In the radicals section we noted that we won't get a real number out of a square root of a negative number. For example √-9 isn't a real number as there is no real number which we can square & get -ve 9.
We now also saw that if a and b were both positive then √(ab) = √a√b .For a second let's forget that limitation and do the following.
√-9 = = √9 √ -1 = 3 √-1
Now, √-1 is not a real number, however if you think about it we can do this for any square root of a negative number. For example,
Thus, even if the number isn't a perfect square still we can always reduce the square root of a -ve number down to the square root of a +ve number (which we or a calculator can deal with) times √-1 .
Thus, if we only had a way to deal with √-1 we could really deal with square roots of negative numbers. Well the reality is that, at this level, there only isn't any way to deal with
√-1. Thus rather than dealing with it we will "make it go away" so to speak using the following definition.
Note that if we square both of sides of this we get,
i2 = -1
It will be significant to remember this later on. It shows that, in some way, i is the only "number" which we can square and acquire a negative value.
By using this definition all the square roots above become,
√-9 = 3i √-100=10i
√-5=√5i √-290 = √290 i
These all are examples of complex numbers.