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Complex Numbers Explained | An Introduction to Complex Numbers
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In this video, we look at an introduction to complex numbers. We show why it is necessary to consider i = sqrt(-1) and complex numbers, based on Gauss' fundamental theorem of algebra. We work through examples of adding complex numbers, multiplying complex numbers, dividing complex numbers, graphing complex numbers. We define the complex conjugate and show how to convert complex numbers into polar form. A quick proof of Euler's identity is given. De'Moivre's theorem is discussed, and we show how to compute the roots of complex numbers.
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0:00 Introduction -- What are complex numbers?
2:04 Example 1 -- Adding complex numbers.
2:26 Example 2 -- Subtracting complex numbers.
3:07 Example 3 -- Multiplying complex numbers.
3:42 Explaining complex conjugates.
4:18 Example 4 -- Dividing complex numbers.
5:24 Introducing polar coordinates.
6:35 Example 5 -- Conversion to polar form.
8:24 Euler's identity (with a quick proof).
10:50 Graphing of complex numbers (Example 6).
12:45 Properties of polar form.
14:15 Example 7 -- Dividing complex numbers (review).
15:18 Example 7B -- Dividing complex numbers using polar form.
16:30 Principal argument.
18:05 Finding exact values of trigonometric functions.
19:30 Roots of Complex Numbers.
20:21 De'Moivre's theorem.
21:00 Example 8 -- Finding the roots of complex numbers.
22:31 Outro.
👍 If you like the video, hit the like button and subscribe to the channel if you want to show your support/want to learn some mathematics.
Ways to support the channel:
FYI, these are affiliate links 💪🙏
These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.
👍 To support the channel, hit the like button and subscribe.
0:00 Introduction -- What are complex numbers?
2:04 Example 1 -- Adding complex numbers.
2:26 Example 2 -- Subtracting complex numbers.
3:07 Example 3 -- Multiplying complex numbers.
3:42 Explaining complex conjugates.
4:18 Example 4 -- Dividing complex numbers.
5:24 Introducing polar coordinates.
6:35 Example 5 -- Conversion to polar form.
8:24 Euler's identity (with a quick proof).
10:50 Graphing of complex numbers (Example 6).
12:45 Properties of polar form.
14:15 Example 7 -- Dividing complex numbers (review).
15:18 Example 7B -- Dividing complex numbers using polar form.
16:30 Principal argument.
18:05 Finding exact values of trigonometric functions.
19:30 Roots of Complex Numbers.
20:21 De'Moivre's theorem.
21:00 Example 8 -- Finding the roots of complex numbers.
22:31 Outro.
👍 If you like the video, hit the like button and subscribe to the channel if you want to show your support/want to learn some mathematics.
Ways to support the channel:
FYI, these are affiliate links 💪🙏
These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.
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