Adding Complex Numbers

We add Complex numbers in a component-wise fashion exactly like vector addition, i.e. add the Real parts of each number together, the x components, and add the Imaginary parts of each number together, the y components, to form a new Complex number with new Real and Imaginary parts

If z_1=a+bi and z_2=c+di, then their sum is a new Complex number z with z=z_1+z_2

z=z_1+z_2=(a+bi)+(c+di)=(a+c)+(b+d)i

where (a+c) is the Re (z)=Re (z_1+z_2)=Re (z_1)+Re (z_2) and

(c+d) is the Im (z)=Im (z_1+z_2)=Im (z_1)+Im (z_2) multiplied by i.

Notice that

(a,0)+(0,b)=(a+0i)+(0+bi)=(a+0)+(0+b)i=a+bi

Example

z_1=2+3i and z_2=1+4i with z=z_1+z_2

z=(2+3i)+(1+4i)=(2+1)+(3+4)i=3+7i

Here z=3+7i and Re (z)=3 with 3=Re (z_1)+Re (z_2)=2+1 and

Im (z)=7 with 7=Im (z_1)+Im (z_2)=3+4

Point in the Complex Plane

Example

z_1=5-3i and z_2=2-i with z=z_1+z_2

z=(5-3i)+(2-i)=(5+2)+(-3-1)i=7-4i

Here Re(z)=7 and Im(z)=-4

The new number is associated with a point (x,y)

(x,y)=(Re(z),Im(z))=(7,-4)

in the 2nd Quadrant of the Complex plane.

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Properties of Complex Number Addition

Additive Identity

Example

z_1=3+2i and z_2=0+0i with z=z_1+z_2

z=(3+2i)+(0+0i)=(3+0)+(2+0)i=3+2i

This shows that adding the Complex number 0+0i to any number gives us back that number.

0+0i is known as the Additive Identity of the Complex Numbers.

Additive Inverse

Example

z_1=2+4i and z_2=-2-4i with z=z_1+z_2 then

z=(2+4i)+(-2-4i)=(2-2)+(4-4)i=0+0i

This shows that adding the negative of a Complex number to that number gives us back the Additive Identity.

The opposite or negative of a Complex number is that Complex number's Additive Inverse.

If z=a+bi then -z=-a-bi and

z+(-z)=(a+bi)+(-a-bi)=(a-a)+(b-b)i=0+0i

z+(-z)=z-z=0+0i

Adding Multiple Complex Numbers

Adding 3 or more Complex numbers is performed in the same manner. We just need to keep track of the components of each number

Ex. z_1=1+2i, z_2=3-i, z_3=4+3i and z_4=2-2i with z=z_1+z_2+z_3+z_4 then

z=(1+2i)+(3-i)+(4+3i)+(2-2i)=(1+3+4+2)+(2-1+3-2)i=10+2i

Subtracting Complex Numbers

We subtract one Complex number from the other by adding the opposite of the number we are subtracting

z_1-z_2=(a+bi)-(c+di)=(a+bi)+(-c-di)=(a-c)+(b-d)i

Example

z_1=4-3i and z_2=3-2i with z=z_1-z_2 then

    \[z=(4-3i)-(3-2i)=(4-3)+(-3-(-2))i=1-i\]

Commutative and Associative Properties

The addition of Complex numbers follows the rules of Commutativity and Associativity for addition

Commutative Property: z_1+z_2=z_2+z_1

Example

z_1=4+3i and z_2=3+i

z_1+z_2=(4+3i)+(3+i)=(4+3)+(3+1)i=(3+4)+(1+3)i=(3+i)+(4+3i)=z_2+z_1

Associative Property: z_1+(z_2+z_3)=(z_1+z_2)+z_3

Example

z_1=3-2i, z_2=4+2i and z_3=1-i then

z_1+(z_2+z_3)=(3-2i)+((4+2i)+(1-i))=(3-2i)+((4+1)+(2-1)i)=(3-2i)+(5+i)=(3+5)+(-2+1)i=8-i

and

(z_1+z_2)+z_3=((3-2i)+(4+2i))+(1-i)=((3+4)+(-2+2)i)+(1-i)=(7+0i)+(1-i)=(7+1)+(0-1)i=8-i

Visualization of the Addition of Complex Numbers

1-Dimensional Addition

We can look at the 1-Dimensional case of the addition or subtraction of Real Numbers on the number line for insights into what happens when we add or subtract vector quantities.

When we add or subtract Real numbers, we are taking an initial position, the first number, on the number line and moving the position to the right, by the addition of another number, or to the left, by the subtraction of another number. This is called displacement.

Distance between 2 Real Numbers

The distance between two numbers x_1 and x_2 on the Real Number Line is the absolute value of their difference

    \[|x_2-x_1|=|x_1-x_2|\]

Distance between 2 Complex Numbers

The distance between 2 points (x_1, y_1), (x_2, y_2) in the plane is found by the distance formula

    \[d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}\]

Complex Numbers as a 2-Dimensional Displacement

One vector in the plane with coordinates (x,y) is a 2-Dimensional displacement from the origin. We are finding a spot in the plane that is a distance

    \[\sqrt{(x-0)^{2}+(y-0)^{2}}=\sqrt{(x)^{2}+(y)^{2}}\]

from the origin.

Displacement from the Origin

Points in the Complex Plane

A Complex number marks a point, which is a place in the plane that is perpendicular to both the x and y axes (unless it lies on one of the axes or the origin, in which case it is coincident with one or both axes and is only perpendicular to one of the axes or lies at the origin).

The Modulus of a Complex Number is the Hypotenuse of a Right Triangle

Perpendicularity allows us to make a right triangle and use the Pythagorean Theorem to find the modulus of the vector, the hypotenuse of the right triangle.

Modulus

The Parallelogram Law for the Addition of Complex Numbers

Adding one Complex number to another Complex number is a 2-Dimensional displacement from the initial position of the first Complex number to the final position of the second Complex number.

We add the x components and the y components of the first and second number together to find a new place in the plane. This is displacement from an initial position to a new and final position, which is 2 total displacements from the origin.

The order that we add the 2 Complex numbers together is irrelevant. The end of the path we take to the new position is the same whether we traverse the first vector and then the second vector or vice versa.

This is known as the Parallelogram Law for Addition, showing that  we have 2 different independent routes to get to the vector sum.

This is also a consequence of the Associativity of Complex number addition.

2-Dimensional Displacement

 

The new position is an end path that starts at the origin, goes to a new position and ends at the final position.

Example

z_1=2+3i and z_2=2+i then

z=(2+3i)+(2+i)=(2+2)+(3+1)i=4+4i

Complex Addition of 2 Numbers

z_1+z_2=4+4i=(4,4) is the final position whether we travel to z_1=(x_1,y_1)=(2,3) first and add z_2=2+i or we travel to z_2=(x_2,y_2)=(2,1) first and then add z_1=2+3i

Adding Triangles

Finding the new position by adding a Complex number is equivalent to adding a right triangle, with sides equal to the x and y components and hypotenuse equal to the modulus of the second vector, to the tip of the initial vector

Example

z_1=3+2i and z_2=2+4i then

z=(3+2i)+(2+4i)=(3+2)+(2+4)i=5+6i

Right Triangle Tip

This is a 2:4:2\sqrt{5} right triangle added to the tip of the initial Complex number at the point z_1=(3,2)

The point (x,y)=(5,6) is a distance of

d=\sqrt{(5-3)^{2}+(6-2)^{2}}=\sqrt{(2)^{2}+(4)^{2}}=\sqrt{20}=2\sqrt{5}

from the point (x,y)=(3,2)

Triangle Inequality

The modulus of the Complex number z_1+z_2=5+6i is

|z_1+z_2|=|5+6i|=\sqrt{(5)^{2}+(6)^{2}}=\sqrt{61}

which is around 7.8

and this is less than

    \[|z_1|+|z_2|=|3+2i|+|2+4i|=\sqrt{(3)^{2}+(2)^{2}}+2\sqrt{5}=\]

    \[\sqrt{13}+2\sqrt{5}\]

which is around 8.1

The modulus of the sum of 2 Complex numbers is always less than or equal to the modulus of the first number plus the modulus of the second number, with equality only holding when the 2 numbers are in a straight line

    \[|z_1+z_2|\leq|z_1|+|z_2|\]

This is the Triangle Inequality, where the sum of the lengths of any 2 sides of a triangle are greater than or equal to the length of the third side.

Visualizing Adding 3 or more Vectors: Commutative Property

Example

z_1=3+4i, z_2=2-2i and z_3=-3-5i with z=z_1+z_2+z_3

z=(3+4i)+(2-2i)+(-3-5i)=((3+4i)+(2-2i))+(-3-5i)=((3+2)+(4-2)i)+(-3-5i)=(5+2i)+(-3-5i)=(5-3)+(2-5)i=2-3i

Adding 3 Complex Numbers

Notice that this is the same sum as if we would add the vector z_1+z_2=5+2i to the vector z_3=-3-5i

z=(z_1+z_2)+z_3=(5+2i)+(-3-5i)=(5-3)+(2-5)i=2-3i

3 Vector Sum to 2 Vector Sum

and also the same sum if we would add vector z_1=3+4i to the vector z_2+z_3=-1-7i

z_1+(z_2+z_3)=(3+4i)+(-1-7i)=(3-1)+(4-7)i=2-3i

Complex 3 Addition to 2 Addition

This shows that the Commutative Property holds true. The final sum is independent of the path taken.

Visualizing Subtraction of 2 Complex Numbers

When we subtract Complex numbers, we are subtracting components from each other, instead of adding them.

Using the parallelogram form we used before for Complex addition, we can see that the other diagonal connecting the vectors z_1 and z_2 gives the vector z_1-z_2 directed towards z_1 or the vector z_2-z_1 directed towards z_2

Complex Subtraction

Example

z_1=5+3i and z_2=1+3i with z=z_1-z_2

z_1-z_2=(5+3i)-(1+3i)=(5-1)+(3-3)i=4+0i

Complex Subtraction

The number z_1-z_2=4+0i is directed towards the point (5,3) because we are subtracting z_2 from z_1.

Subtraction is Addition of the Opposite of a Complex Number

Example

z_1=-3+2i and z_2=2+4i, then its negative is -z_2=-2-4i

z=z_1+(-z_2)=(-3+2i)+(-2-4i)=(-3-2)+(2-4)i=-5-2i

Complex Subtarction

z=-5-2i is the blue dotted line, while z_2=2+4i, the negative of -z_2 is the green dotted line and the sum of z_1+z_2=-1+6i is the purple dotted line

We can see that adding the negative of a Complex number is similar to the subtraction of the original Complex number but the vector is directed towards -z_2 instead of z_1.

The Relationship Between a Complex Number and its Conjugate

Real and Imaginary Parts of a Complex Number

We will show that Re(z)=\frac{z+z^{*}}{2} and Im(z)=\frac{z-z^{*}}{2i}

Example

z=3-2i and z^{*}=3+2i with z+z^{*}

(3+2i)+(3-2i)=(3+3)+(2-2)i=6+0i=2Re(z)

\frac{6}{2}=3=Re(z)

so Re(z)=\frac{z+z^{*}}{2}

Example

(3-2i)-(3+2i)=(3-3)+(-2-2)i=0-4i=2Im(z)

\frac{-4i}{2i}=-2=Im(z)

so Im(z)=\frac{z-z^{*}}{2i}

Conjugate of a Sum is the Sum of the Conjugates

z_1=2-i and z_2=5+3i with z=z_1+z_2

z=(2-i)+(5+3i)=7+2i

then z^{*}=(z_1+z_2)^{*}=7-2i

z_1^{*}=2+i and z_2^{*}=5-3i

z_1^{*}+z_2^{*}=(2+i)+(5-3i)=7-2i and

    \[z^{*}=(z_1+z_2)^{*}=z_1^{*}+z_2^{*}\]

This shows that the conjugate of the sum of 2 Complex numbers is equal to the sum of the conjugates of each Complex number.

 

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