Master the Method for Dividing Complex Numbers

Most attempts at dividing complex numbers fail because people try to treat i as a variable and separate it into two fractions. That is not how it works. The key step is multiplying both the top and bottom by the complex conjugate of the denominator. That single operation turns the denominator into a real number and lets you split the result into a clean real part and imaginary part. Here is the exact method and the mistakes that will ruin it.

The complex conjugate of a number a+bi is a−bi. You flip the sign of the imaginary part only. Geometrically, the conjugate reflects the point across the real axis. Its practical purpose: when you multiply a number by its conjugate, the result is always a real, non-negative number equal to a²+b², which is the square of the modulus. For the denominator c+di, multiplying by c−di gives c²+d², a quantity you can divide into without worrying about imaginary denominators.

This is the single tool that makes complex division possible in rectangular form. Without it, you are stuck.

To divide (a+bi) by (c+di), write the division as a single fraction. Multiply the top and the bottom by the conjugate of the denominator, c−di. Expand the top using normal algebra (remember i² = −1). The bottom becomes c²+d², a real number. Now take the real terms and the imaginary terms from the top, and divide each by the real denominator. The result is a new complex number in a+bi form.

The formula is exactly (a+bi)/(c+di) = ((a+bi)(c−di))/(c²+d²). The whole process takes about 30 seconds once you have practiced it. The most common error at this stage is forgetting to distribute the minus sign when you subtract the imaginary parts in the expansion. Write every term explicitly, then combine.

(3+4i) ÷ (1−2i)

Multiply top and bottom by the conjugate of the denominator, which is 1+2i. The top becomes (3+4i)(1+2i) = 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i. The bottom is 1² + 2² = 1 + 4 = 5. So the result is (−5+10i)/5 = −1 + 2i. To verify, multiply (−1+2i) by (1−2i). (−1+2i)(1−2i) = −1 + 2i + 2i − 4i² = −1 + 4i + 4 = 3 + 4i, which matches the top. The result is correct.

(1+i) ÷ (1−i)

The conjugate of the denominator is 1+i. Multiply top and bottom by 1+i. The top is (1+i)(1+i) = 1 + 2i + i² = 1 + 2i − 1 = 2i. The bottom is 1² + 1² = 2. So the result is 2i/2 = i. Check by multiplying i by (1−i). i(1−i) = i − i² = i + 1 = 1+i, which matches the original top. The division is correct.

(2+3i) ÷ (4+5i)

Conjugate of the denominator is 4−5i. Multiply: (2+3i)(4−5i) = 8 − 10i + 12i − 15i² = 8 + 2i + 15 = 23 + 2i. Bottom: 4² + 5² = 16 + 25 = 41. Result is (23+2i)/41. To verify, multiply (23+2i)/41 by (4+5i). (23+2i)(4+5i) = 92 + 115i + 8i + 10i² = 92 + 123i − 10 = 82 + 123i. Then divide by 41: (82+123i)/41 = 2+3i. That is the top. The verification works: the quotient times the denominator equals the original top.

Dividing by a Pure Imaginary

When the denominator is a pure imaginary, like 3i, the conjugate is −3i. For (2+5i)/(3i), multiply by (−3i)/(−3i). The top is (2+5i)(−3i) = −6i − 15i² = −6i + 15 = 15 − 6i. The bottom is 3i * (−3i) = −9i² = 9. Result is (15−6i)/9 = (15/9) − (6/9)i = (5/3) − (2/3)i. You can also divide directly by writing each term over the denominator: (2)/(3i) + (5i)/(3i) = (2)/(3i) + 5/3. Then rationalise (2)/(3i) by multiplying top and bottom by i: (2i)/(3i²) = (2i)/(−3) = −(2/3)i. Combine: 5/3 − (2/3)i. Same result.

If you have both numbers in polar form, r(cos θ + i sin θ), division becomes simpler than the rectangular method. You divide the r values and subtract the angles. For two numbers z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), the quotient is (r1/r2)[cos(θ1 − θ2) + i sin(θ1 − θ2)].

This works because multiplication and division preserve the structure of the polar representation, as De Moivre's theorem (for integer exponents) shows. The polar method is faster for division but requires that you can convert between rectangular and polar forms fluently. If the angle difference falls outside the principal argument range (−π, π], adjust by adding or subtracting 2π.

Engineers using phasor notation (r∠θ) do exactly this: divide magnitudes, subtract phases. The result is the phasor of the quotient. This is standard practice in AC circuit analysis as described in Alexander and Sadiku's Fundamentals of Electric Circuits.

The Quadrant Mistake

The most frequent error in complex division is the quadrant mistake. When you try to find the argument of a complex number by using arctan(b/a), that only works when the real part a is positive. If a is negative, you must add π (or 180°) to the result from arctan. For example, the argument of −1+i is 3π/4 (135°), not −π/4 (−45°). The atan2 function on calculators handles this, but doing it manually without the quadrant adjustment gives a 180° error.

Conjugate and Sign Errors

Another common slip is writing the conjugate incorrectly. For c+di, the conjugate is c−di, not c+di. That mistake leaves the denominator still complex, and the division never works. Also, students sometimes forget that i² = −1 when expanding, turning 8i² into 8 instead of −8. In the example (3+4i)/(1−2i), missing that sign gives 3+10i+8=11+10i, which is wrong.

Angle Mode Mismatch

When using polar form, the main error is subtracting the angles in degrees when the calculator is in radian mode, or vice versa. The result will be rotated by the wrong amount. Always check the calculator's angle mode before you enter trigonometric functions. The Casio fx-991EX and TI-84 Plus CE both have a dedicated complex mode, but the argument they return depends on the principal range convention. The Casio uses 0 to 2π; the TI uses −π to π. Know which one your exam expects.

Don't Assume Commutativity

Do not assume division is commutative. (a+bi)/(c+di) is not the same as (c+di)/(a+bi). And never try to split the fraction into a/(c+di) + bi/(c+di) unless you rationalise each part separately, which is more work. The conjugate method on the whole fraction at once is the correct approach.