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CCSS Math: HSN.CN.A.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 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.