Today's concept requires a bit of imagination, because it introduces the idea of imaginary numbers. An imaginary number is the opposite of a real number, and is represented by i, which is √(-1). So i x i = -1. A complex number is some real number plus an imaginary number. It is expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
This concept appeared 15 times last year, with a median placement at question # 52.
Number Dojo Level: TBD
Most of these questions require a bit of FOILing. I will jump right into the examples so you can see how they work.
Example 1: (5 + 6i)(5 - 6i) = (a + bi). Find (a + b). ___
- Simplify (5 + 6i)(5 - 6i) by FOILing. You get (5 x 5) + (5 x -6i) + (5 x 6i) + (6i x -6i).
- The middle two terms cancel each other out, so you have (5 x 5) + (6i x -6i). Remember that i x i = -1, so you are left with 25 - 36(-1) = 25 + 36 = 61.
- You were asked for (a + b). a = 61 and b = 0 (there are no i's left), so (a + b) = 61.
Example 2: (2 + 3i)(4 - 5i) = (a + bi). Find ab. ___
- Simplify by FOILing. You get (2 x 4) + (2 x -5i) + (4 x 3i) + (3i x -5i).
- This reduces to 8 - 10i + 12i - 15ii, which is 8 + 2i - 15(-1).
- You now have 8 + 2i + 15, or 23 + 2i. a = 23 and b = 2. ab = (23)(2) = 46.
Example 3: (4 + 2i)(1 - 3i) = a + bi. Find b. ___
- Since you are asked only for b, worry about FOILing only the terms that have i in them (which would be OI).
- (4 x -3i) + (1 x 2i) = -12i + 2i = -10i. b = -10.
Example 4: If (5 - 2i)(3 - 4i) = a + bi, then a = ___
- Since you are asked only for a, and the OI terms will not cancel, worry only about FL.
- (5 x 3) + (-2i x -4i) = 15 + 8ii = 15 + (8)(-1) = 15 - 8 = 7.
Example 5: (2 + 3i) ÷ 5i = (a + bi). Find (a + b). ___
- This one is pretty tricky. Since we are dividing by 5i, let's also multiply everything by 5i/5i (which is equal to 1) to keep i only in the numerator.
- (2 + 3i) ÷ 5i x 5i/5i = (2 + 3i)(5i) divided by (5i)(5i). This simplifies to (10i + 15ii)/(25ii).
- Simplified further, we get (10i - 15)/(-25), which is the same as (-15 + 10i)/(-25).
- Factor out a -5 from the top and bottom. You are left with (3 - 2i)/5. This is the same as 3/5 - 2/5 i.
- You were asked for (a + b). a = 3/5 and b = -2/5. (a + b) = 3/5 - 2/5 = 1/5.