1

00:00:00,060 --> 00:00:06,480

Once again, it's really difficult to see how this factor is related to the two initial factors, and



2

00:00:06,480 --> 00:00:09,090

we will find out now how they are related.



3

00:00:09,390 --> 00:00:12,000

Before we can do this, we need another trick.



4

00:00:12,600 --> 00:00:16,820

We need to first understand how we can calculate exponential growth.



5

00:00:17,490 --> 00:00:20,190

So exponential of complex numbers.



6

00:00:21,580 --> 00:00:28,960

And I'm not sure if you know this, but the exponential function is actually defined as a role, so



7

00:00:28,960 --> 00:00:32,860

it's not really clear how the exponential function is defined.



8

00:00:32,860 --> 00:00:40,160

And in very simple terms, you really have to consider this role, which is a sum of infinite terms.



9

00:00:40,450 --> 00:00:45,370

So you start by using this expression here and set an equal to zero.



10

00:00:45,370 --> 00:00:52,300

So you get here the number to the power of zero, which is one, and then you get to zero facultative,



11

00:00:52,300 --> 00:00:53,290

which is also one.



12

00:00:53,290 --> 00:00:59,560

So you get one and then you go to the next term where you use an equal one, then another one where



13

00:00:59,560 --> 00:01:04,510

you use an equal two and so on, and you add up an infinite number of terms.



14

00:01:05,380 --> 00:01:12,300

But due to this denominator here, these individual terms will decay and they will decrease to zero.



15

00:01:12,460 --> 00:01:19,330

So this means you add up an infinite number of terms, but the terms will get smaller and smaller.



16

00:01:19,570 --> 00:01:23,320

So the whole row converges to a finite value.



17

00:01:23,500 --> 00:01:26,390

And this is really the definition of the exponential function.



18

00:01:27,160 --> 00:01:31,420

And this is how it is defined in in real space for real numbers.



19

00:01:31,780 --> 00:01:36,910

But we can just use the same definition also for complex numbers as as is written down here.



20

00:01:38,350 --> 00:01:47,170

So let's use this for purely imaginary numbers, so instead of writing down X plus I y, I just used



21

00:01:47,170 --> 00:01:51,030

to I y this was just a special case for imaginary numbers.



22

00:01:51,040 --> 00:01:58,780

And this is actually all we need to understand how we can geometrically multiply and divide complex



23

00:01:58,780 --> 00:01:59,200

numbers.



24

00:02:00,100 --> 00:02:00,930

So let's do this.



25

00:02:00,970 --> 00:02:09,070

Let's use Iwai instead of Z so we get Y to the power of end and I to the power of an ant.



26

00:02:09,070 --> 00:02:15,520

Now just to get a feeling, let's write down the first six terms so we get one, as I already mentioned.



27

00:02:15,880 --> 00:02:25,600

Then we get y times I divide it by one faculty and then we get here minus we get first of all, why



28

00:02:25,600 --> 00:02:28,630

square and then we get here I square which is minus one.



29

00:02:28,810 --> 00:02:30,180

This is why we have this one here.



30

00:02:30,940 --> 00:02:31,900

So it's really simple.



31

00:02:31,900 --> 00:02:34,630

The only thing that you have to take care of this is I.



32

00:02:35,380 --> 00:02:42,970

And here we find that I to the power of three is AI times AI to the power of two which is minus AI.



33

00:02:43,970 --> 00:02:50,240

And then when we have eye to depart for, we have I squared Times Square, which is minus one times,



34

00:02:50,240 --> 00:02:52,080

minus one, which is plus one.



35

00:02:52,940 --> 00:02:57,790

So the exponential of this, I hear they go in circles.



36

00:02:58,520 --> 00:03:03,940

You start with one, then I and minus one minus I.



37

00:03:03,980 --> 00:03:11,240

And then again one this you can see here one eye minus one minus I, one eye and so on and so on.



38

00:03:12,980 --> 00:03:18,350

So now we have this terms and now we can all of them, we can order them by the real part and by the



39

00:03:18,350 --> 00:03:24,290

imaginary part, and you will see that every second term will be real and every second term will be



40

00:03:24,290 --> 00:03:24,950

imaginary.



41

00:03:25,910 --> 00:03:32,240

So here we have all the real parts, which are the terms for N equals zero for two, for four and so



42

00:03:32,240 --> 00:03:32,490

on.



43

00:03:32,510 --> 00:03:37,450

So for even numbers and here we have two imaginary parts for art.



44

00:03:37,580 --> 00:03:40,220

And so just for one, three, five and so on.



45

00:03:40,890 --> 00:03:47,630

And now our next step is to use these real parts and these imaginary parts and we will re express them



46

00:03:47,630 --> 00:03:54,520

in terms of rows, because as you can see here, we are still adding up an infinite number of terms



47

00:03:54,530 --> 00:03:55,690

and here as well.



48

00:03:56,240 --> 00:04:01,430

So we can once again write it as a row, which is a sum that goes to infinity.



49

00:04:03,110 --> 00:04:09,410

Now, here, since we have only even numbers in the exponent and in the denominator, we write down



50

00:04:09,410 --> 00:04:10,190

to end.



51

00:04:10,730 --> 00:04:14,510

And he has a prefect to account for this plus minus, plus minus.



52

00:04:14,510 --> 00:04:21,140

We write minus one to the power of end because this gives us exactly a positive sign, a negative sign



53

00:04:21,140 --> 00:04:22,000

and so on.



54

00:04:23,270 --> 00:04:25,700

And for the other row we have the same thing.



55

00:04:25,700 --> 00:04:31,670

But just instead of writing down to the end, we write down to and plus one so that we get the odd numbers.



56

00:04:33,650 --> 00:04:41,180

So this if you know about trigonometric functions, this is the real expression of the cosine function



57

00:04:41,180 --> 00:04:41,820

of Y.



58

00:04:42,560 --> 00:04:48,140

And this here is the sign expression or the the real expression of the sign of Y.



59

00:04:49,070 --> 00:04:58,490

So what we have here is that the exponent of Iwai is the cosine of Y plus the sign of Y times eye.



60

00:04:59,390 --> 00:05:02,510

And this is a very, very famous equation.



61

00:05:02,930 --> 00:05:10,160

This is called the Solar Equation, named after Oilor, which is the same guy after which the E constant



62

00:05:10,160 --> 00:05:12,560

is named this two point seven something.



63

00:05:13,880 --> 00:05:23,570

And it allows us essentially to express a complex number in terms of the length times some E to the



64

00:05:23,570 --> 00:05:24,450

I find.



65

00:05:25,260 --> 00:05:32,480

So here we know now that this each of the five corresponds to this term and we have here a real part



66

00:05:32,480 --> 00:05:33,760

and an imaginary part.



67

00:05:34,460 --> 00:05:38,240

And the ratio of these two parts can take any value.



68

00:05:38,540 --> 00:05:43,530

So the ratio of cosine divided by sine can take any value, positive and negative.



69

00:05:43,880 --> 00:05:49,490

So this means we can express any complex number in terms of such a relation here.



70

00:05:50,120 --> 00:05:56,690

And then we just have to scale the length because the absolute value of this expression here is always



71

00:05:56,690 --> 00:06:03,030

one, because the absolute value is cosine squared plus sine square, which is always one.



72

00:06:04,430 --> 00:06:06,010

So this is our main result here.



73

00:06:06,500 --> 00:06:15,830

We have expressed our Z, our complex number in terms of its length times, some exponential function



74

00:06:15,830 --> 00:06:18,950

with a purely imaginary argument.



75

00:06:20,300 --> 00:06:22,340

And I go I go one step back.



76

00:06:22,370 --> 00:06:24,200

So here you can see this angle.



77

00:06:24,200 --> 00:06:28,490

Phi is determined by the ratio of these two terms, as I mentioned.



78

00:06:28,790 --> 00:06:34,700

And the ratio of these two terms is the Arcus tangency function Y divided by X.



79

00:06:35,840 --> 00:06:38,240

And now I will show you this later.



80

00:06:38,240 --> 00:06:44,540

In an example, the across tangent function is not really well defined for defining this angle here



81

00:06:44,960 --> 00:06:52,670

because it gives the same value for, let's say, positive and the positive Y and X and also for a negative



82

00:06:53,630 --> 00:06:56,760

Y index because these two negative signs would cancel out.



83

00:06:57,380 --> 00:07:03,260

So by using just this across Tongans, we would get the same angle for these two examples, even though



84

00:07:03,470 --> 00:07:06,380

the real angle would be different by one hundred and eighty.



85

00:07:07,640 --> 00:07:09,590

So there is something missing here.



86

00:07:10,040 --> 00:07:17,270

And a better way to define this angle is to use this across tokens to function, which is often defined



87

00:07:17,460 --> 00:07:19,010

in programming.



88

00:07:19,310 --> 00:07:25,050

For example, if you use Python, you have to turn to function already available.



89

00:07:25,070 --> 00:07:26,370

You can just go ahead and use it.



90

00:07:27,590 --> 00:07:28,120

All right.



91

00:07:28,130 --> 00:07:31,820

So this is a very important result, as you will see now.



92

00:07:32,690 --> 00:07:37,310

So we are now able to express any complex number in terms of this relation here.



93

00:07:38,510 --> 00:07:43,240

Now we can revisit our multiplication from previously.



94

00:07:44,420 --> 00:07:48,410

So let us calculate once again the product of C1 and C2.



95

00:07:49,130 --> 00:07:56,390

So we have already done this previously using this standard equation here and we got the result minus



96

00:07:56,390 --> 00:07:57,680

five, minus five.



97

00:07:59,030 --> 00:08:06,550

So let's use our new Arless relation where we write down the one as the absolute value of C1 times this



98

00:08:07,140 --> 00:08:07,300

yeah.



99

00:08:07,430 --> 00:08:08,780

This complex E function.



100

00:08:09,260 --> 00:08:11,410

And for Z2 two we do the same thing.



101

00:08:11,420 --> 00:08:14,540

So we have now this relation here.



102

00:08:14,720 --> 00:08:24,020

So we have to do is we have to multiply the absolute values and we have to add the the angles five one



103

00:08:24,020 --> 00:08:24,710

and five two.



104

00:08:25,790 --> 00:08:30,950

So we must check now that this term here gives the same thing as previously.



105

00:08:30,960 --> 00:08:33,260

So we need to arrive at minus five, minus five.



106

00:08:34,640 --> 00:08:39,230

So of course, we need, first of all, the absolute values of both numbers.



107

00:08:40,070 --> 00:08:43,430

So the absolute value of C one is the square root of.



108

00:08:44,930 --> 00:08:51,260

The square of the real part, which is one square, which is one and square of the imaginary part,



109

00:08:51,260 --> 00:09:00,950

which is three square, which is nine, so we get a square root of ten for the two, we get a square



110

00:09:00,950 --> 00:09:08,480

root of the square of minus two, which is three apart, which is four plus the square of the imaginary



111

00:09:08,480 --> 00:09:11,420

part, which is one square, which is one.



112

00:09:11,780 --> 00:09:14,220

So we have to square root of five.



113

00:09:15,260 --> 00:09:22,070

So our product C one times two is the square root of the 50.



114

00:09:23,230 --> 00:09:30,340

And we can write 50 as the product of 25 times to and this is good because we can calculate the square



115

00:09:30,340 --> 00:09:32,160

root of 25, which is five.



116

00:09:32,530 --> 00:09:36,190

So we have a square root of two times five.



117

00:09:38,350 --> 00:09:40,750

OK, now we need the angles.



118

00:09:41,180 --> 00:09:45,520

This is something you can calculate when you take your calculator.



119

00:09:45,670 --> 00:09:47,770

You can of course, not do it by hand.



120

00:09:48,100 --> 00:09:51,250

It's it's difficult because you have Tsakos tangents.



121

00:09:51,850 --> 00:09:56,710

And here I will use the old relation that I had previously to show you where the problem is.



122

00:09:57,610 --> 00:10:03,670

So we take four or five one, we take Y one divided by X one.



123

00:10:04,660 --> 00:10:09,430

And so this is equal to Archos Tongans.



124

00:10:12,180 --> 00:10:23,970

Of three, divided by one, which is three, and if we put it in the calculator, we get a value of



125

00:10:23,970 --> 00:10:34,050

one point two for nine, which is equal to if you transform it to an angle, you have to multiply it



126

00:10:34,050 --> 00:10:37,320

by 180 degree and divide by PI.



127

00:10:37,810 --> 00:10:44,460

And what you get is seventy one point six degree.



128

00:10:44,820 --> 00:10:47,680

And that is, of course, the angle here for this blue vector.



129

00:10:49,080 --> 00:11:00,330

Now, four, five, two, we can do the same thing and this time we get our cost tangency of Y two divided



130

00:11:00,330 --> 00:11:01,710

by X two.



131

00:11:02,670 --> 00:11:13,910

So we get the Arcus tangents of um, one divided by minus two, which is minus one half.



132

00:11:15,480 --> 00:11:29,130

And here we get the value minus zero point four six four, which gives us an angle of minus twenty six



133

00:11:29,130 --> 00:11:30,690

point six degrees.



134

00:11:31,530 --> 00:11:37,950

And here you see, this is actually wrong because our angle, as we can see, is somewhere between 90



135

00:11:37,950 --> 00:11:39,570

degree and 180 degree.



136

00:11:40,020 --> 00:11:46,380

So we have to calculate the angle from the positive real access to this green vector and minus twenty



137

00:11:46,380 --> 00:11:48,340

six would be exactly the opposite.



138

00:11:48,990 --> 00:11:54,240

So this is the problem of the Yarkas tanginess function because it is not sine sensitive here for these



139

00:11:54,240 --> 00:11:55,020

two arguments.



140

00:11:55,680 --> 00:12:00,630

So if we would have used the acquis tannin's two, we would have gotten the correct result and we can



141

00:12:00,630 --> 00:12:05,660

get the correct result here by adding PI or by adding 180 degree.



142

00:12:06,240 --> 00:12:12,960

So the correct answer is two point six, seven, eight, which is this value here.



143

00:12:12,960 --> 00:12:22,920

But adding PI and this is equal to this, this corresponds to one hundred fifty three point four degrees.



144

00:12:24,870 --> 00:12:30,210

OK, so now we can just we already have already calculated the product of the absolute values of the



145

00:12:30,210 --> 00:12:30,970

One and Z two.



146

00:12:31,350 --> 00:12:36,300

Now we calculate the sum of five one and five two.



147

00:12:36,990 --> 00:12:40,670

So let's calculate five one plus five two.



148

00:12:41,820 --> 00:12:43,790

Um, this is equal to.



149

00:12:44,010 --> 00:12:47,060

So please you have to take the calculator.



150

00:12:47,850 --> 00:12:49,500

You shouldn't do it by hand.



151

00:12:49,800 --> 00:12:50,920

Would be very difficult.



152

00:12:51,540 --> 00:12:57,210

So you have here the, the, you know, the value and radials and.



153

00:12:59,010 --> 00:13:08,850

This corresponds to an angle of exactly 225 degree, if you would take here all the following numbers



154

00:13:08,850 --> 00:13:12,870

into account, and this is equal to.



155

00:13:14,350 --> 00:13:23,500

The stun guns or this tend to function of minus five, minus five, which is exactly what we wanted,



156

00:13:23,500 --> 00:13:26,370

because we have here minus five, minus five.



157

00:13:27,670 --> 00:13:39,610

So this means our numbers, the one Z two is equal to square root two times five times E to the power



158

00:13:39,880 --> 00:13:41,200

of EI.



159

00:13:41,530 --> 00:13:44,020

And let's just write it down in terms of angles.



160

00:13:44,290 --> 00:13:45,750

And I like this a bit more.



161

00:13:45,760 --> 00:13:46,440

It's a bit easier.



162

00:13:46,750 --> 00:13:53,860

And if you calculate this, you get minus five, minus five EI, which is exactly what we wanted.



163

00:13:54,730 --> 00:13:57,820

So now let's look at what happens here once again.



164

00:13:59,020 --> 00:14:01,750

So I take the laser pointer to show you.



165

00:14:03,040 --> 00:14:08,290

So here we have our two numbers, C one and Z two in blue and green.



166

00:14:08,590 --> 00:14:10,920

And the length is, of course, the absolute value.



167

00:14:11,110 --> 00:14:16,480

And then you have here these angles that have to be measured from the positive, real access to the



168

00:14:16,480 --> 00:14:17,260

vector itself.



169

00:14:17,660 --> 00:14:19,180

So you have five one five two.



170

00:14:19,870 --> 00:14:24,040

And you can see here, if you add these two angles up, you get this one.



171

00:14:25,000 --> 00:14:28,330

And for the length, you just have to multiply these length.



172

00:14:28,330 --> 00:14:31,780

So C one time, Z two and you get this vector.



173

00:14:32,810 --> 00:14:41,120

So now we finally have found out how we can really visualize multiplication of two reciprocal, two



174

00:14:41,120 --> 00:14:45,080

complex numbers, and for this we need the oil gas relation.



175

00:14:45,080 --> 00:14:50,990

We need to express the numbers in terms of the absolute value and then this argument here.



176

00:14:52,580 --> 00:14:58,760

So while this may look a bit difficult, a bit of a long calculation because of all of these Arkless



177

00:14:58,760 --> 00:15:03,760

tangents, it is actually really, really easy if you just do it.



178

00:15:04,640 --> 00:15:04,770

Yeah.



179

00:15:04,910 --> 00:15:06,620

By looking at these vectors.



180

00:15:07,130 --> 00:15:13,070

So if you would not have known what the product is and you don't want to calculate this, you can just



181

00:15:13,250 --> 00:15:18,170

draw these two vectors in the complex plane and roughly estimate the length.



182

00:15:18,180 --> 00:15:20,930

You say, OK, this is a bit longer than three.



183

00:15:20,940 --> 00:15:22,450

This is a bit longer than two.



184

00:15:22,910 --> 00:15:28,220

So yeah, we have here the product of these two, so it has to be a bit longer than six.



185

00:15:28,920 --> 00:15:34,060

OK, this is actually a bit even longer, but yeah, approximately six, maybe seven.



186

00:15:34,550 --> 00:15:40,850

And then here for the angle, you say, OK, this is maybe approximately 70 degrees.



187

00:15:41,310 --> 00:15:44,570

This is approximately I don't know, maybe.



188

00:15:45,140 --> 00:15:46,590

Yeah, 150.



189

00:15:46,940 --> 00:15:51,680

So you take to some one probably 220 degrees.



190

00:15:51,680 --> 00:15:52,370

So yeah.



191

00:15:52,630 --> 00:15:53,890

Goes along this direction.



192

00:15:54,200 --> 00:15:58,370

So you have already a very good feeling what the product of these two numbers will look like.



193

00:15:58,700 --> 00:16:02,450

And you see, if you really do it carefully, it turns out correct.



194

00:16:03,680 --> 00:16:06,400

Now we can do the same thing for the division.



195

00:16:06,980 --> 00:16:09,200

So here we already had our result.



196

00:16:09,210 --> 00:16:13,010

It was one of five minus seven over five times I.



197

00:16:13,460 --> 00:16:21,560

And here we again can use the only representation where we express our number as the absolute value



198

00:16:21,590 --> 00:16:23,660

times this complex function.



199

00:16:24,350 --> 00:16:31,490

So this time we need to calculate the quotient of those two lengths and we need to calculate the difference



200

00:16:31,490 --> 00:16:32,600

in these two vectors.



201

00:16:33,800 --> 00:16:38,350

So first, let's start with calculating the quotient of the two length.



202

00:16:38,660 --> 00:16:44,870

So we have C1 divided by Z to see one more square root of ten.



203

00:16:45,290 --> 00:16:48,670

This one was the square root of five.



204

00:16:49,070 --> 00:16:58,850

So we get the square root of two as our length and for the angles we have five one minus five two and



205

00:16:58,850 --> 00:17:14,990

this is equal to minus one point four to nine, which corresponds to an angle of three minus eighty



206

00:17:15,000 --> 00:17:18,310

one times nine degree.



207

00:17:18,530 --> 00:17:26,330

So it looks a bit ugly, minus eighty one point nine degree, or this is also the same as two hundred



208

00:17:26,330 --> 00:17:28,930

seventy eight point one degree.



209

00:17:30,590 --> 00:17:37,790

And this one is equal to the Archos tangent to function of.



210

00:17:38,980 --> 00:17:44,980

Minus seven over five and one over five.



211

00:17:46,510 --> 00:17:53,590

And now if we if we if we write down our quotients, the one divided by Z two, we get a square root



212

00:17:53,590 --> 00:18:03,870

of two three to the power of IE two hundred seventy eight point one degree, which is exactly one of



213

00:18:03,880 --> 00:18:09,190

a five minus seven over five high, which is what we want.



214

00:18:10,210 --> 00:18:15,250

So now we can yeah, we can do the same argumentation as before.



215

00:18:15,610 --> 00:18:21,640

I will once again switch to the laser pointer and here we can see that we have these two A length C



216

00:18:21,640 --> 00:18:23,650

one, zoo two and these two angles.



217

00:18:24,250 --> 00:18:27,450

And what we have to do is we have to subtract the angles.



218

00:18:27,460 --> 00:18:35,830

So you see, since FI two is larger than five one, we get into negative territories and since these



219

00:18:35,830 --> 00:18:43,420

have almost the same length, but this one is a bit larger, we get the length that is just a bit larger



220

00:18:43,420 --> 00:18:43,960

than one.



221

00:18:44,350 --> 00:18:47,680

But in fact, it's a square root of two, which is one point for approximately.



222

00:18:48,610 --> 00:18:52,060

So once again, you do not have to do this difficult looking calculation.



223

00:18:52,060 --> 00:18:55,240

It would probably be easier to just use this calculation here.



224

00:18:55,720 --> 00:18:58,200

But as you can see, this is also correct.



225

00:18:58,210 --> 00:19:00,700

And this allows us to just.



226

00:19:01,480 --> 00:19:07,480

Yeah, make these considerations so you can really estimate what the quotient of these two numbers will



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look like without having to actually calculate it.



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Because I think that's really, really nice, you have now learned what complex numbers are, we have



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understood how we can do basic mathematical operations and we have learned about this very important



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Euler's relation and how you can multiply and divide two numbers in a graphical way.



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So this is something that would not have been possible by using vectors.



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So now we are good to go to use complex numbers in our project that follows up.