



Remember in Algebra using something called the Quadratic Formula?




It goes like this:









There is a special part of this formula called the "Discriminant."












When b
^{2}
 4ac turns out to be a positive number, we get two answers: 











When b
^{2}
 4ac turns out to be zero, we get one answer: 











But when b
^{2}
 4ac turns out to be negative, we were stuck. 





A square root
radical means: 


"what number times itself equals the number under
this radical sign." 


No real number
times itself equals a negative number. 


It was a
problem that couldn't be solved. 





Math people
hate it when that happens. 


They hate it so much that
they usually go out and invent 


a whole new kind of math to solve the
problem. 





What they did
here was just invent a number 


and say that when you multiplied it by
itself, you got 1. 





They could
have called it "the made up number," 


but they wanted
something that sounded fancier. 





They called it, the IMAGINARY
NUMBER and used the letter i to stand for it.






So:



i x i = 1 


and we can
also say: 








Now that we have i, if we
have something like:












we can deal with it.






What you do, is factor to get
the square root of 1 by itself like this:












We know that: 





So:









POP QUIZ! 


Is 3i a real number or an
imaginary number? 





Hey, real
number/imaginary number, it's just a name. 


WHAT'S THE BIG DEAL??? 





It's not that
important unless you get asked that on a test 


or someone tries to embarrass
you if you don't know. 


But here's the
rule. 


If you multiply a real number and an imaginary number
together, 


you get an imaginary number. 


It's like the real number
gets infected with imaginariness! 





ENOUGH ALREADY, LET'S DO ONE! 





Solve this for X: 


X^{2} + X + 1 = 0 





OK, use the quadratic equation and solve for X: 





The standard
form of a quadratic is: 





aX^{2} +
bX + c = 0 





So given the
equation in our example: 











Putting these values into the
quadratic equation and solving, we get: 











This is all
correct and everything, 


but most of the time math people like to see
it written a little differently. 











Take a minute
and make sure you see how we did that. 





Whenever you have an answer
that includes an imaginary number, 


math people want it written a
special way. 





If we let a
and b stand for any numbers we might have in an answer, the form is: 





a + bi 





This is called a complex number. 


Hey, they're just trying to
intimidate you with that name 


and make themselves look so smart. 


Don't buy it. It's no big
deal. 





Sometimes you
might see complex numbers written in fractions, like this: 











You might be asked to change
one of these puppies



into something that looks like a + bi.






Can we do
that? 





No problem.
Remember something called "the difference of two squares"? 


It went like this: 





(a + b) x (a 
b) = a^{2}  b^{2} 





The nice thing is that the
middle terms cancel out



and you just get left with squared terms.






We need to multiply the term by
something



with the other factor of the difference of two squares thing



to get rid of the i in the denominator.






Since the
thing we are working on is an expression, not an equation, 


the only
thing we can multiply it by is a fancy name for 1. 


That is, some
fancy term divided by itself. 





Here we go: 





3
+ 4i 
x 
7+
3i 


7
3i 
7+
3i 






3
+ 4i 
x 
7+
3i 
= 
21
+ 9i + 28i + 12i^{2} 
= 
21
+ 37i + 12i^{2} 




7
3i 
7+
3i 
49
+ 21i  21i 9i^{2} 
49
 9i^{2} 






i ^{2} = 1 so: 





21
+ 37i + 12i^{2} 
= 
21
+ 37i + 12(1) 
= 
9
+ 37i 



49
 9i^{2} 
49
 9(1) 
58 






Now we
"split things up" like we did when we were solving for X. 











Let's review.
We had 7  3i in the denominator. We used 7 + 3i to simplify it. 





In general
with these things, if we have a + bi, we can use a  bi to simplify
it. 





In math talk,
these are called "Complex Conjugates" of each other. 


(these names
are killing me!) 





So i equals
the square root of negative 1 and i ^{2} = 1. 





What about
stuff like i ^{3} and i ^{4} and like that, eh? 





Well: 





i
= 

i^{2}
= 
1 
i^{3}
= 
1 x i
= i 
i^{4}
= 
i^{2}
x i^{2}
= 1 x 1 = 1 
i^{5}
= 
1 x i
= i 
i^{6}
= 
i x i
= i^{2}
= 1 
i^{7}
= 
1 x i
= i 
i^{8}
= 
i x i
= i^{2}
= (1) = 1 






Look closely
at those values above: 





i = i^{5}
i^{2} = i^{6} i^{3} = i^{7}
i^{4} = i^{8} 





The values of
"i" repeat every 4 powers. 





There is a
little trick we can do with that info: 






= 
i 
i^{5} 
i^{9} 
i^{13} 
i^{17} 
i^{21} 
i^{25} 
i^{29} 
1 
= 
i^{2} 
i^{6} 
i^{10} 
i^{14} 
i^{18} 
i^{22} 
i^{26} 
i^{30} 
i 
= 
i^{3} 
i^{7} 
i^{11} 
i^{15} 
i^{19} 
i^{23} 
i^{27} 
i^{31} 
i 
= 
i^{4} 
i^{8} 
i^{12} 
i^{16} 
i^{20} 
i^{24} 
i^{28} 
i^{32} 






Here's the
deal. 


What if
someone asks: "What is the cube root of i? 


The way to answer that is to say: 


Well, i = i9, and the cube root of i9 = i3, and i3  i. 


So the cube root of i is i 











If you look
close, you will see that the powers of i that are equal to itself 


are ALL ODD.
(as in odd and even) 





That means you
can use this trick to find odd roots of i but not even roots of i. 





OK, so what's the square root of i? 





When you have
even roots, you use fractional exponents to solve the problem. 


The
square root of i can be written as i to the one half power. 


That is:
i ^{1/2} 





Now: 


i = (1)^{1/2} 


So: 


i^{1/2} = ((1)^{1/2})^{1/2} 





i^{1/2 = }(1)^{1/4} 





Writing
this with root radicals instead of fractional exponents we get: 











This
trick will work with odd roots root too, 


but for them we know
something trickier! 





copyright 2008 Bruce Kirkpatrick

