Hey kids. Welcome to my math class. Let's start today's lesson.
There is no such thing as an imaginary number or "i" the imaginary unit. Wipe these terms from your memory.
There is no solution to sqrt(-1) in the real numbers. End of story.
Mathematicians sometimes like to group numbers together into a single term called a vector. For example (1,2,3) is a vector. It has 3 numbers but it's one expression. Isn't that neat? Soak it in. [show some examples of vectors and basic vector arithmetic]
A while ago mathematicians invented a new class of numbers called "Complex numbers." Again the word complex can be misleading, it doesn't mean they are any harder to understand it just they're composed of two values instead of one. A complex number is just a vector of two real numbers. That certainly doesn't sound very "complex" to me. [pause for laughter]
Let's state formally that a complex number is an expression of the form (a,b) where a and b are real numbers. Now let's review some of the operations we can do on these numbers.
Let's start with the simple ones.
If we have two complex values (a,b) and (c,d) we define (a,b) + (c,d) = (a+c, b+d). We define (a,b) - (c,d) = (a-c, b-d). Simple ain't it?
Multiplication is a bit trickier.
If we have two complex values (a,b) and (c,d) we define (a,b) x (c,d) = (ac-bd, ad+cb)
Look, you can show (-1, 0) = (0, 1) x (0, 1). Neat isn't it? Be careful Some people confuse this as being a way to solve sqrt(-1) but it's not really the case because this takes place in a completely separate class of numbers.
That's enough for today.
