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Dividing complex numbers worksheets

You cannot divide by a complex number directly, but you can make the bottom real first. These free printable worksheets give six divisions like (−2 + 3i) ÷ (2 + 3i), and the answer key multiplies above and below by the conjugate and writes each answer as a + bi with exact fractions.

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Dividing complex numbers
Dividing complex numbers worksheet — By the conjugate, fraction answers, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The conjugate of 2 + 3i is 2 − 3i, and it works because of a difference of two squares: (2 + 3i)(2 − 3i) is 4 − 9i², and since i² = −1 that is 4 + 9 = 13, a plain whole number. Multiplying the top by the same conjugate leaves the value unchanged, because together the two are a multiplication by (2 − 3i)/(2 − 3i), which is 1. So the division becomes a multiplication on top and a real number underneath.

Why this sheet works

The key writes that bottom as 2² + 3² = 13 rather than as a subtraction, because the usual slip is 4 − 9 = −5 with the i² forgotten. The other is multiplying only the bottom by the conjugate. Every divisor on the sheet has two non-zero parts, so the bottom always comes out between 10 and 72 and the conjugate is never a step that can be skipped.

How this one works

(−2 + 3i) ÷ (2 + 3i)

conjugate of2 + 3iis2 − 3i
(−2 + 3i)(2 − 3i)over(2 + 3i)(2 − 3i)
bottom = 2² + 3² = 13
=+i
  1. You cannot divide by 2 + 3i directly — the bottom has an i in it. The move is to make the bottom a plain number, and the conjugate is what does it: same real part, opposite sign on the imaginary one.
  2. Multiply the top AND the bottom by 2 − 3i. Multiplying by the same thing top and bottom does not change the value — it is the same trick as writing a fraction over a common denominator.
  3. The bottom is where the work pays off: the two middle terms cancel and the last one turns real, so it comes to 2² + 3² = 13. No i left downstairs.
  4. Expand the top the usual way, remembering i² = −1, then divide each part by 13. That gives + i.

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