i as the Principal Root of -1 (a little technical) i as the principal square root of -1
i as the Principal Root of -1 (a little technical)
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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, "Okay. Well, let's just start with -1.
- We know from definition that -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 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 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 square root a times b
- so based on this property of the radical of the principal root, they'll say this over here is the same thing as
- the squrae root of negative 1 times negative 1 if I have the principal root of the product of 2 things that's the same
- thing as the the product of each of their principal roots I am doing this in 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 squre root of 1, remember this radical means
- principal squre root, positive squre 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 therefore you can't make the subtitution that we did in this step and you should then point out is
- that, this was not the incorrect step that it is true that negative 1 is not equal to 1 but the faulty line of reasoning here was
- in using this propperty when both a and b are negative, if both a and b are negative this will
- never be true, so a and b both can not be negative infact normaly when this property is given,
- sometimes is given a little bit in footnotes you might not even notice it because its not relevant when you
- learning it in the first time but usually they give a little bit of construct there, they usually say for
- a and b greater than or equal to zero so thats where they listes property this is true for a and b be
- greater or equal to zero and in particular it's false if both a and b are negative, Now I've said
- that, I've just spend lat three minutes saying that people who tell you this is wrong are wrong but with
- that I said I do say you have to be a little bit careful about it, when we take traditional principal
- square roots so you take thi principal square root of 4, we know this is positive 2 that 4 actually
- has two square roots, negative 2 ia also a square root of 4, if you have negative 2 times negaive 2
- is also equal to 4, this radical symbol here means principal square root or when we just dealing with
- real numbers non imaginary non complex numbers you can really ??? as positive
- square root, this is 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 you'll do imaginary numbers and complex numbers and all the rest you have
- to expend the definition of what this radical means, so when you are taking the square root
- of really of any negative number you'll really saying this is no longer the traditional principal
- square root function you've now talking this is the principal complex square root function, this is now
- to find for complex inputs or the domain it can also generate imaginary or complex output or you
- should call that the range and if you assume that, then really straight from this you get that 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 can apply this we can apply this we can apply 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 can be zero, it's a negative number and then we can apply this right over here if x was less than
- zero then we'll be doing all of this nonesense up here and we will start to get nonesense equal
- answeres and if you look at it this way you'll say hey look i can be the square of negative 1 if we were
- taking the if it's the principal branch of the complex square root function, then you could
- rewrite this right over here as 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 faughlt is
- using this property, when both a and b when both of these are negative numbers that will come up with
- something that is unambiguously false, if you expend the definition of complex or expend the definition of
- principle root include negative numbers in the domain and including and to include imaginary
- numbers then you can do this you can say the the square root of negative x is the qsuare root of
- negative 1 times or instead (say) the principle square root of negative x, I should be particulare in 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 zero and I don't want confuse you, if x is greater than or equal to
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