Complex Numbers
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,
√-100 =√100√-1=10√-1
√-5=√5√-1
√-290=√290√-1 etc.
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.
i =√-1
Note that if we square both of sides of this we get,
i^{2} = -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.