Equal Complex Numbers

Two Complex numbers z_1=x_1+iy_1 and z_2=x_2+iy_2 are equal if and only if their Real  and their Imaginary parts are equal

z_1=z_2\iff x_1=x_2 and y_1=y_2

Their arguments are also equal

arg(z_1)=arg(z_2)

Example

z_1=4-4i with argument arg(z_1)=\theta_1=\frac{7\pi}{4} and

z_2=4-4i with argument arg(z_2)=\theta_2=\frac{15\pi}{4}

We have equality of the Real and Imaginary parts

x_1=4=x_2 and y_1=-4=y_2

and their arguments are also equal

arg(z_1)=\frac{7\pi}{4}=\frac{15\pi}{4}-2\pi=\frac{15\pi}{4}-\frac{8\pi}{4}=\frac{7\pi}{4}

z_2 has traversed an angle larger than 2\pi but its principle argument is \frac{7\pi}{4} in the 4th Quadrant of the Complex plane.

z(1)=4-4i arg=7pi/4                                                               z(2)=4-4i arg=15pi/4
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Conjugate Complex Numbers

Two Complex numbers z_1=x_1+iy_1 and z_2=x_2-iy_2 are conjugates if they have equal Real parts and opposite (negative) Imaginary parts

z_1=x_1+iy_1 and z_2=x_2+iy_2 with x_1=x_2 and y_1=-y_2

z_2=z_1^{*}=x_2+iy_2=x_1-iy_1 and z_1=z_2^{*}=x_1+iy_1=x_2-iy_2

Example

If z_1=2+5i then z_2=z_1^{*}=2-5i

x_1=2=x_2 and y_1=5=-(-5)=-y_2

We will denote the conjugate of a Complex number z as z^{*}

Complex Conjugates

The Modulus of a Complex Number and its Conjugate

The modulus of z is equal to the modulus of z^{*}

z=2+5i and z^{*}=2-5i

|z|=\sqrt{(2)^{2}+(5)^{2}}=\sqrt{4+25}=\sqrt{29} and

|z^{*}|=\sqrt{(2)^{2}+(-5)^{2}}=\sqrt{4+25}=\sqrt{29}

Their product zz^{*} is equal to the square of their modulus

zz^{*}=(2+5i)(2-5i)=(4+25)+(-10+10)i=29 and

|z|^{2}=|z^{*}|^{2}=(\sqrt{29})^{2}=29

Quadrant Location

We saw from the example above that if a Complex number is located in the 1st Quadrant, then its conjugate is located in the 4th Quadrant. The opposite is also true. If a Complex number is located in the 4th Quadrant, then its conjugate lies in the 1st Quadrant.

The same relationship holds for the 2nd and 3rd Quadrants

Example

If z=-3-2i then z^{*}=-3+2i

z is located in Quadrant 3, so z^{*} is located in Quadrant 2

Complex Conjugates

The Argument of a Complex Number and its Conjugate

The arguments of a Complex number z and its conjugate z^{*} add up to 2\pi.

If arg(z)=\theta then arg(z^{*})=-\theta because of the symmetry across the x-axis due to the Quadrant symmetries of 1 and 4 and 2 and 3

with -\theta=2\pi-\theta

The sum of their arguments is (2\pi-\theta)+\theta=2\pi.

Opposite Complex Numbers

Two Complex numbers z_1=x_1+iy_1 and z_2=x_2+iy_2 are opposite if they have opposite (negative) Real and Imaginary parts

x_1=-x_2 and y_1=-y_2 or

z_2=-(z_1)

Example

z_1=4-i then its opposite is z_2=-4+i=-z_1

Complex Opposites

The Modulus of a Complex Number and its Opposite

The modulus of a Complex number z and its opposite -z are equal in magnitude

Example

If z=-3-4i then its opposite is -z=-(-3-4i)=3+4i

|z|=\sqrt{(-3)^{2}+(-4)^{2}}=\sqrt{9+16}=\sqrt{25}=5 and

|-z|=\sqrt{(3)^{2}+(4)^{2}}=\sqrt{9+16}=\sqrt{25}=5

Then |z|=|-z|.

Quadrant Location

There is also quadrant symmetry between a Complex number and its opposite.

If a Complex number z=x+iy lies in the 1st Quadrant, then its opposite -z=-x-iy lies in the 3rd Quadrant and if a Complex number z=x-iy lies in the 4th Quadrant, then its opposite -z=-x+iy lies in the 2nd Quadrant.

Example

z=2+i is in the 1st Quadrant because both x and y are positive, then -z=-2-i lies in the 3rd Quadrant because both x and y are negative.

Example

z=-2+i is in the 2nd Quadrant because x is negative and y is positive, then its opposite -z=2-i is in the 4th Quadrant because x is positive and y is negative.

The Argument of a Complex Number and its Opposite

The arguments of a Complex number z and its opposite -z differ by a value of \pi because the numbers lie along a straight line in the plane due to the Quadrant symmetries between 1 and 3 and between 2 and 4, and a straight line has a measure of 180^{\circ}, which is equivalent to \pi radians.

Reciprocal Complex Numbers

The reciprocal of a Complex number z is z^{-1}=\frac{1}{z} with z\neq0.

The reciprocal of a Complex number is its Multiplicative Inverse because zz^{-1}=1.

Example

If z=3-2i then z^{-1}=\frac{1}{3-2i}

We must rationalize the denominator to make the reciprocal into a form that we can Mathematically manipulate because having a Complex number in the denominator

z^{-1}=(\frac{1}{3-2i})(\frac{3+2i}{3+2i})=\frac{3+2i}{(3-2i)(3+2i)}=\frac{3+2i}{9-8i+8i+4}=\frac{3+2i}{13}

z^{-1}=\frac{3+2i}{13} is the conjugate of z=3-2i divided by the square of the modulus. It's just a scaled version of the conjugate of the original Complex number.

Multiplying a Complex number by its conjugate divided by the square of the modulus will yield 1 because the product of the Complex number and its conjugate is just the square of the modulus

zz^{-1}=(3-2i)(\frac{3+2i}{13})=\frac{9+6i-6i+4}{13}=\frac{13}{13}=1

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Patrick