Addition and subtraction represent translations on the complex plane. In this video we first go through the basics of adding subtracting complex numbers. A process that works as you would expect if you treat "i" as a simple constant. Then we start investigating adding simple constants to some functions. In the process we discover learn more about reading phase portraits and 3D modular surfaces. We also come across the fundamental theorem of algebra. My aim for this series is to introduce complex analysis in a visually intuitive manner, starting with the basics of complex numbers. We will start also start visualizing some simple functions very early in the series. Don't expect much in the way of rigorous proofs and definitions, there are plenty of text books for that. Instead, this series will aim to give you some visual intuition, that I hope will make any future study easier and more enjoyable. In this video: 0:00 Introduction 0:18 Addition of Complex Numbers 1:49 Subtraction of Complex Numbers 2:49 Inequalities 3:26 Adding a constant to a function 4:32 Phase portrait: z+1+i 5:07 3D Modular Surface: z+1+i 6:24 3D Modular Surface: z² 7:35 Phase Portrait: z² +1 and polynomials 12:41 Adding constants to cos(z) In this series: 1 - https://www.youtube.com/watch?v=jU7QW6AjUf4 Introduction to Complex Numbers. 2 - https://www.youtube.com/watch?v=nT3WYFxvPLk Adding and Subtracting Complex Numbers 3 - https://www.youtube.com/watch?v=O3aJCGbyfR8 Polar Coordinates of Complex Numbers 4 - [Coming Soon] Multiplication of Complex Numbers and Functions 5 - [Coming Soon] Division of Complex Numbers and Functions 6 - [Coming Soon] Complex Differentiation and Analytic Functions Extra Visuals (No Commentary): 1 - https://www.youtube.com/watch?v=3qEJeP6qQGA Trigonometric Functions Visualised I don't like to interrupt my videos with ads or promos, so you won't see any mid-roll ads. You can support this channel directly by joining my Patreon page. It takes me a long time to develop these videos, so any support will mean I can spend more of my time producing animations. As a bonus you will get downloadable access to the Maths Town fractal videos. https://www.patreon.com/mathstown