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Title: Factorial
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Series: Advent of Mathematical Symbols
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YouTube-Title: Factorial
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Bright video: https://youtu.be/Y4yZaEuoThs
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Dark video: https://youtu.be/rogCR0ZJrTU
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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: aoms04_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,500 –> 00:00:03,788 The mathematical symbol of today is the factorial.
2 00:00:03,988 –> 00:00:06,059 Which is written with an exclamation mark.
3 00:00:07,157 –> 00:00:10,136 and usually you see the definition for a natural number n.
4 00:00:11,357 –> 00:00:18,243 This is because this n with an exclamation mark, represents a product with exactly n factors.
5 00:00:18,929 –> 00:00:23,626 We start with the number n itself. Then we reduce the number by 1
6 00:00:24,457 –> 00:00:29,016 and then we continue this whole procedure until we reach the number 1.
7 00:00:29,929 –> 00:00:36,762 This means when you see this definition, you know we need a positive integer n such that this makes sense.
8 00:00:37,714 –> 00:00:40,471 For example when you see 4!
9 00:00:40,571 –> 00:00:45,343 This just means you have 4 times 3 times 2 times 1.
10 00:00:46,214 –> 00:00:49,068 Which is, as one can remember 24.
11 00:00:50,486 –> 00:00:54,770 also you should remember 1! is just 1.
12 00:00:55,614 –> 00:01:01,282 Now what you see here with the factorial is what in mathematics we call a recursive definition.
13 00:01:02,000 –> 00:01:07,228 This means that we first fix a starting value and indeed we can do this with 0.
14 00:01:08,200 –> 00:01:11,287 Here 0! should be defined as 1.
15 00:01:12,500 –> 00:01:17,443 This makes sense for a lot of formulas, so this is indeed something you really should remember.
16 00:01:18,343 –> 00:01:24,977 and then the next step in the recursive definition is saying what happens with the successor of a given natural number.
17 00:01:25,714 –> 00:01:30,284 So n! should be n times (n-1)!
18 00:01:30,943 –> 00:01:36,378 So when we know what the factorial does to (n-1), we also know what it does to n.
19 00:01:37,157 –> 00:01:43,212 and because we have a starting value, we now could calculate the values for all other natural numbers n.
20 00:01:44,229 –> 00:01:50,069 Ok, this is the definition for the factorial. Which you can apply for natural numbers n.
21 00:01:50,886 –> 00:01:53,468 Thanks for listening and see you next time.
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Quiz Content
Q1: You have 10 different books in front of you and a bookshelf where exactly three of them fit in. How many possibilities do you have if you put all books in and the order in the bookshelf matters?
A1: $10 \cdot 9 \cdot 8$
A2: $10!$
A3: $90$
A4: $10!/3!$
Q2: You have 10 different books in front of you and a bookshelf where exactly these 10 fit in. How many possibilities do you have if you put all books in and the order in the bookshelf matters?
A1: $10 \cdot 9 \cdot 8$
A2: $10!$
A3: $90$
A4: $10!/3!$
Q3: What is the value of $4!$ if you use the formula from video?
A1: $24$
A2: $12$
A3: $4 \cdot 3 \cdot 1 $
A4: $50$
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Last update: 2024-11
