The imaginary unit i
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In your mathematical careers you might encounter people who say it is wrong to say that i is equal to the principal square root of negative 1. And if you ask them why is this wrong, they'll show up with this kind of line of logic that actually seems pretty reasonable. They will tell you that, OK, well let's just start with negative 1. We know from definition that negative 1 is equal to i times i. Everything seems pretty straightforward right now. And then they'll say, well look, if you take this, if you assume this part right here, then we can replace each of these i's with the square root of negative 1. And they'd be right. So then this would be the same thing as the square root of negative 1 times the square root of negative 1. And then they would tell you that, hey, look just from straight up properties of the principal square root function, they'll tell you that the square root of a times b is the same thing as the principal square root of a times the principal square root of b. And so if you have the principal square root of a times the principal square root of b, that's the same thing as the square root of a times b. So based on this property of the radical of the principal root, they'll say that this over here is the same thing as the square root of negative 1 times negative 1. If I have the principal root of the product of two things, that's the same thing as the product of each of their principal roots. I'm doing this in the other order here. Here I have the principal root of the products, over here I have this on the right. And then from that we all know that negative 1 times negative 1 is 1. So this should be equal to the principal square root of 1. And then the principal square root of 1-- Remember, this radical means principal square root, positive square root, that is just going to be positive 1. And they'll say, this is wrong. Clearly, negative 1 and positive 1 are not the same thing. And they'll argue therefore, you can't make this substitution that we did in this step. And what you should then point out is that this was not the incorrect step. That it is true negative 1 is not equal to 1, but the faulty line of reasoning here was in using this property when both a and b are negative. If both a and b are negative, this will never be true. So a and b both cannot be negative. In fact normally when this property is given-- sometimes it's given a little bit in the footnotes or you might not even notice it because it's not relevant when you're learning it the first time-- but they'll usually give a little bit of a constraint there. They'll usually say for a and b greater than or equal to 0. So that's where they list this property. This is true for a and b greater than or equal 0. And in particular, it's false if both a and b are both, if they are both negative. Now, I've just spent the last three minutes saying that people who tell you that this is wrong are wrong. But with that said, I will say that you have to be a little bit careful about it. When we take traditional principal square roots. So when you take the principle square root of 4. We know that this is positive 2, that 4 actually has two square roots. There's negative 2 also is a square root of 4. If you have negative 2 times negative 2 it's also equal to 4. This radical symbol here means principal square root. Or when we're just dealing with real numbers, non imaginary, non complex numbers, you could really view it as the positive square root. This has two square roots, positive and negative 2. If you have this radical symbol right here principal square roots, it means the positive square root of 2. So when you start thinking about taking square roots of negative numbers-- or even in the future we'll do imaginary numbers and complex numbers and all the rest-- you have to expand the definition of what this radical means. So when you are taking the square root of really of any negative number, you're really saying that this is no longer the traditional principal square root function. You're now talking that this is the principal complex square root function, or this is now defined for complex inputs or the domain, it can also generate imaginary or complex outputs, or I guess you could call that the range. And if you assume that, then really straight from this you get that negative, the square root of negative x is going to be equal to i times the square root of x. And this is only-- and I'm going to make this clear because I just told you that this will be false if both a and b are negative. So this is only true-- So we could apply this when x is greater than or equal to 0. So if x is greater than or equal to 0, then negative x is clearly a negative number, or I guess it could also be 0. It's a negative number. And then we can apply this right over here. If x was less than 0, then we would be doing all of this nonsense up here. And we start to get nonsensical answers. And if you look at it this way and you say hey, look, i can be the square root of negative one, if it's the principal branch of the complex square root function. Then you could rewrite this right over here as the square root of negative 1 times the square root of x. And so really the real fault in this logic when people say, hey, negative 1 can't be equal to 1, the real fault is using this property when both a and b, where both of these are negative numbers. That will come up with something that is unambiguously false. If you expand your definition of the complex or expand your definition of the principal root to include negative numbers in the domain and to include imaginary numbers, then you can do this. You can say the square root of negative x is the square root of negative 1 times-- Or you should say the principal square root of negative x-- I should be particular my words-- is the same thing as the principal square root of negative 1 times the principal square root of x when x is greater than or equal to 0. And I don't want to confuse you, if x is greater than or equal to 0, this negative x, that is clearly a negative, or I guess you should say a non positive number.