Wren pins three index cards to the Glass House chalkboard: 3 + 2i, 1 - 5i and 2 - 3i.
"If these are numbers," Comet says, "we should be able to add them. What can we make of 3 + 2i plus 1 - 5i?"
"Like terms," Wren says. "Real parts together, i parts together. 4 - 3i. What do you notice?"
"Like adding binomials," Comet says. "So multiplying is like multiplying binomials?"
Nova projects a grid. "Would you like a hint? Every part times every part, then one more step."
Comet fills the grid for (2 + 3i)(2 - 3i). "4, minus 6i, plus 6i, minus 9i². The i terms cancel."
"And -9i² is -9 times -1," Wren says. "Which is plus 9. So the answer is 13. A real number."
"A complex number times a complex number can be real," Comet says. "That is going in the log."
Treat i like a letter when adding or subtracting. Combine the real parts, then combine the parts with i.
(3 + 2i) + (1 - 5i) = 4 - 3i. And (3 + 2i) - (1 - 5i) = 2 + 7i.
When subtracting, subtract both parts of the second number. The sign of its imaginary part flips.
Way one: multiply like binomials, then replace i² with -1. Way two: use the grid Nova projected. Both give the same product.
| Problem | Real parts | Parts with i | Answer |
|---|---|---|---|
| (3 + 2i) + (1 - 5i) | 3 + 1 | 2i - 5i | 4 - 3i |
| (3 + 2i) - (1 - 5i) | 3 - 1 | 2i + 5i | 2 + 7i |
| (2 - 3i) + (2 + 3i) | 2 + 2 | -3i + 3i | 4 |
| (2 + 3i)(2 - 3i) | 4 + 9 | -6i + 6i | 13 |
| (1 + i)(1 + i) | 1 - 1 | i + i | 2i |
Look at the fourth row. (2 + 3i)(2 - 3i) = 4 + 9 = 13, with no i left. A pair like 2 + 3i and 2 - 3i always multiplies to a real number.
Look at the last row. (1 + i)² = 2i. Squaring a complex number can leave only an imaginary part.
| Statement | True or false? |
|---|---|
| (2 + 3i) + (1 - i) = 3 + 2i | ? |
| (5 - 2i) - (3 - 2i) = 2 - 4i | ? |
| (1 + i)(1 - i) = 2 | ? |
| (3i)(3i) = 9 | ? |
| The product of two complex numbers is never a real number. | ? |
Clean work. Tomorrow is Lab day: you measure a water arc and write its rule yourself.