 I am Zor. Welcome to Unizor education. Today's topic will be complex numbers. This is a continuation of our saga about numbers. We started from natural numbers, added negative ones, then added fractions to get the rational and real numbers came to story. And right now we are still expanding our universe of numbers. But this is, by the way, the last expansion. This is the frontier behind which we do not really go at all. So how would that saga about numbers unraveled? If you remember, first we were dealing with natural numbers only, like one, two, three. And then they added negative numbers, but actually first they added zero and then negative numbers to make it a group, something which is a beautiful harmonious and closed in itself set of objects which we can operate upon. So the operation of addition among integer numbers is completely closed. Any two numbers can be added together and you can get another integer number. And for every number there is a reverse number which will give zero if added together. What it actually means is that any equation which is related to addition, something like a plus x equals b where a, x and b are any integer numbers, it always has a solution. What we have to do is we have to reverse a, that's number one, add it to both parts of the equation because we know that equation will still help if we add the same number. So it will be minus a plus a plus x equals two minus a plus b. This is obviously zero because that's the definition of minus a and zero plus x is x and that's the definition of zero actually, right? So zero plus x equals minus a plus b and so this is definition of zero, it's x. Using commutative law of addition we can say that this is b plus minus a and then we basically define the operation of subtraction to subtract b from a actually means adding a reverse to a. So this is by definition of subtraction b minus a. So this is always true for any integer number so this equation always has a solution. This is very important because if we don't add negative numbers to our set of natural numbers we will not be able to solve something like five plus x equals three, right? Okay, so the ability to solve equations within certain set of numbers is very important. Actually, that's why we have added rational numbers for instance. Well, rational numbers were added because the multiplication wasn't always solvable. If we will multiply x by a to get b for certain integer a and b it's solvable. It's something like x times three equals six has a root two. But for others it's not solvable in the set of integer numbers. So we have to introduce rational numbers and x would be five over three a new number, rational numbers to make this equation possible. We continue this theme of solving ability to solve equation as the necessity to add new numbers. So the next step was if you remember square root of two. We did not have a rational number which was a square root of two. We do not have a rational number p over q which being squared gives us two. There is no such thing as a rational number p over q where p and q are integers. So that's why we have introduced a brand new rational number square root of two and many others. Now, there are still certain equations within the area within the set of real numbers which cannot be solved. And the primary one of them is that there is no such thing as a square root of a negative number. Why? Well, obviously because if the rational number a then being squared it should be equal to minus one. But we know that any rational number positive or negative if squared will always give positive number. So this is not solvable among the real numbers. Same thing as always, what mathematicians do in this case, they invent new numbers. Okay, fine. So this is our definition of a new number. Traditionally, it's denoted as a lower case i. It stands for imaginary. It's an opposition to real numbers. Now we have imaginary numbers. So i is an imaginary number and the definition is exactly what we could not do before. This is a number being squared gives minus one. Or if you wish, which is the same thing is square root of minus one. That's the same thing. Now, we can introduce a new number and now we can solve the equation x square equals minus one. We know that the solution is x equals five because we have defined it this way. There's nothing to it. We didn't solve this basically, we just defined it. This is a number which gives the solution of this equation. Alright, fine. How about the rest of the things and what can we do with this number? Well, if you remember any set, any good set of numbers is supposed to satisfy certain rules, axioms, whatever. Now, for instance, addition has two very, very important rules. First is commutative and another is associative or which is the same thing. So the order of addition first to the second and then the result of the third or first we will do second and third and then add the first to the result of this operation is insignificant and the commutative means we can add in any order. So this is true and similar law exists for multiplication, again the same thing, commutative for multiplication and associative, exactly the same thing. So when we introduce a new number, we would like our universe of numbers to be expanded in such a way that these laws are preserved because these laws are basically the foundation of any kind of equation solving, any kind of calculations which we do, etc., etc. So no matter how fantastic numbers we can really come up with, we still have to be within certain boundaries, within certain framework of these laws. So by introducing a new number i, I really have to be able to expand the universe a little bit further than just this because for instance, if I introduced a new number, I have to be able to know what is something like this or what is this or what is to multiply by i. So all these numbers have to be defined as well because i is something which is completely artificial, the only thing which we define is i squared basically. So everything else, all other operations, we really have to define somehow. Well, how can we define it? Well, the simplest way which people have decided to do is the following. We will define once and for all this number. So multiply i by any number and the result at any number where a and b are real numbers. So these are real. So for any two a and b, for any two pairs of real numbers, we will define this operation. Well, how can we define it? Well, easily. If this is a number then it should satisfy certain laws. So basically what I'm saying is this is a new number and these are the laws which I have to really define about this number. I have to be able to add two such numbers to get the number of the same set. I don't know how to multiply, how to add or how to add these two together. I don't know that at all. All I'm saying is I'm just defining this new number by basically defining how it operates on a similar number. And this is how a small mistake is a d. So if I have two numbers of that complex form, by the way, this is a general form of complex number, I have to define an operation of addition using just exactly this. So to add two complex numbers where we have an imaginary part and the real part, I have to use the coefficient one and coefficient another. Again, I made another mistake. I'm sorry. This is c. I add together imaginary parts and add together real parts. Okay. So imaginary to imaginary a to c and b to d. Okay. So I know how to add two real numbers. So basically for a general representation of any complex number, which is this, using this axiom, I know how to add them together. Now why these actually are good definitions? Because since addition is commutative and associative among real numbers, let's see if it will be held for a complex number. Well, very easily. Let's say commutative. a times i plus d. One complex number. c times i plus d is equal to, as I said, a plus c. i plus d plus d. Now, let's do it in reverse. c i plus d and a i plus b. I just reversed the sequence. Well, according to definition of addition among the complex numbers, it should be c plus a, right? Times i and d plus b. Imaginary to imaginary, real to real. Now, but we know that a plus c and c plus a are equal to each other because among real numbers, a and c are real numbers. All these coefficients are real numbers. It's only i which is imaginary. So among real numbers, commutative law is held. Same thing here. So basically these are equal. Since these are equal, we have exactly the same numbers as a result of this addition. Now, let's go over to multiplication. We will do exactly the same thing. We will define an operation on two different complex numbers. I don't want to write this answer right now. I would like to think about how it should look like. Obviously, the operation of multiplication should also satisfy commutative and associative laws. So I really have to define it in a reasonable manner. Let's assume that i is not an imaginary number which we don't know how to multiply or add or anything like that. What if a is just some number, real number? Well, in which case we can open the parentheses to my member and what we will get. Let's just write down what we will get. First component to first component will be a times i times c times i. Well, second component. First and second. a times i times d plus b times c times i and b times d. So I'm using the rules which I really not supposed to, but I'm using them to derive with some definition which will be reasonable and these rules therefore will be held for that definition. Now, let's think about what we got. Again, multiplication should be commutative. So whatever transformations I will do I really should think about the preservation of commutative and associative laws. So this can be multiplied in different order and I will have a times c times i squared. Now this can be, again, change the order, a d times i and i can be factored out and I will have a g plus b c times i and then b g remains the same. What is it equal? Now you remember that i squared was defined as minus one. That's the definition of i. i is the square root of minus one, so i squared is minus one. So this thing becomes minus a c plus the same thing and the whole thing becomes equal to b g minus a c plus a g plus b c i. Okay, now what I do I wipe out this and say that by definition this is equal to this. So it's a d plus b c i b d minus a c. So this is a definition of multiplication. If I define it this way, then as we will make sure and it definitely will be true all the different laws about multiplication will be preserved. Just as an illustration, let's do, for instance, let's check that it's a commutative. Well, very easy. Let's do c i plus d multiply by a i plus b. Using this, we have to basically substitute c for i and d for b and reverse. So it will be c b plus d a substituted a for c and d for b. b for d and c for a. Same thing here. b will be changed to d, d will be changed to b, c, a. Now, but these two expressions are exactly the same, right? It's just different order, but among real numbers this is exactly equal because b c is equal to c b because multiplication is commutative and then addition is also commutative. So these are the same and these are the same because of the same reason. So we have come up with exactly the same number. Using just this definition. I have not done any calculations. I'm just using this as a definition of the multiplication among two complex numbers. So there is a very interesting circumstance here which I would like you to pay attention to. This is a multiplication, operation of multiplication. These, this, that and this plus and this, that and this plus are really not because we don't know how to multiply i by a or how to add i, a to b. This is not defined yet. All I'm saying is that this is generalized notation for a complex number. So probably, what probably would be even mathematically more strict I would say is to say this is just a conditional expression for some operation which will look like addition among real numbers. And this also will be some kind of a sign, if you wish, which actually will mean exactly the same thing as multiplication. And let me explain why. So being equipped with these definitions of multiplication and addition, we can actually say that this I will use dot in the circle and plus in the circle is a generalized notation for a complex number. But let me just tell you why these multiplication and addition signs in circles are really like the real ones, like the real multiplication and addition. Let's start with addition. For obvious reason, any rational number b I can represent as 0 times a plus b in this complex notation. Why? Because this is 0 and 0 basically nullifies the imaginary part. So I have only this number which is retained. So I can say that if operations upon b are basically in sync with operations on this complex number, then basically b is represented as the complex number with imaginary part equal to 0. Well, and actually it actually is because if we will take another number, let's say d, and I will also represent it as this. Now, for obvious reason I would expect that b plus g would be represented as this. This is the real addition. But let's check whether it's true or false. Remember how our complex numbers were defined as far as operation of addition. You remember that if I have this then the operation of addition is actually this. Using this definition, we can say that a and c are 0 in this case. So a plus c will be 0 as well. So as we see, this general representation of a real number as 0 dot in a circle which is kind of a pseudo-multiplication. 0 is pseudo-multiplied by i plus pseudo-plus b. Now this representation is really held for any real number and the laws of addition, let's say, exactly result in the same thing. As sum of two real numbers will be, according to this rule, will be again a complex number with 0 pseudo-multiplied by i. And the real part will be still b plus d. So using this notation, my real numbers are preserved with all their laws, commutative, associative. So we can basically do any kind of calculation which we used to do with real numbers by themselves. We can do these calculations using their complex representation. Now, how purely complex number, let's say i, should be represented in the generalized notation? Well, obviously, i is represented as this. As 1 pseudo-multiplied by i plus 0. So a will be equal to 1 and b is equal to 0. Now, let's see what happens. This is an imaginary number now. And now the question is how is my multiplication, for instance? Well, let's just multiply i by i what happens. Let's use our law of multiplication and I will write here again a i plus b real multiplication not by c i plus d is equal to b g minus a c pseudo-multiplied by i no, sorry this is a real part. b d minus a c, this is a real part and the imaginary part would be a times d plus b times c pseudo-multiplied by i. Now, using this rule and using this representation for i let's see what happens if I multiply i by i. So a equals 1, b equals 0 c equals 1 g equals 0. Well, if I substitute this into this let's see b times d is 0 a times c is 1 with a minus sign. So i times i is i times i is minus 1 that's a c. Now, a d a g is 0 b times c is also 0 so we have 0 0 pseudo-multiplied by i. This is a generalized notation for any real number. So this is basically minus 1. i times i which is i squared is minus 1. We came to a definition of i squared. So everything seems to be dancing together. Using these definitions for addition and multiplication it's actually working among the complex numbers and all the laws associated for preserved and we still are within the framework of the complex numbers. If you multiply one complex number by another complex number we get another complex number. So the universe of complex numbers again is closed. It has all the operations and everything seems to be okay and now what's important is we can expand it our universe of numbers with this ability to solve certain equations which we were not able to solve before. What kind of equations? This is the equation. This is not solvable among the real numbers but among complex numbers I can say the solution is x equals y where y is just imaginary number. Now one more little thing about to the multiplication and to the addition as I was using them before. I said that the generalized form of complex number is this I have basically invented new number, new signs for multiplication and addition because I could not really multiply real number by imaginary or add to imaginary numbers. However, I can definitely say that let's talk about multiplication first. Let's talk about addition first. I can definitely say that this is a real plus. Why? Because if I will add this number complex number and this number also represented as a complex you will see that I will get this thing which is the same as the real plus. Look at this. A times I image pseudo multiplied by I, sorry plus 0 that is basically the same thing as this. D can be represented as 0 pseudo multiplied by I pseudo plus B. This is this. If I will add them together sorry if I will add them together according to the rules of addition among complex numbers what will I get? Well, obviously this is the real addition. Obviously I will get this. Imaginary parts are added together so it will be A plus 0 pseudo multiplied by I and real parts are added together which is now this is the real plus and this is the real plus so I can use my real numbers arithmetic. So this is A times I plus B. Well, we come basically to exactly the same thing. Right? So by really adding these two numbers together really adding them together I should really use circle here. So plus is a real addition addition which we have defined for complex numbers we basically come up with the same result which means that if we use this notation or we use this notation it's exactly the same thing because this represented separately in a complex way as this and added according to the complex arithmetic will give us exactly the same thing. So this plus, pseudo plus and the real plus are exactly the same. Now let's talk about multiplication as you expect to get exactly the same thing. So the result of multiplication what I'm saying is that the multiplication among the complex numbers is exactly equivalent to pseudo multiplication which we used for this type of thing. Let's think about what this actually is. This is a number complex number which can be actually expressed as zero pseudo plus zero, right? So I would like to prove that this is a real multiplication. How can I prove it? Very easily. If I start it with this number with this number let me do it differently. Let me start with number A and number I and multiply, really multiply them together according to the rules of complex arithmetic. So I is actually in the complex form would be zero pseudo multiplied by I plus A and I is one pseudo multiplied by Y plus zero, right? Now if I will multiply them together well, if you remember I should probably put that formula again for the let's use X I plus Y pseudo plus multiply U I plus V So according to the definition it will be imaginary part would be Y times U plus X times V pseudo multiplied by I and the real part would be Y V minus X U Okay? Now using this formula so this is X this is Y this is U and this is V Y times U Y times U it's A times one so it's A X plus V X times V, X times V that's zero pseudo multiplied by I sorry, this is the definition so I should use the pseudo sign here pseudo plus Y times V Y times V is zero X times U is zero so basically zero which is, as you see is what we have started this so really multiplying A times I in their full complex representation we got exactly the same number which we used before so the real multiplication so A times A pseudo multiplied by I is basically the same thing as if you really multiply A by I so both plus and minus in their circular representation actually mean exactly the same thing as without the circle basically the same sign as before and say that my generalized complex form is this where A and B are real and I is this well that's the introduction to complex numbers what was the reason to invent these numbers the reason was the most important reason was that we cannot solve this equation among the real numbers and we wanted to not only this actually any equation which is related to polynomial has certain solutions roots and there are many polynomials basically one of them which do not have a solution so if you have any polynomial something like 2x cubed plus 2x square minus 5 x plus 7 equals to 0 if you have any kind of polynomial equation it's not always that we can find the roots of this if we are dealing with these real numbers only introduction of complex numbers allows to actually find roots for any kind of a polynomial and what's interesting is that the number of roots is very much closely related to the degree of this polynomial but that's a completely different story anyway we have expanded our universe of the real numbers towards complex numbers we know how to write them down we know how to multiply them how to add them basically that's the beginning and you probably would find very interesting to solve certain problems which will be accompanying this lecture and just one of the properties maybe just to finish it up in a good note let's just think about what will be if you will multiply three times i multiplied three times to get i cubed well again we all know that we have defined our new numbers in such a way that all the laws are supposed to be preserved right so this is i times i times i and since we are preserving the associative law it means i times i in parenthesis times i which is i squared so it's minus one times i or just minus i but that's just one of the examples of complex arithmetic by the way i to fourth degree obviously will be i squared times i squared which is one minus one times minus one etc so we can multiply and add any complex numbers without any problems we can use the square root of minus one well anything possible right now so in this universe of complex numbers which we have expanded our universe of real numbers to all kinds of equations as i was saying are possible to solve and that's probably a good point where i would like to stop here and there will be certain interesting problems as i said and i believe that these problems will will dig further into the depths of the complex numbers complex numbers are very interesting in some way they are much more i would say harmonious even than complex numbers there are many operations which are not possible among the real numbers which mathematicians like to have and complex numbers really expand the universe towards it the last note which i would like to make about complex numbers is their geometrical representation if you remember we can always have a straight line put zero somewhere have some kind of scale complex one and using that scale we can basically place integer numbers and in between all rational and irrational numbers etc so all the real numbers are represented on the straight line how complex numbers can be geometrically represented well the way how people usually do it is the following let's have two perpendicular lines one for real part of the complex number and another will be for imaginary part so this is B and this is A and this point on the plane now so we have come up from the line where only real numbers are represented to the plane where complex numbers are represented and now every point on the plane represented by certain coordinates down to real axis you will have the real part which is B and to the left or to the right projection on the vertical axis will give you the imaginary part so you can say that this point or this vector if you wish represent represents the complex number and what's interesting is and this is related to vector algebra you can actually prove that addition of complex numbers and additions of vectors are exactly the same and this geometrical representation is really much deeper than just placing a dot on the plane but this is again a completely different story we will go into vector algebra and some other lecture thank you very much that's it for today