on the complex plane

By Pythagoras’ theorem it is clear that each point is the same distance from the origin and when we multiply a complex number by i the new complex number is rotated about the origin in the counterclockwise direction. It can be seen that each rotation is equivalent, and after the fourth rotation we return to the original value or starting point. Multiplying a complex number by i will give a new complex number which is the same distance from the origin but has been rotated 90° counterclockwise. This is useful in the language of transformations of the plane.

Division

The division of two complex numbers in rectangular form can be performed by multiplying the numerator and denominator by the conjugate of the denominator, and then applying the FOIL method.

Polar Representation

Rectangular polar coordinates relationship
Figure 7. The relationship between rectangular and polar coordinates

To convert rectangular coordinates (x, y) to polar coordinates (r, θ),

The value of θ is called the argument of the complex number z and written arg (z). We can further define a unique value

to avoid multiple values due to the periodic nature of the circular functions and hence arg (z).

If we have been given the polar coordinates of a point, we can use trigonometric relations to find the equivalent rectangular coordinates.

Polar coordinates use rotation from an axis and distance from an origin. These are valuable attributes when we consider the multiplication of complex numbers because when complex numbers are multiplied, points are both dilated from/collapsed into and rotated about the origin. This process is better represented in terms of the variables in polar coordinates as the resulting dilation is explicitly given by the distance (r) and the rotation about the origin is given by the argument (). When multiplying (and dividing) complex numbers in cartesian coordinates, the FOIL method can be prone to errors, while the same process is straightforward in polar coordinates.

If you have two complex numbers in polar form

and

z1z2 is given by

and z1 / z2 is given by

To obtain the product, the lengths are multiplied and the arguments are added. To obtain the quotient, the lengths are divided and the arguments are subtracted. This should encourage us to multiply and divide complex numbers expressed in polar form.

In this tutorial we briefly reviewed complex numbers and their overlapping concept with vectors. The applications of complex vector algebra are essential in electromagnetic theory and in the dynamics of elementary particles with extended structure. Complex vector notation simplifies expressions and provides clarity of geometrical understanding of the basic concepts.

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