High School: Number and Quantity
The Complex Number System HSN-CN.A.2
2. Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Addition and subtraction of complex numbers is easy. Just because it says "complex" doesn't mean it is. Some ice cream cartons say, "nonfat," but do you really think it doesn't have the same effect on your waistline as regular ice cream? Please.
The i looks just like a variable and in this context, it acts just like one too. The basic rule for adding and subtracting is the one your students might imagine: we add and subtract like terms. Just make sure they remember to express their answers in a + bi form. In other words, we want the i term to come after the "non-i" term.
So, for example, if we want to add 4 + 10i and 7 – 2i, we combine the like terms. That means the 4 and the 7 go together, as do the 10i and -2i. (We can do this thanks to a generous donation from the Commutative Property. Remember to send it a thank you card.) The answer, then, is 11 + 8i.
If we want to subtract, we simply remember to distribute the negative sign (à la Distributive Property).
(21 + 4i) – (16 – i) becomes 21 + 4i – 16 + i, or even more simply, 5 + 5i.
Multiplication also follows the rules of algebra. Many people use the "FOIL" method to multiply binomials (First, Outer, Inner, Last). We can also think of it as the "double distributive" property. First, distribute the real portion of the first complex number, then the imaginary part.
When multiplying, it's worth noting that i2 = -1. This will help a lot when we simplify. Don't be surprised when a real number results from us fiddling around with these imaginary numbers. It happens sometimes. We just gotta roll with it.