# Addition & Division of Complex Numbers

**Question description**

Explain how complex numbers combine algebraically and graphically (solely using the graph, meaning just graphing the result of the algebraic computation is not sufficient) under the following operations:

a. Addition

b. Division

*Hints/Notes Given on Task:*

**For Addition**

Algebraically: Give the general formula for adding 2 complex numbers z1 = a + bi and z2 = c + di

Give a simple example demonstrating the addition

Graphically: Show how to add 2 complex numbers graphically. It is not necessary to provide any numbers. This is easily done when you use the complex plane and draw the complex numbers as vectors.Your method should allow someone who is given z1 and z2 to “eye-ball”/sketch where z1 + z2 will be.

**For Division**

Algebraically: Give the general formula/explanation for dividing 2 complex numbers z1 = a + bi and z2 = c + di

Give an appropriately worked out example of division.

Graphically: Instead of expressing a complex number as z = a + bi, we can define it in terms of the angle θ it makes with the x-axis going counter-clockwise and its radius (also called modulus, radius, or distance from the origin)

r=√(a^2+b^2 ) θ=〖tan〗^(-1) (b/a)

Division is a very similar operation to multiplication (meaning it involves a rotation and scaling).

•Search online for a definition of division in polar form.

•Take 2 simple complex vectors in the first quadrant (radii = 1, 2, 3, … and angles such as 90, 60, 45, 30, etc)

•Divide them algebraically: z1/z2 (you can use the same example you have for the first part of the problem)

•Convert that answer to polar form to understand what dividing by Z2 does to Z1

•Draw an appropriate picture demonstrating your understanding of division in polar coordinates.

## Tutor Answer

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors