 I am Zor. Welcome to you, Zor Education. Today, we will talk about equations and how to solve them using invariant transformations. First of all, what is an equation? Well, everybody now, something like x squared equals to 1 represents an equation. Right, that's absolutely correct. But I would like to generalize this concept and basically talk about equations in general, what are equations in general. But look at this one. On the left side, you have some function, basically. x squared represents a function. So you can say that on the left side of the equation, we always have some kind of function. Now, what is a function? Function is some kind of a combination of domain where arguments are taken from, co-domain, where the function takes value, and some kind of a rule which explains how to get from the domain to the corresponding variable. So basically, these arrows represent the rules. So if you take this element of the domain, the function transforms it into this element of a co-domain, from this to this, et cetera. That's the concept of function, a domain, co-domain, and certain set of rules. So in this particular case, we usually are talking about some algebraic expression of the function, which basically assumes that domain and co-domain are numerical. Now, usually, the most frequently, I would say, case in algebra is when the function is considered to be defined on the domain and co-domain real numbers. Sometimes it's complex numbers. Sometimes equation is defined on a set of integer numbers. Actually, one of the most famous equations in the history of mathematics, which presented by Thames's French mathematician Fermat, was actually an equation in integer numbers, a great theorem of Fermat. But anyway, usually in school, in algebra, people are considering most often the equations where both domain and co-domain are real numbers, and f of x is represented by some kind of algebraic expression on x-ray. So that's on the left. On the right, since function takes values in the area of co-domain, and we're talking about ecology, then basically the element which is supposed to be on the right is an element of the co-domain. Well, basically, that's what the equation is. It's the algebraic expression of function defined on certain numerical domain, usually real or complex numbers. And it takes values in corresponding real or complex numerical numbers. And on the right, we have one particular element from the co-domain. And that's the ecology between them, and it represents an equation. So what is an equation? We understand. Next is basically what is the solution of this equation, because the purpose of the equation is basically to solve it. So what is a solution? Well, let's talk about this graphically again. If this is an element B, which I'm talking about right now, then to solve equation means to find all elements from the domain which are mapped by this function into this element B. Let's say it's A1 and A2. By the way, it's not necessarily that we have any solution to the equation in this domain. Maybe there is none. So there are certain values in the co-domain which are not values of this particular function defined in a particular argument. Another case is when we have only one element of the domain which is transformed by the function into this element B. Or we can many. Let me exemplify it. This particular equation, x where it goes 1, has two solutions in the area of complex number, in the area of real numbers even, plus and minus 1, because 1 squared is 1 and minus 1 squared is 1. So we have two solutions in the area of real numbers. Now, if I will tell you that I would like to solve this equation in the area of positive numbers, then I have only one solution, only one, because minus 1 doesn't belong to my domain. Function was defined originally in a set of positive numbers only, so there are no solutions. Another example would be if I have an equation of x squared equals to minus 1, which obviously has no solutions in the area of real numbers, but it has two solutions among the complex numbers, plus i and minus i, both squared gives you minus 1. So it's very important from the very beginning that we are talking about equation to talk about the function and how it's defined. This particular function on the left should be completely defined, which means there is a domain and there is a codomain and there are rules. In this particular case, if we are saying that the domain is real, no solutions. If we are saying that the domain is complex, set of complex numbers, we have two solutions. OK, done with that. So we know what is an equation. It's an algebraic expression of the equality between the certain function and certain element of its codomain. We know what the solution is. That's the 0, 1, or many different values of the domain which are transferred into this particular element of the codomain. So now, probably the most important question is how to solve equations. And I'm sure you all know that there are tons of different ways to solve different equations. There are certain techniques. And the goal which I'm trying to achieve here is to generalize the process of solving equation. And this generalization I would like to present using the concept of invariant transformations. All right, so let's think about what is invariant transformation. And let me start from the example. If you have something like x plus 10 equals 13, and I would like to solve this particular equation, let's talk for definiteness that we are talking about real numbers. Well, again, I'm sure everybody knows that what we can do is we can subtract 10 from both sides of the equation, and we will get basically x equals 3. Wow, but what did we do, actually? And that's what's very important. I said we subtracted 10 from both sides of the equation. Well, we transformed one equation, original one, into a new one, which is simpler. So this transformation of the equation is the basis which I'm just trying to talk about when I'm talking about solutions general solutions to the equations. We are transforming equations into a different type, which might be simpler, like in this particular case, and deliver solution. So that's what transformation is. We are transforming original equation into a different one. Now, how can it be generalized? OK, let's consider that, again, we have this domain, codemain, and the function f. This is domain. This is codemain. Now, in this case, let's consider that both of them are real numbers. Again, it doesn't really matter. So what we did, we applied the same transformation to both sides. Now, both sides are elements of the codemain, right? Because this is the function which has values in the codemain, and this is the element of the codemain. So both sides have been transformed, and we actually were still within the codemain, because this is still codemain, and this is still an element of the codemain. So what did we do? From one element of the codemain, this or this, we basically moved to another element of the codemain. Well, this is a transformation. Well, another word for this is a function, by the way. So we applied another function on both sides of the equation. In this case, the function is what? Basically, the function is x minus 10. That's the general algebraic expression of the function which we have applied to both sides. x, in the first case, is this x plus 10, and the minus 10, it will be only x. On the right side of the equation, x is 13, and x minus 10 is 3. So we have applied a function which is defined in the codemain, and takes values in the codemain, and it's an algebraic expression at x minus 10. So what basically it means from the general standpoint is, let's consider we have a function t of x, which transforms the same codemain which was here into itself somehow. This is my function x minus 10. From any element of the codemain, which happened to be real numbers in this case, my function value is this element minus 10. So that's the transformation which I have applied. Now, what kind of transformations can be applied? A, well, not quite. And here is why. What we would like to have is, we would like to have transformed equation to be absolutely equivalent to the original one. Equivalent in what way? Well, every solution of this one should be a solution of this, and every solution of this should be a solution of that. So these two equations are supposed to be completely equivalent. What's required of this transformation for the solutions to be completely equivalent? Again, let me give you an example. What if instead of this transformation x minus 10 from this equation, I would have another transformation of this type, x minus 10 squared? It's a transformation. It's a function which is defined on the same set of real numbers. Now, if I define this function to the left part of the equation, so instead of x, I will substitute x plus 10. I will have on the left x squared, right? x minus x minus plus 10 minus 10 squared. On the right, I will have what? I will have 13 minus 10, which is 3 squared 9. So that's my transformed equation. Well, I can say, hey, this is an easy equation. I know that the square of 3 is 9. But well, the square of minus 3 is also 9. So this equation has two different solutions, 3 and minus 3. Original equation, obviously, has only 3. Minus 3 doesn't fit. So this transformation is not good. This transformation changes the equation to something which is not equivalent. So somewhere, somehow, I have to generalize the concept of transformation, which transforms my original equation into another one, which is completely equivalent. And here's how it's done, how this generalization is done. Let's consider this function t of x. What I'm stating right now is that if t of x is representing a function which is one-to-one correspondence between its arguments and its values, one-to-one correspondence, it means that I can always not only go from argument to a value, but also from the value back to the same argument, then the function t of x is a valid transformation for any equation, actually. So only functions which represent one-to-one correspondence between arguments and values fit to our transformations which we can use to solve equations. Now, what it means is the following. I can take any value of this function. And what I'm saying is that I can always find a regional argument of this function which resulted in this value if applied the function t of x. What it means from another standpoint is that I have another function which usually is symbolically represented t to the power of minus 1. It's just a symbolic, has nothing to do with power. It's a symbolic representation of inverse function. So the function t minus 1 of x actually works this way. It's defined where the t of x has values, which means here. And its values are where the arguments of the original function is. And it works in a completely reverse fashion. So the arrows, it's the same arrows but directed differently to the opposite side. So these two functions are inverse to each other. And all those functions t of x, which have inverse function, are allowed to be called invariant transformations of our equations. Function x minus 10 is obviously inversible. The inverse function is x plus 10. Function x minus 10 squared is obviously not inversible because there are always two different values of x of argument x for any value of this function. For instance, to get to 9, I can have either x equals to 13. So it will be 3 squared is equal to 9. Or I can get 7, 7 minus 10 minus 3 squared again 9. So I have two different arguments, 7 and 13, which are transformed by this function into 9. And if I have two different arguments which are transformed into the same value, then there is no inverse, obviously, because I cannot say what's the inverse function of 9. It can be either 7 or 13. And if I have two different values, it's not a function anymore. Function is always something which you can definitely say what exactly is the result of this function. And reverse function does not really exist in this particular case. So when we are talking about invariant transformation, we are talking about those functions, t of x, defined on the codemain of our original function f, which have inverse. Now what I'm saying is that applying this transformation transforms our original equation like this to equivalent. Now, if I apply my transformation t on the left bot, I will have this. On the right part, I have this. So if my original equation is this, and transformation is x minus 10, for instance, then I apply x minus 10 to this. And I get x plus 10 minus 10, which is x. And on this, 13 minus 10 will be 3. And that's how I get x equals to 3. Using the transformation t of x equals x minus 10. That's an example. Now, let me prove that this particular equation is exactly equivalent to the original one, because that's the purpose of this transformation. I have to transform into something simpler. But that should be equivalent. OK, so that's what we're getting into right now. We have to prove that these two equations are completely equivalent, which means every solution of this is a solution of this. And vice versa. If I have a solution of this equation, then it will be a solution of that equation. All right, let's consider that A is a solution of original equation. That means that f of A is equal to d. OK, now, what it means is this is A is an element of the domain of function f. f of A is an element of a column A. And v is also an element of column A. If I have two different elements of the column A, I can always apply function g. But this is exactly the same element. So applying the function t to the same argument will obviously give me the same value. So I can say that t from f of A is obviously equal to t of b, because f of A and b are two equal arguments. And that's why the function results in the same value. So the direct theorem is very simple. It's trivial, actually. What's important is to convert it. So let's go back. This is not even a theorem. This is just a joke. Now, the real one is, what if A is a solution of this equation? So I know that g of f of A equals to g of b. So that means that the A is the solution of this equation. And here, the invariant transformation, the property of invariant transformation to always have inverse, that's where it's very important. And that's where it plays the most important role. Here's half. Now, I know that A is a solution. So these are two identical elements of the codemy. And I also know that t has an inverse function, which we call t of minus 1. So if I will apply t minus 1 to t of f of A, I will have exactly the same as t minus 1 applied to t of b. Well, since these are identical, then applying a function, t to the minus 1, which is inverse function to t, will give me the same values. But now, let's think about what this represents. Well, f of A is certain element of the codemy of A. t transforms it to something. But I know that t minus 1 is inverse to t, which means it returns back from the value to the argument. So again, if this is my codemy, function t goes this way, function t minus 1 goes back. That's the definition of the inverse function. It's defined from here to here. And the result of t minus 1, the inverse function to t, is the original argument from which we started. So by definition of the inverse function, left part is equal to f of A. And the right part, for the same reason, by definition of the inverse function is b. That actually should be very clear. The fact that inverse function, inverse t of t of b, is b, basically by definition of the inverse function. And we have assumed that inverse function exists for this transformation t. So from existence of this inverse function, we have concluded, basically, that if A is a solution of this transformed equation, then A is a solution of the original equation. So we have proved right now that both equations have exactly the same set of solutions. If solution is here, then it's here. Or if it's here, then it's there. So that basically kind of explains what we do when we try to solve equations using our, I would say, digital ways which everybody is learning at school. We know that we can subtract, for instance, the same number from both sides of the equation, like in this particular case. We can subtract 10, we are saying. Well, what does it mean we can subtract 10? It means we can apply an invariant transformation of type t of x equals x minus 10. If we apply this transformation to this equation, we will get x equals 3. And that's basically a trivial solution of this thing. And this transformation is obviously reversible, and the reversible function is x plus 10. One is subtract 10, another is else 10. And obviously, these two transformations are inverse to each other. Because think about this, t minus 1 of t of x. I was saying that by definition it should give x, right? Well, let's just think about it. From t of x, I can substitute x plus 10, sorry, minus 10. But t minus 1 of this means this plus 10. Obviously, I get x as a result. That's what it means that the functions are inverse to each other. So basically, again, what I'm saying is that the rules which we have learned in school that we can subtract the same number for both sides of equation are actually a particular representation of this invariant transformation of this type. Now, another rule. For instance, we have an equation 2x equals 8. Well, everybody knows we can divide both sides of the equation by some number, which is not equal to 0, obviously. If you divide it by 2, we will get x equals 4. What does it mean from the general standpoint? It means that we apply transformation equals to 1 half of x. So if we apply this transformation to the left part, we will have 2 times 1 half of x, which is x. And on the right, 8 applied to the paper. Now, what's the inverse function in this ace? Obviously, it's 2x. If you apply first this and then that, you will return back to x. First from x to 1 half of x, then multiply by 2 will be x again. So both sides of equation can be multiplied or divided by not equal to 0, or multiplied by not equal to 0 by the same number. Oh, by the way, why is it not 0? Why can I say that I can multiply both sides of the equation by 0? Well, for obvious reason. If function is this 0, then there is no t minus 1 of x. There is no inverse function. There is no such thing as a function which returns back to the original. Because if you take, for instance, value of 2, for instance, you get 0. So from 2, you get 0. But if you get value of 22, it will also give you 0. So this is not a one-to-one correspondence. t of x does not represent a one-to-one correspondence. And that's why many different, actually infinitely many different value arguments are resulting in the same value. So there is no way I can go back. I don't know from 0 where the original argument came from. So that's why we cannot multiply, and obviously cannot divide, by 0 as an invariant transformation. OK, what if you have an equation of a different sort? Let's say x squared equals to 4. Well, you know that 2 and minus 2 both are solutions to this particular equation, right? So let's apply a transformation, which is a mistake in this particular case, and extract the square root from both sides. Well, I can say that square root of x2 is equal to square root of 4 x equals 2. Is it right? No, it's absolutely incorrect. It's absolutely wrong. Why? Because the square root is not an invariant transformation in this particular case. And as a result of applying a non-invariant transformation, we can lost one solution, which is minus 2. This equation has only solution x equals 2. This solution obviously has 2 and minus 2. So by applying non-invariant transformation, we can lost one of the roots, one of the solutions of the equation. So square root is not an invariant transformation. How about cubicle root? Well, actually it is. And here is why. Example 5 on that graph. Y equals cubicle root of x, which is the same as x to the minus 1, 1, 3, power of 1, 1, 3, has a graph which looks like this. This is 1. This is 1 minus 1. So this is a graph. And it's very easy to see from the graph that it represents a one-to-one correspondence between arguments. These are arguments on the x-axis and values on the y. Because for every x, I have one and only one y. And for every y, I have only one x. So that's one-to-one correspondence. And that's very, very important. And square root is not such a thing. Because if this is y is equal to x, then it's very important, actually. For positive y, this function is defined. And you can definitely get the positive value of square root of it. But again, if you do the negative part, this would also be exactly the same thing. Like this is 4 on y. This is minus 2. And this is 2. Both will give you the actual, I shouldn't say it this way. We're talking about function which looks like this. Then we can go from y back to x. From x we square it, we get y. From y, we go down by using x is equal to square root of y. So the shape is like this. But in any case, whatever the shape is, the same y has basically two different value of x which correspond to this function. And since we have two different values, we don't have one-to-one correspondence. So in this particular case, we do have a one-to-one correspondence. And that's why this function can be applied as an invariant transformation. And this function cannot. This is not a good one. Anyway, there are many different examples of invariant transformations. And basically, now I can answer the question, how to solve equations. OK, this is the general answer. You have to try to find invariant transformations which will transform your original equation into something simpler. And the simpler one can probably be solved. If you can transform it down to something like x is equal to 3, then this basically represents a solution to your original equation. Sometimes it's easier to solve equations using these invariant transformations by basically breaking the domain into different parts. And in certain parts, use one particular transformation. And in another part, use another. And I will just exemplify in this particular case of this equation. How can I solve these equations without actually using the square root? I said that we cannot use the square root. But here's how. I know the domain of this is all real numbers, right? So let's just divide this domain into two halves. One is positive numbers, and another is negative numbers. If x is only positive numbers, then function is invariant transformation. If x is only the positive number, y is a positive number, and we can always square y to get the positive number, and we will get x. And there is no problem with this. Now, in the negative part, we can always apply a transformation minus x. So let's divide. If x greater than 0, or if it doesn't matter, we apply this, and we get x is equal to 2. We're talking about positive numbers only. Now, in the negative numbers, if x is negative, then what I can say is I can apply square root of x, of minus x, and what will I have? If x is not, that's probably not exactly correct. I think let's just use a different function. Let's use this function. Here, this transformation. Square root of x square is x, but now we have a minus, right? Here. So minus x is equal to 2. That's what we'll do. x square goes to this, and square root of x square gives you x, and this minus in front will give you this. Or from this, we can multiply both by minus 1. We get x equal to minus 1. So now, since we have divided our domain into two parts and used two different invariant transformations in each part, then we are OK. And we have two solutions, and we have not lost things. Well, I was trying to be as smooth as possible. There are certain things which I really had to think twice about. Well, but that just proves that this is not like other things from the very beginning. And as long as you understand that invariant transformation is the generalized method to solve equations, that's great. And a technique of dividing the domain into different parts where you can define different invariant transformations, that's the proper way of solving equations of this type where you don't really have an easy invariant transformation which can be applied to an entire domain. Well, that's it for today, and thank you very much.