 So the important expansion of the real number system emerges when we introduce what are called complex numbers. And this emerges as follows. So if I have the square root, if I write this particular symbol, what I mean to refer to here is it's the non-negative number whose square is n, whatever the radicand is. So for example, square root of 16 equals 4, this symbol equals 4 because 4 squared is 16. More importantly, and this can't be emphasized enough, square root of 16 is not equal to negative 4. This symbol is never to be interpreted as including the negative. Even though negative 4 squared does give you 16, it fails the second part which is I have to have the non-negative number. Negative 4 is not non-negative. So in answer what is this thing, the only correct answer is 4. It is not true that this is equal to negative 4. And if you remember in the quadratic formula, we have to include plus or minus the square root. The reason we have to have that plus or minus the square root is the square root itself is a positive number. If I want to include the other one, I need to include the negative of the square root. And what this means is if I'm dealing with a variable expression like square root of x squared, I have to write this as the absolute value of x because I don't know if x is positive or negative. So I'll put the absolute value symbols or indicators around x just to make sure that I do have the non-negative value. What about something like square root of negative 16? Well, here we run into a problem. The square of any non-negative number is positive and the square of any negative number is positive. So it turns out that no number squared will give you negative 16. And well, mathematicians don't like that. So what we do is we say, all right, well, let's invent something new. And so it's convenient to invent a new imaginary unit where we're going to define the square root of negative one. Well, it's a new thing. We're going to call that i as a short word for imaginary. And equivalently, because square root of negative one is i, that's also the same as saying that the square of i, i squared is equal to negative one. And what this does is this allows us to rewrite square roots of negative numbers because we can factor out that minus one. So for example, let's take that square root of negative 16 again. So what I'm going to do is I'm going to factor out this negative one. So I split off the negative portion. That's negative one times the square root of 16. And that square root of negative one, by definition, I can write square root of negative one equal to i. So instead of writing this, I'll write i, i squared 16. Well, that's an ordinary real number. That's in fact, equal to four. And it's traditional, conventional, but not absolutely required that if we have a imaginary i multiplied by a real number, we typically put the real number first. Now, here's an important thing to be very, very careful with. We have this rule, square root of a times square root of b equals the square root of the product a times b. And the important thing to recognize about this rule is we introduce this rule when we didn't have to worry about negative square roots. In fact, this rule only applies. This is only a guarantee if a and b are both positive. If a and b are negative, if one of them or both is negative, we can't guarantee that this rule holds. What we actually have to do is we have to treat the complex quantities as they show up. So, for example, why you might have square root nine times square root of negative four. Well, I have to consider that this is a complex number. This is a complex number. This is an imaginary number. I have that square root negative one there. So I'll factor out my square roots of negative one. And I am using this in the one case it can apply. I can factor out the negative one. And let's see square root negative one is i. Square root nine is three. Square root four is two. And again, I'll combine my whole number terms. I get six i squared. And there is a useful simplification I can make at this point. Remember, i is the square root of negative one. So i squared. i is the thing whose square is negative one. This i squared value here, that's going to be negative one. And I can make a final simplification. Six times negative one is negative six as our final answer. Except for this funny rule that i squared equals negative one, which allows us to simplify expressions that contain powers of i, we can treat i as an ordinary algebraic variable. So, for example, if I had the expression three plus four x plus two minus x, I could handle that by adding the real numbers three plus two, that gives me five. And I can add the x as four x plus minus x, that gives me three x. And if I have the complex numbers, that's a real number plus an imaginary part, three plus four i plus two minus i, I can add those in the same way. Three plus two gives me five, four i minus i gives me three i. Now, the only twist is that when I expand a product, one plus x times three plus x, I expand this, that's three plus four x plus x squared. Well, I can do exactly the same thing with complex numbers, three plus four i plus i squared, except I do have this other option here, i squared equals negative one. So, I don't have to and I really shouldn't carry around i squared. I could replace that with negative one and get my final answer three plus negative one. That's two plus four i is my product. Now, this is important when we talk about the conjugate of a complex number. And this appears when we look at rational expressions. So, for example, let's take three plus i over two minus i. I have a complex number divided by another complex number. And so, I have this complex quotient here. And if I want to simplify this, what I'm going to do is I'm going to multiply by the conjugate of the expression. That conjugate of the denominator of any complex number in general, but of the denominator in particular, is it's the same values except we're going to add instead of subtract or subtract instead of add. So, it's the same terms, but we just change the operation from minus to plus or from plus to minus. So, I have to multiply numerator and denominator by the same thing if I want to maintain equality. So, I'll multiply it by two plus i over two plus i. So, again, same terms, two and i, two and i, but this one's a subtraction, this one's an addition. So, I now have denominator multiplied by a conjugate, numerator multiplied by something. And, again, I can treat i as an algebraic variable with the one extra simplification that i squared is negative one. So, in the numerator, I have three plus i times two plus i. In the denominator, three plus i times two minus i, I have a product. I'll expand out that product. Numerator is six plus five i plus i squared. Denominator, this is a sum and difference. This is going to be four minus i squared. And i squared is negative one, so I'll replace those. And I'll do the arithmetic I can do. Six minus one, that's five. Four minus negative one, that's also five. And I can go one step farther. And this is a rule from the arithmetic of fractions. You know that if I have two fractions with the same denominator, I can add the numerators and get a single fraction. Well, all such rules also work in reverse. If I have a numerator, that's the sum of two things, I can rewrite it as the sum of two fractions with the same denominator. So, this five plus five i over five, I'm going to rewrite that as two fractions, added one with numerator five, denominator five, and the other with numerator five i, denominator five. So, that's going to give me this, and I can simplify five over five is one, five over five is one, and that simplifies to one plus i. Now, why would we ever want to deal with square roots of negative numbers? Well, this goes back to the quadratic formula. And in general, where we run into complex numbers, the very first time that mathematicians ever ran into a complex number, was dealing with the solution to a quadratic equation. So, again, we have our quadratic equation. We have our quadratic formula, 2x squared minus 4x plus 10a, coefficient of x squared, that's two, b is the coefficient of x, negative four, c is the constant term, is 10, and I can substitute these into my quadratic formula. And after all the dust settles, what I end up with is the square root of a negative number. Now, previously, we said this corresponds to no real solutions. And now we actually have a way of working with them. These are some imaginary solutions. So, again, that square root negative 64 is going to be split off negative one times square root 64, that's i times eight, and I have a fraction, four halves, that's two, plus or minus four i. And here's my two solutions.