What this lesson covers
- i squared equals negative one (replace it everywhere)
- To divide, multiply top and bottom by the denominator's conjugate
- Powers of i cycle every 4: use the remainder mod 4
Chapters
- 0:00The one fact that runs everything
- 0:23i squared and the cycle of four
- 0:55Multiply: FOIL, then swap i squared
- 1:41Divide: use the conjugate
- 2:31Your turn: powers of i
- 3:09Three traps
- 3:43Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:00The one fact that runs everything
Welcome to SAT Climb. Complex numbers look scary because of that letter i. But here's the secret. There is really only one new fact. i squared equals negative one. Master that single substitution, and every complex-number question on the digital SAT becomes ordinary algebra.
0:23i squared and the cycle of four
Start with the definition. i is the square root of negative one, so i squared is negative one. From that, the powers of i fall into a cycle that repeats every four. i to the first is i. i squared is negative one. i cubed is negative i. i to the fourth is one. Then it starts over. That four-step cycle, and the substitution i squared equals negative one, are the entire toolkit.
0:55Multiply: FOIL, then swap i squared
Let's use it. Multiply three plus two i, times one plus four i. FOIL it exactly like a normal binomial. Three times one is three. Three times four i is twelve i. Two i times one is two i. Two i times four i is eight i squared. Now the only new move. Replace i squared with negative one. So eight i squared becomes negative eight. Combine the real parts, three minus eight, negative five. Combine the imaginary parts, twelve i plus two i, fourteen i. The answer is negative five plus fourteen i. Same FOIL you already know, one substitution at the end.
1:41Divide: use the conjugate
What if you're dividing? You can't leave i in the denominator. The trick is the conjugate. Take the bottom, one minus two i, and flip the sign on the i term to get one plus two i. Multiply top and bottom by that. Watch the bottom. One minus two i, times one plus two i, is one minus four i squared, which is one plus four, five. A real number. The i vanished. The top works out to negative four plus seven i. So the answer is negative four fifths plus seven fifths i. The conjugate always clears the bottom, because a minus b i times a plus b i is a squared plus b squared.
2:31Your turn: powers of i
Your turn. Simplify i to the twenty-second power. Don't multiply it out twenty-two times. The powers cycle every four, so all that matters is the remainder when you divide the exponent by four. Twenty-two divided by four is five, remainder two. So i to the twenty-second is the same as i squared, which is negative one. Remainder two, answer negative one.
3:09Three traps
Three traps. One, forgetting the substitution. If i squared is still sitting in your answer, you're not done. Replace it with negative one. Two, botching the conjugate. Flip the sign on the imaginary term only. One minus two i becomes one plus two i, not negative one plus two i. Three, powers of i. Divide the exponent by four and use the remainder, never the exponent itself. Everything traces back to i squared equals negative one.
3:43Recap
Complex numbers, solved. Multiply like normal, then swap every i squared for negative one. To divide, hit the bottom with its conjugate. And powers of i just cycle every four. Start practicing free at satclimb.com. Your SAT is closer than you think.
Read the written version: the Equivalent expressions strategy guide, then try 25 hard Equivalent expressions questions with full explanations. Both are free, no account needed.