THE COMPOUND // FIELD NOTES // WEEK 22
Variation in four steps and complex conjugates
Field Book
Lessons 80 to 83
Due
Test 19 due ยท Supplement 22
Method
Paper first, every step
Review 3 runs this week. Lesson 80 hands over the four-step variation method: write it, find k, rewrite it, answer it.
What you have to be able to do
- Work every variation problem in the same four steps
- Divide complex numbers with conjugates
- Push exponent rules onto variable exponents
The procedures
Variation in four stepsField Book Lesson 80
- Write the variation equation. Substitute the given pair to find k.
- Rewrite the equation with k, then answer the actual question. Ratios work for direct and inverse alike.
Complex conjugatesField Book Lesson 81
- The product of a + bi and a - bi is real: a2 + b2.
- To divide complex numbers, multiply top and bottom by the denominator's conjugate.
Simplification laddersField Book Lesson 82
- Long simplifications run add, divide, add, divide: one rung at a time.
- Keep each intermediate line as a single clean fraction.
Variable exponentsField Book Lesson 83
- The product, power, and quotient rules apply when exponents are variables.
- xax2a = x3a: add the exponents exactly as with numbers.
Worked: a conjugate product
- Multiply (2 + i)(2 - i).
- 4 - 2i + 2i - i2 = 4 - (-1).
- The product is 5: real, as conjugates promise.
Terms on the record
| Term | Meaning |
|---|
| conjugate pair | a + bi and a - bi, whose product is real |
| variable exponent | an exponent that is itself a variable |
| constant of variation | the k that scales a variation equation |