
Title: Natural Numbers

Series: Advent of Mathematical Symbols

YouTubeTitle: Advent of Mathematical Symbols  Part 33  Natural Numbers

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Subtitle in English
1 00:00:00,429 –> 00:00:03,409 Hello and welcome back to the next mathematical symbol.
2 00:00:03,410 –> 00:00:05,945 Which is the set symbol N
3 00:00:06,145 –> 00:00:11,251 and throughout mathematics it always denotes the so called natural numbers.
4 00:00:11,671 –> 00:00:15,814 They are called natural, because you use them for counting.
5 00:00:16,400 –> 00:00:21,371 However, sadly there is no consensus with which number you should start counting.
6 00:00:21,782 –> 00:00:25,624 Therefore you usually find 2 competing definitions.
7 00:00:25,824 –> 00:00:30,571 On the one hand the one where we first count 1 object, then 2 and so on
8 00:00:31,343 –> 00:00:37,443 and on the other hand we have the definition where also no objects at all are represented with a 0.
9 00:00:37,929 –> 00:00:41,221 So there we have 0, 1, 2 and so on.
10 00:00:41,843 –> 00:00:49,030 Therefore if you start reading a mathematical book, you always should check and make sure which of both definitions is used.
11 00:00:49,514 –> 00:00:53,371 For example if the first notation for the natural numbers here is used,
12 00:00:53,510 –> 00:01:00,094 then also the second set is important and usually gets a new name. Like N with index 0.
13 00:01:00,900 –> 00:01:08,542 This happens, because the natural numbers including the 0 element represent a very important mathematical structure.
14 00:01:09,314 –> 00:01:11,890 Namely it is a so called monoid.
15 00:01:12,257 –> 00:01:16,365 More precisely it’s a monoid with respect to the addition.
16 00:01:16,565 –> 00:01:22,382 So there is an important fact. In the set of the natural numbers we know what the addition means.
17 00:01:22,971 –> 00:01:28,253 This I don’t have to tell you. I assume that you know what it means when you have to add numbers.
18 00:01:28,771 –> 00:01:32,786 However if you want an explanation for this, an axiomatic approach,
19 00:01:32,957 –> 00:01:37,094 I have a whole video course about it called “start learning number”.
20 00:01:37,657 –> 00:01:43,916 But most importantly after you know how the addition is defined, we find 2 properties for it.
21 00:01:44,500 –> 00:01:49,826 First we find it’s an associative operator when we add 3 numbers.
22 00:01:50,200 –> 00:01:58,968 So we have a,b and c as natural numbers and then we get the equality for these 2 expressions, where we just add the 3 numbers.
23 00:01:59,414 –> 00:02:04,920 So it means we can set parentheses as we want, without changing the result
24 00:02:05,457 –> 00:02:09,133 and now exactly this property is called associativity.
25 00:02:09,671 –> 00:02:13,758 Ok and this is the first part we have for a so called monoid
26 00:02:14,257 –> 00:02:17,882 and the second part is that a so called neutral element exist.
27 00:02:18,757 –> 00:02:25,824 Often it’s also called the identity element and it just means that it doesn’t change the element in the addition
28 00:02:26,400 –> 00:02:30,711 and now for our natural numbers here, this is of course the number 0.
29 00:02:31,343 –> 00:02:37,672 So “a” + 0 is “a” again and also from the other side. 0 + “a” is also “a”
30 00:02:37,871 –> 00:02:41,377 and this holds no matter which natural number “a” we choose
31 00:02:41,577 –> 00:02:46,935 and with this you now know the definition of the mathematical concept called a monoid
32 00:02:47,135 –> 00:02:53,362 and moreover you also know an important example. The natural numbers including the 0 element
33 00:02:53,814 –> 00:02:58,183 and now if you want to learn more about the natural numbers, you can watch my other videos
34 00:02:58,383 –> 00:03:02,054 or otherwise we meet in the next video about mathematical symbols.
35 00:03:02,254 –> 00:03:04,971 Have a nice day and bye!