Lesson Addition and subtraction of complex numbers in complex plane

Addition and subtraction of complex numbers in the complex plane

In this lesson you will learn about the geometrical interpretation for addition and subtraction of complex numbers.

Let me remind you that the set of complex numbers and arithmetical operations over them were introduced in the lesson Complex numbers and arithmetical operations in this module. The geometry presentation of complex number in the complex plane was introduced in the lesson Complex plane in this module.

Addition of complex numbers in the complex plane

Let us consider complex numbers u=a+bi and v=c+di, and let vectors OM and ON represent these complex numbers in the complex plane XOY (Figure 1). So, the vector OM has components a and b (a=OR, b=RM), and the vector ON has components c and d (c=OQ and d=QN). From the point M draw the vector MP equal to ON (so, the vector MP has the same length and same direction as ON). Let us consider the vector OP. We state (and will proof it now) that the vector OP represents the sum of the given complex numbers u and v. Indeed, from the one side, the sum of complex numbers u and v is the complex number z=(a+c)+(b+d)i. From the other side, X-component of the vector OP is equal to OS = OQ + QS = OQ + OR = a+c, and Y-component of the vector OP is equal to SP = ST + TP = RM + QN = c+d. Here we used the fact that right triangles OQN and MTP are congruent (have equal hypotenuse and adjacent angles). You see that the vector OP has the same components as the complex number z. The proof is completed.

Figure 1. Summing complex numbers as vectors in the Complex Plane

The vector OP constructed in this way above, is termed the geometric sum of vectors OM and ON. This operation, geometric sum of vectors, is an analog of summing velocities of moving bodies, of summing forces applied to the point, and many other vectorial physical quantities.

So, the geometrical interpretation of addition of complex numbers is read as follows:
the sum of complex numbers is depicted in the complex plane as the sum of vectors representing the given complex numbers.

Subtraction of complex numbers in the complex plane

It was noted in the lesson Complex numbers and arithmetical operations in this module that to subtract the complex number v=c+di from the complex number u=a+bi is the same as to add the opposite complex number w=-v=-c-di to the complex number u.

From the other side, it is natural way to define the difference between two vectors as the sum of the first given vector and the opposite to the second one. Here the opposite vector is understood as having opposite component values relative to the original vector.

Note that this is the standard way how the difference of vectors is defined in physics for velocities, forces and so on.

By doing so, the geometrical interpretation of subtraction of complex numbers is read as follows:
the difference of complex numbers is depicted in the complex plane as the difference of the vectors representing the given complex numbers.



For your convenience, below is the list of my relevant lessons on complex numbers in this site in the logical order. They all are under the current topic Complex numbers in the section Algebra II.
Complex numbers and arithmetical operations over them
Complex plane
Addition and subtraction of complex numbers in complex plane                          (this lesson)
Multiplication and division of complex numbers in complex plane
Raising a complex number to an integer power
How to take a root of a complex number
Solution of the quadratic equation with real coefficients on complex domain
How to take a square root of a complex number
Solution of the quadratic equation with complex coefficients on complex domain