Information about Algebra - Part 1

1 00:00:00,271 –> 00:00:04,815 Hello and welcome to the video course called Algebra.

2 00:00:05,015 –> 00:00:12,252 This is a new video series here on the channel. Which nicely puts my other courses closer together.

3 00:00:12,800 –> 00:00:19,452 Of course you might know that algebra is a huge part of mathematics and it can mean a lot of different things.

4 00:00:19,652 –> 00:00:25,866 Therefore let me tell you that here we have an introduction to algebra in a video course.

5 00:00:26,271 –> 00:00:31,504 So you don’t need a lot of prerequisites here to understand the first videos.

6 00:00:31,704 –> 00:00:39,033 However since this is a very abstract topic, I would say this is not one of the easiest courses I have.

7 00:00:39,233 –> 00:00:46,293 Still you might know the topics we learn here are important and useful for other parts of mathematics.

8 00:00:46,771 –> 00:00:54,213 Ok and here you see the only prerequisite we have is that you know the stuff in my “start learning mathematics” series.

9 00:00:54,729 –> 00:00:59,516 In particular you should be familiar with the construction of the number sets,

10 00:01:00,000 –> 00:01:05,014 because that’s something we will immediately generalize in our algebra setting.

11 00:01:05,429 –> 00:01:13,417 Ok, here you also see my other basic video courses are not required and are more or less on the same level.

12 00:01:13,617 –> 00:01:22,140 However they are less abstract and therefore I would say it’s still helpful to have some knowledge of calculus and linear algebra.

13 00:01:22,500 –> 00:01:28,072 Indeed, this helps to understand why we even ask some questions in algebra,

14 00:01:28,700 –> 00:01:32,771 but of course your personal taste could be completely different than mine.

15 00:01:32,886 –> 00:01:39,068 Maybe you don’t like this concrete level in calculus, but really like this abstract level in algebra.

16 00:01:39,586 –> 00:01:44,541 In this case you did the correct choice watching this video series here

17 00:01:44,741 –> 00:01:50,322 and this is a good point in the video to thank all the nice people, that made this video course possible.

18 00:01:50,929 –> 00:01:59,354 As a supporter on Steady, on Patreon or here on YouTube, you make it possible that I can create a lot of math videos

19 00:02:00,000 –> 00:02:06,890 and you might already know, as a reward you get PDF versions, quizzes and exercises for all the videos.

20 00:02:07,090 –> 00:02:13,996 Ok, then the next question might be: “What can you expect of this video course about algebra, here?”

21 00:02:14,196 –> 00:02:19,108 and indeed this is easy to answer, because I have a nice summary.

22 00:02:19,308 –> 00:02:24,710 We will first talk about groups, then about rings and in the end about fields.

23 00:02:25,586 –> 00:02:28,943 In fact you might already know all these concepts here,

24 00:02:29,021 –> 00:02:35,614 because they were needed and defined when we constructed the number sets in the “start learning mathematics” series.

25 00:02:36,429 –> 00:02:42,657 A very important example of a group would be the integers together with the addition.

26 00:02:43,257 –> 00:02:50,579 So we need a set together with an operation and most crucially all the inverses have to lie in the set.

27 00:02:50,914 –> 00:02:57,155 In other words we can generalize this concept of the integers and talk about an abstract group

28 00:02:57,355 –> 00:03:05,491 and then it turns out that this new concept is more powerful and can do way more than just describing numbers.

29 00:03:05,691 –> 00:03:11,953 Now, on the other hand the integers are also a good example for a so called ring.

30 00:03:12,153 –> 00:03:19,434 There the integers need more structures. Namely besides the addition we also need the multiplication.

31 00:03:19,771 –> 00:03:23,709 Hence a ring is just a set together with two operations.

32 00:03:23,909 –> 00:03:28,722 Therefore for the one operation, we don’t need all the inverses in the set.

33 00:03:28,922 –> 00:03:33,958 In fact by having more inverses, we reach the concept of a field

34 00:03:34,429 –> 00:03:41,077 and there the typical example would be the real numbers with all the calculation rules you know.

35 00:03:41,277 –> 00:03:48,759 Therefore a field generalizes all these rich calculation rules we have for the real numbers in an abstract setting.

36 00:03:49,200 –> 00:03:58,665 Ok and at this point I should tell you that in this video course we will learn that all these abstract concepts have nice applications

37 00:03:59,029 –> 00:04:02,594 and they go beyond just calculating with numbers.

38 00:04:02,914 –> 00:04:10,557 For example if you want to solve a Rubik’s cube, it turns out that all the moves together form a group.

39 00:04:11,047 –> 00:04:15,869 The crucial part here is that every move you can do can be reversed

40 00:04:16,486 –> 00:04:22,182 and this guarantees that the Rubik’s cube can be solved, if you started with a solved one.

41 00:04:22,757 –> 00:04:29,515 However one interesting fact here I can tell you is that this group is not a commutative one.

42 00:04:30,157 –> 00:04:36,529 This means in contrast to adding integers, here the order of operations matters.

43 00:04:36,729 –> 00:04:41,871 Now, on the other hand we also get applications on a more abstract level.

44 00:04:42,071 –> 00:04:47,683 In particular this whole algebra concept tells us about solutions of equations.

45 00:04:48,329 –> 00:04:55,176 For example you might want to have the real or complex solutions of this equation.

46 00:04:55,829 –> 00:04:59,617 You see it’s a polynomial equation with degree 5

47 00:04:59,817 –> 00:05:06,925 and now with algebra we can show that we have at most 5 different solutions of this equation.

48 00:05:07,229 –> 00:05:17,002 However our abstract algebra concepts here also show that we can not have a general solution formula for these solutions.

49 00:05:17,371 –> 00:05:26,236 More precisely it means that the solutions can not be expressed by using the integers, roots and the basic calculation symbols.

50 00:05:26,436 –> 00:05:33,024 This is something that is indeed possible, if the degree here is less or equal than 4.

51 00:05:33,486 –> 00:05:40,539 However at degree 5 this breaks down, which algebra or more precisely Galois theory shows.

52 00:05:41,314 –> 00:05:47,971 Now, in order to check this interesting fact let’s put this particular equation into Wolfram Alpha

53 00:05:48,329 –> 00:05:55,405 and there we see Wolfram Alpha is not able to give more than just an approximation of the solutions

54 00:05:55,871 –> 00:06:04,508 and indeed we also see more. The graph of this polynomial is given here and we have only 3 real solutions.

55 00:06:05,143 –> 00:06:10,493 The other 2 of the possible 5 are now found in the complex domain.

56 00:06:10,857 –> 00:06:18,959 However, the result here is that for none of these complex solutions we have a formula that is expressed with roots.

57 00:06:19,543 –> 00:06:27,452 Now, at this point you might criticize me and tell me that in Wolfram Alpha you find this button “exact forms”,

58 00:06:28,143 –> 00:06:36,053 but after pressing it, we still don’t have a formula with just roots, because there are very complicated functions involved

59 00:06:36,329 –> 00:06:42,159 and at this point it might be helpful to compare this to an equation with degree 4.

60 00:06:42,929 –> 00:06:46,516 So let’s change the power to 4 and let’s see what we get.

61 00:06:47,414 –> 00:06:53,885 Again we just get approximations, but now let’s see what the exact forms are

62 00:06:54,371 –> 00:07:05,191 and there you see, now we can use integers, roots of higher degree and just the basic calculation operations to formulate this solution

63 00:07:05,391 –> 00:07:11,250 and indeed, this formula with roots works for each of the 4 solutions

64 00:07:12,043 –> 00:07:18,016 and the reason why we have this nice fact, the Galois theory will tell us later,

65 00:07:18,216 –> 00:07:22,764 but before we do that we will first start discussing groups.

66 00:07:23,129 –> 00:07:29,743 So let’s do that in the next video and I really hope that I see you there. Have a nice day and Bye.

Q1: Consider the polynomial equation $X^2 + p X + q = 0$ over the complex numbers. What is the correct solution formula?

A1: $$ x_{\pm} = -\frac{p}{2} \pm \sqrt{ \Big( \frac{p}{2} \Big)^2 - q } $$

A2: $$ x_{\pm} = \pm \frac{p}{2} - \sqrt{ \Big( \frac{p}{2} \Big)^2 - q } $$

A3: $$ x_{\pm} = \pm \frac{p}{2} - \sqrt{ \Big( \frac{p}{2} \Big)^2 - q^2 } $$

A4: $$ x_{\pm} = \frac{p}{2} \pm \sqrt{ \Big( \frac{p}{2} \Big)^2 + q^2 } $$

Q2: Consider the the cubic equation $ X^3 - 5 X^2 + 6X - 10 = 0$ over the real numbers. Is it possible that there is no solution at all?

A1: No, this is not possible.

A2: Yes, for this example, we don’t have any real solutions.

A3: One cannot say because it’s not possible to write down the solutions in this case.