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Title: Quaternions
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Series: Advent of Mathematical Symbols
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YouTube-Title: Quaternions
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Bright video: https://youtu.be/r1qH9GBlX-0
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Dark video: https://youtu.be/oDJhz7iLvtE
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: aoms23_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,714 –> 00:00:02,286 Hello and welcome.
2 00:00:02,443 –> 00:00:05,686 and the mathematical symbol of today is given by the quaternions.
3 00:00:05,986 –> 00:00:07,724 Denoted with such an H.
4 00:00:08,771 –> 00:00:14,900 Seeing that double line here, you might already guess that this symbol stands for a special set.
5 00:00:15,643 –> 00:00:20,743 and indeed it’s a number set that extends the complex numbers C.
6 00:00:21,329 –> 00:00:28,914 It’s an extension in the sense that it’s a bigger number set, with almost the same rules as we have it in the complex numbers.
7 00:00:29,786 –> 00:00:36,920 In fact in H we lose one important calculation rule, namely that the multiplication is commutative.
8 00:00:37,757 –> 00:00:42,614 This means that if you multiply numbers in H the order matters now.
9 00:00:43,586 –> 00:00:49,076 However if you want to multiply numbers you need to know about which numbers we talk.
10 00:00:49,914 –> 00:00:54,067 and i can tell you a quaternion consists of 4 parts.
11 00:00:54,814 –> 00:00:57,851 So maybe lets call them a, b, c and d.
12 00:00:58,329 –> 00:01:00,923 and these are ordinary real numbers.
13 00:01:02,029 –> 00:01:07,238 Now as you might know it from the complex numbers, we can connect these parts with +
14 00:01:07,438 –> 00:01:10,429 and then introduce a new strange number “i”.
15 00:01:11,700 –> 00:01:15,874 This means that we have that i^2 is -1.
16 00:01:17,000 –> 00:01:20,301 However now you can see we have 2 parts left.
17 00:01:20,757 –> 00:01:25,596 Therefore we need to introduce two new numbers we can call j and k.
18 00:01:26,271 –> 00:01:31,243 and indeed j^2 and k^2 should also be -1.
19 00:01:32,371 –> 00:01:37,414 Moreover we also have the property that if we multiply all 3 together,
20 00:01:37,814 –> 00:01:39,788 So first “i”, then j, then k.
21 00:01:40,429 –> 00:01:42,154 We also get out -1.
22 00:01:43,171 –> 00:01:48,186 Ok and there you have all the rules we need to calculate with quaternions.
23 00:01:48,986 –> 00:01:53,121 Indeed this construction here is what we would call a quaternion.
24 00:01:54,129 –> 00:01:59,226 Now using these rules and some other basic assumptions, like associativity
25 00:01:59,426 –> 00:02:03,361 we can show that the multiplication is in fact not commutative.
26 00:02:04,200 –> 00:02:10,456 For example we get that “i” multiplied with j is - the other order.
27 00:02:11,043 –> 00:02:13,903 So j multiplied with “i”.
28 00:02:14,729 –> 00:02:21,861 Indeed if you combine “i” with k or j with k, you will get the same relation here with the - sign.
29 00:02:22,743 –> 00:02:27,997 So in general you can remember, if you multiply quaternions the order matters.
30 00:02:28,829 –> 00:02:37,658 and now to close this video i can tell you, the letter H for the number set is chosen to honor the mathematician William Rowan Hamilton.
31 00:02:38,629 –> 00:02:42,514 So i hope that this was helpful and that i see you next time.
32 00:02:42,614 –> 00:02:43,329 Bye!
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Quiz Content
Q1: What is the common form for a general quaternion $x \in \mathbb{H}$?
A1: $x = a + i b + j c + k d$
A2: $x = a + i b + j c $
A3: $x = a + i b $
Q2: Do the quaternions $\mathbb{H}$ contain the complex numbers $\mathbb{C}$?
A1: Yes, they are the quaternions $x = a + i b + j c + k d$ where $c = d = 0$.
A2: Yes, they are the quaternions $x = a + i b + j c + k d$ where $a = c = d = 0$.
A3: No, they don’t.
Q3: Do the quaternions $\mathbb{H}$ form a so-called field?
A1: No, because the multiplication is not commutative.
A2: Yes, they do like the real numbers and the complex numbers.
A3: No, because there is no addition for quaternions.
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Last update: 2024-11
