February 28, 2020
- Adding Complex Numbers
- Properties of Complex Number Addition
- Commutative and Associative Properties
- Visualization of the Addition of Complex Numbers
- The Parallelogram Law for the Addition of Complex Numbers
- Adding Triangles
- Visualizing Adding 3 or more Vectors: Commutative Property
- Visualizing Subtraction of 2 Complex Numbers
- Subtraction is Addition of the Opposite of a Complex Number
- The Relationship Between a Complex Number and its Conjugate
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 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 .
Here and with and
Point in the Complex Plane
The new number is associated with a point
in the 2nd Quadrant of the Complex plane.
Properties of Complex Number Addition
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.
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 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. 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
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.
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
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
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
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
The number is directed towards the point because we are subtracting from .
Subtraction is Addition of the Opposite of a Complex Number
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
Conjugate of a Sum is the Sum of the Conjugates
This shows that the conjugate of the sum of 2 Complex numbers is equal to the sum of the conjugates of each Complex number.