Introducing the complex numbers and complex analysis. This is the first video in a series covering the topic of complex analysis. We begin by introducing a complex number. Then we investigate the effects of multiplying any number by the imaginary number i. Finally, we take a look at some of the visualisation tools that we will use in later videos; phase portraits and modular surfaces. Please subscribe! My aim for this series is to introduce complex analysis in a visually intuitive manner, and we will be starting with the very basics of complex numbers. We will 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: 00:00 Introduction 00:30 A complex number 02:17 The imaginary number "i" 03:53 Visualising a complex number 05:49 Multiplying a number by i 06:36 Powers of i 08:53 Introducing complex analysis 10:43 Visualisation tools - phase portraits 12:42 3D phase portraits (modular surfaces) 13:42 cos(z) and cosh(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 Credits: Monkey Image: https://pixabay.com/vectors/monkey-marmoset-banana-chimpanzee-4698962/