Complex Conjugates and Division

Algebra 2 · Axiom Academy

LESSON Complex Conjugates and Division Flip the sign of the imaginary part and you can divide any complex numbers — the conjugate turns the denominator real. 1. What Is a Complex Conjugate? The conjugate of a complex number keeps its real part exactly the same and flips the sign of its imaginary part . Geometrically that is a reflection across the real axis : the point hops straight down (or up) to its mirror image, its distance from the axis unchanged. 2. The Key Property: the Product Is Real Here is why the conjugate is the perfect tool. Multiply a complex number by its conjugate and the answer is always a real, non-negative number — every imaginary term cancels. Expand with FOIL and watch the middle terms disappear: 3. Using the Conjugate to Divide To divide, multiply the numerator and denominator by the conjugate of the denominator. Because that is the same as multiplying by 1 , the value does not change — but the denominator becomes real. Why does it work? Multiplying by is multiplying by 1 — we change the form, not the value. And by the key property the denominator lands on a real number: Let us divide from start to finish — then check the answer by multiplying back. Step 1 — Identify the conjugate. The conjugate of 1 + 2i is 1 - 2i . Step 2 — Multiply top and bottom by that conjugate. Step 3 — Expand the numerator. Step 4 — Simplify the denominator. (1 - i)(1 + 2i) = 3 + i , the number we started with, so 1 - i is correct. ✓︎

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