Curriculum placement
Precalculus. This describes the skill, not a learner estimate.
The conjugate of a complex number
Use conjugates to compute complex quotients and moduli
Problem types
- Conjugates and complex division
- Modulus from a conjugate
Builds on
- Calculating with complex numbers
- Proving a polynomial identity · Useful idea
The difference of squares identity gives (a + bi)(a - bi) = a squared + b squared, the product of a number and its conjugate.
Leads to
- Distance and midpoints in the complex plane · Extend the idea
The squared modulus of a complex number is the number times its conjugate. The distance between z and w is the modulus of z − w, so its square is (z − w) times that difference's conjugate.
- Zeros and factors of a polynomial · See it applied
Nonreal zeros of a polynomial with real coefficients come in conjugate pairs.
Related skills
- Rational exponents and radicals · Related idea
Multiplying a complex quotient by the conjugate of the denominator parallels rationalizing a radical denominator with its conjugate.
- Factoring over the complex numbers · Shared foundations
- The complex plane · Shared foundations
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