Calculator for finding the Area of a circle
   
 Remember in Algebra using something called the Quadratic Formula?
Related Chapters
   
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:
X2 + X + 1 = 0
 
 OK, use the quadratic equation and solve for X:
 
 The standard form of a quadratic is:
 

aX2 + bX + c = 0

 
 So given the equation in our example:
 
a = 1 b = 1 c = 1
 
 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:
 
3 + 4i

7- 3i
 
  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) = a2 - b2

 
 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 + 12i2 = 21 + 37i + 12i2




7- 3i 7+ 3i 49 + 21i - 21i -9i2 49 - 9i2
 
 i 2 = -1 so:
 
21 + 37i + 12i2 = 21 + 37i + 12(-1) = 9 + 37i



49 - 9i2 49 - 9(-1) 58
 
 Now we "split things up" like we did when we were solving for X.
 
9 + 37

 i



58 58
 

 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  =
i2 =  -1
i3 =  -1 x i = -i
i4 =  i2 x i2 = -1 x -1 = 1
i5 =  1 x i = i
i6 =  i x i = i2 = -1
i7 =  -1 x i = -i
i8 =  -i x i = -i2 = -(-1) = 1
  
 Look closely at those values above:
 

i = i5    i2 = i6    i3 = i7    i4 = i8

 
 The values of "i" repeat every 4 powers.
  
 There is a little trick we can do with that info:
 
= i i5 i9 i13 i17 i21 i25 i29
-1 = i2 i6 i10 i14 i18 i22 i26 i30
-i = i3 i7 i11 i15 i19 i23 i27 i31
i = i4 i8 i12 i16 i20 i24 i28 i32
 
 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:
i1/2 = ((-1)1/2)1/2
 

i1/2 = (-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