i as the principal root of -1

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 one and if you ask them why is this wrong they'll show up with this this kind of line of logic that actually seems pretty reasonable they will tell you that okay well let's just start with negative one we know from definition that negative one 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 eyes with the square root of negative one 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 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 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 had 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 this radical needs principal square root positive square root that is just going to be positive 1 and they'll say this is wrong clearly clearly negative 1 and positive 1 are not the same thing and they'll argue therefore you can't make this substitution that I 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 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 a and B both cannot cannot be 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 for a a and B greater than or equal to zero so that's where they list this property this is true for a and B greater than equals zero and in particular its false if both a and B are both if they are both negative now I've said that 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 do I will say that you have to be a little bit careful about it you have to be a little bit careful about it when we take traditional principle square roots so when you take this principal square root of four we know that this is positive two that four actually has two square roots there's negative two is also a square root of also is a square root square root of four if you have negative two times negative two it's also equal to four this radical symbol here means principal square root or when we're just dealing with real numbers not imaginary non complex numbers you can really view it as the positive square root this has two square roots positive and negative two if you have this radical symbol right here principal square roots it means the positive square root of two so when you 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 when you are taking the square root really of any 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 and can also generate imaginary or complex outputs or I guess you could call that you could call that the range and if you assume that then really straight from 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 of both a and B or negative so this is only true so we can apply this we can apply we can apply this we can apply when when X when X is greater than or equal to zero so if X is greater than or equal to zero the negative X is clearly a negative number or I guess it could also be zero it's a negative number and then we can apply this right over here if X was less than zero then we would be doing all of this nonsense up here and we would 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 we're taking the if it's the principle branch of the complex square root function then you could rewrite this right over here as the square root of negative one x times the square root times the square root of x and so really the real fault in this logic why what when people say hey one negative one can't be equal to one the real fault is using this property 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 expand your definition of the principal root to include negative numbers in the domain and including and to include imaginary numbers then you can do this you can say the square root of negative x is negative the square root of negative 1 times the or you say that the principal square root of negative x I should be particularly 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 is clearly this negative X then is clearly a or I guess you say a non positive number