• Title: Integers

• Bright video: https://youtu.be/ZjhoVgkK3hw

• Dark video: https://youtu.be/eXKQrjLqajM

• Timestamps
• Subtitle in English

1 00:00:00,429 –> 00:00:06,156 Hello and welcome back to the mathematical symbol which is given by the set letter Z

2 00:00:06,543 –> 00:00:11,963 and this symbol represents the integers, also sometimes called the whole numbers.

3 00:00:12,486 –> 00:00:17,603 It means it’s a set of numbers that extends the set of the natural numbers.

4 00:00:18,057 –> 00:00:23,969 Namely by also including the negative numbers, like -2, -1 and so on

5 00:00:24,169 –> 00:00:30,305 and important to remember is that we have all the natural numbers including 0 in this set.

6 00:00:30,505 –> 00:00:36,622 Now, indeed this object Z forms a common example for a very important mathematical concept.

7 00:00:37,143 –> 00:00:41,371 It’s essential throughout mathematics and it’s the so called “group”.

8 00:00:41,763 –> 00:00:47,386 However, here it’s important to know that a set can only be a group if it has an additional structure.

9 00:00:47,529 –> 00:00:49,958 Here given by the addition, +

10 00:00:50,158 –> 00:00:57,478 and then if we have this additional operation together with 3 properties we can say that we have a group.

11 00:00:57,678 –> 00:01:01,271 Now the first property is that the operation is associative.

12 00:01:01,471 –> 00:01:09,640 Which means if we add up three elements, then we can set the parentheses as we want without changing the result.

13 00:01:10,057 –> 00:01:17,758 So again here you see a, b, c are integers and this is the formulation of the associativity rule.

14 00:01:17,958 –> 00:01:24,714 Ok and then the next property is that we have an identity element. Also often called a neutral element

15 00:01:25,100 –> 00:01:30,865 and it just means that we have one element that does not change any element while using the addition

16 00:01:31,329 –> 00:01:36,056 and this you already know. It’s the number 0 that fulfills this property.

17 00:01:36,871 –> 00:01:43,973 Ok and then finally, the last part is that for each element in a set, there exists a so called inverse element.

18 00:01:44,500 –> 00:01:50,098 More precisely it means if you add both elements, you get out the neutral element.

19 00:01:50,543 –> 00:01:56,718 So it reads like: " a + (-a) or (-a) + a, gives us 0".

20 00:01:57,300 –> 00:02:02,438 So what you should see here is, the inverse element of 2 is simply -2

21 00:02:02,638 –> 00:02:07,238 and on the other hand the inverse element of -1 is +1.

22 00:02:07,438 –> 00:02:13,034 Moreover you also see that 0 has an inverse element, it’s simply itself

23 00:02:13,234 –> 00:02:19,007 and indeed this is what we need here. All elements in a set must have an inverse element

24 00:02:19,571 –> 00:02:23,206 and only then we can call the whole thing a group.

25 00:02:23,371 –> 00:02:28,171 Now, if you want to know more about how to construct this group of the integers,