Introduction

Complex numbers are multi-part, 2-Dimensional numbers of the form

    \[z=a+bi\]

where a is any Real number and is known as the Real part of the Complex number, Re(z),

and b is any Real number multiplied by i=\sqrt{-1} and is known as the Imaginary part of the Complex number, Im(z).

Example

If z=1+i, then Re (z)=1 and Im (z)=1, which is multiplied by i

The set of all Complex Numbers is denoted by \mathbb{C}.

If b=0 in the Complex number a+bi, then the number reduces to an ordinary Real number a

Example

    \[3+0i=3\]

A number of the form bi is known as a pure Imaginary number

Example

    \[4i\]

    \[\sqrt{-25}=\sqrt{25i^{2}}=5i\]

Equality of 2 Complex Numbers

We say that two Complex numbers are equal if and only if they have the same Real component and the same Imaginary component
z_1=z_2\iff x_1=x_2 and y_1=y_2
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Complex Numbers in the Complex Plane

Complex numbers are multi-part because of the inclusion of both the Real and Imaginary parts in the whole Complex Number.

As for being 2-Dimensional, we equate the ordered pair (a,b) with a point (x,y) in the 2-Dimensional Complex or Argand plane

z=a+bi=x+iy with a=x and b=y

The Argand Plane

The Complex plane is a plane similar to the xy-plane, with 2 axes and 4 quadrants. The x-coordinate is treated as an independent dimension and so is the y-coordinate, which has all of its members multiplied by i.

In the Complex plane, the x-axis is the Real axis and the y-axis is the Imaginary axis.

A Real number is represented by a number on the x-axis and an Imaginary number is represented by a number on the y-axis.

A Complex Number is 2-Dimensional and is represented by a point (x,y) in the plane.

Complex Plane

Example

z_1=x_1+iy_1=1+i is the Complex number represented by the point z_1=(x_1,y_1)=(1,1) in the 1st Quadrant of the Argand plane

Example

z_2=x_2+iy_2=-1-i is the Complex number represented by the point z_2=(x_2,y_2)=(-1,-1) in the 3rd Quadrant of the plane

Points in the Plane

Complex Numbers as Vectors

Multidimensionality gives us the ability to graphically represent a Complex number with a vector quantity and an associated directed angle \theta that the vector makes as measured counterclockwise from the positive x-axis, with \theta measured in radians and

    \[0\leq\theta\leq2\pi\]

The vector associated with each unique Complex number z=x+iy is a vector emanating from the origin and directed to the point (x,y) in the plane.

Example

z=1+i is not only the point (1,1) in the plane, it is also the vector emanating from the origin directed to the point (1,1).

Point (1,1) and Angle

We can see geometrically that along this vector, x=y and the associated angle would be \theta=45^{\circ} or \frac{\pi}{4}, because a line coincident with the vector cuts the plane into 2 parts at a 45^{\circ} angle.

Argument of a Complex Number

The angle \theta is called the argument of the Complex number z and is denoted by \theta =arg (z).

We can find the value of the argument by taking the inverse tangent or arctangent of the angle \theta, which is the angle that has a tangent value of \frac{y}{x}

    \[\theta=tan^{-1}(\frac{y}{x})=arctan(\frac{y}{x})\]

Example

To check that the angle from the previous example is equivalent to \frac{\pi}{4}, we need to find the angle whose tangent is equal to 1, because x=1 and y=1 and their ratio \frac{y}{x} is also equal to 1

arg (1+i)=tan^{-1}(\frac{1}{1})=tan^{-1}(1)=\frac{\pi}{4}

Range of the Argument

All angles will be measured in radians and we limit the range for the directed angle \theta to 0-2\pi so as to have a unique angle measure for each vector associated with a unique point, because the angle can have an infinite number of equivalent values past 2\pi.

Reference Angles

The argument is always a reference angle between 0 and 2\pi and if the angle is larger than 2\pi we determine the reference angle by subtracting multiples of 2\pi from the original angle.

Example

If \theta=\frac{13\pi}{2}, we must subtract 2\pi from it to find the argument

arg(z)=\frac{13\pi}{2}-2\pi=\frac{13\pi}{2}-\frac{12\pi}{2}=\frac{\pi}{2}

Example

If z=2-2i, we know the point is located in the 4th Quadrant because x=2 and y=-2.

We must always be careful to keep track of minus signs and where exactly the point is located, so as not to confuse angles or quadrants.

arg(z)=\theta=tan^{-1}(\frac{-2}{2})=tan^{-1}(-1)=\frac{7\pi}{4}

Point (2,-2)

Negative Arguments

An angle -\theta is an angle measured clockwise from the positive x-axis and it has an equivalent but opposite range from -2\pi\leq-\theta\leq0.

An angle measure that is negative, -\theta, is equivalent to an angle measure of 2\pi-\theta.

Modulus of a Complex Number

We can resolve any vector into its respective x and y components and use the Pythagorean Theorem to find its length.

The length of a vector is the distance from the origin to the tip and is known as its magnitude. The magnitude of the vector associated with each Complex number z is known as its modulus or absolute value and is found by taking the square root of the sum of the squares of its components

    \[|z|=\sqrt {x^{2}+y^{2}}\]

 

Modulus

The modulus is always a non-negative Real number

0\leq|z| with |z|\in\mathbb{R} and |z|=0\iff x=0 and y=0

Example

z=0+0i and |z|=\sqrt{(0)^{2}+(0)^{2}}=0

Example

z=1+i and |z|=|1+i|=\sqrt{(1)^{2}+(1)^{2}}=\sqrt 2

Example

z=1-\sqrt{3}i and |z|=|1-\sqrt{3}i|=\sqrt{(1)^{2}+(-\sqrt{3})^{2}}=\sqrt{1+3}=2

Magnitude of a Complex Number's Opposite

The magnitude of a Complex number and its negative or opposite are equal in value because the square of a number and the square of its negative are always equal

|-z|=|-(x+iy)|=|-x-iy|=\sqrt{(-x)^{2}+(-y)^{2}}=\sqrt{x^{2}+y^{2}}=|z|

 

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Patrick