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

If and , then their sum is a new Complex number with

where is the and

is the multiplied by .

Notice that

#### Example

and with

Here and with and

with

### Point in the Complex Plane

#### Example

and with

Here and

The new number is associated with a point

in the 2nd Quadrant of the Complex plane.

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## Properties of Complex Number Addition

#### Example

and with

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

is known as the Additive Identity of the Complex Numbers.

#### Example

and with then

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 then and

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. and with then

#### Subtracting Complex Numbers

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

and with then

## Commutative and Associative Properties

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

Commutative Property:

#### Example

and

Associative Property:

and then

and

## Visualization of the Addition of Complex Numbers

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 and on the Real Number Line is the absolute value of their difference

### Distance between 2 Complex Numbers

The distance between 2 points in the plane is found by the distance formula

### Complex Numbers as a 2-Dimensional Displacement

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

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 and 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.

## 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 components and the 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.

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

#### Example

and then

is the final position whether we travel to first and add or we travel to first and then add

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

#### Example

and then

This is a right triangle added to the tip of the initial Complex number at the point

The point is a distance of

from the point

### Triangle Inequality

The modulus of the Complex number is

which is around

and this is less than

which is around

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

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

and with

Notice that this is the same sum as if we would add the vector to the vector

and also the same sum if we would add vector to the vector

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 and gives the vector directed towards or the vector directed towards

#### Example

and with

The number is directed towards the point because we are subtracting from .

## Subtraction is Addition of the Opposite of a Complex Number

#### Example

and , then its negative is

is the blue dotted line, while , the negative of is the green dotted line and the sum of 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 instead of .

## The Relationship Between a Complex Number and its Conjugate

### Real and Imaginary Parts of a Complex Number

We will show that and

and with

so

so

### Conjugate of a Sum is the Sum of the Conjugates

and with

then

and

and

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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