All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Dividing complex numbers and equating parts

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

F1 builds on the basics of i and a + bi. Here you divide complex numbers, use the fact that two complex numbers are equal only when their real parts and their imaginary parts match, and find square roots of complex numbers.

Practise this in the app: instant marking and a fix for every mistake. Free, no email needed to start.

Practise this lesson

Dividing

To divide by a + bi, multiply the top and the bottom by its conjugate a - bi. The bottom becomes the real number a2 + b2.

(3 + 4i)/(1 - 2i) = (3 + 4i)(1 + 2i)/((1 - 2i)(1 + 2i)) = (3 + 6i + 4i + 8i2)/5 = (-5 + 10i)/5 = -1 + 2i.

Equal complex numbers

a + bi = c + di (with a, b, c, d real) means a = c and b = d. One complex equation therefore gives two real equations.

For an equation in z and z*, write z = a + bi and z* = a - bi, then equate real and imaginary parts.

Example: z + 2z* = 6 - 3i gives (a + bi) + 2(a - bi) = 3a - bi = 6 - 3i, so a = 2, b = 3 and z = 2 + 3i.

Square roots of a complex number

To find sqrt(5 - 12i), let (a + bi)2 = 5 - 12i with a and b real.

Expanding: a2 - b2 + 2abi = 5 - 12i, so a2 - b2 = 5 and 2ab = -12.

Substitute b = -6/a: a2 - 36/a2 = 5, so a4 - 5a2 - 36 = 0, (a2 - 9)(a2 + 4) = 0. As a is real, a2 = 9.

There are always two square roots, negatives of each other: ±(3 - 2i).

Worked example

Express 10/(3 + i) in the form a + bi.

  1. Multiply top and bottom by 3 - i: 10(3 - i)/((3 + i)(3 - i)).
  2. The bottom is 32 + 12 = 10.
  3. 10(3 - i)/10 = 3 - i.

Answer: 3 - i

Key idea: Divide by multiplying by the conjugate of the bottom. Equal complex numbers have equal real parts and equal imaginary parts. Square roots come from (a + bi)2 and equating parts.

Check yourself

  1. Write (3 + 4i)/(1 - 2i) in the form a + bi. Find a.
    Show answer

    -1. Multiply by (1 + 2i)/(1 + 2i): top = 3 + 6i + 4i + 8i2 = -5 + 10i, bottom = 1 + 4 = 5. So the answer is -1 + 2i and a = -1.

  2. z = 2 - i. Find the real part of 1/z. Give your answer as a fraction in its simplest form.
    Show answer

    2/5. 1/(2 - i) = (2 + i)/((2 - i)(2 + i)) = (2 + i)/5, so the real part is 2/5.

  3. z = a + bi satisfies z + 2z* = 6 - 3i. Find b.
    Show answer

    3. z + 2z* = (a + bi) + 2(a - bi) = 3a - bi. Equate parts: 3a = 6 and -b = -3, so b = 3 (and a = 2).

  4. a and b are real and (a + 2i)(3 - i) = 11 + bi. Find b.
    Show answer

    3. Expand: 3a - ai + 6i - 2i2 = (3a + 2) + (6 - a)i. Real parts: 3a + 2 = 11, so a = 3. Imaginary parts: b = 6 - 3 = 3.

  5. Find the square roots of 8 + 6i.
    • ±(2 + 2i)
    • ±(3 + i)
    • ±(3 - i)
    • ±(4 + 3i)
    Show answer

    ±(3 + i). (a + bi)2 = 8 + 6i gives a2 - b2 = 8 and ab = 3. a = 3, b = 1 works: (3 + i)2 = 9 + 6i - 1 = 8 + 6i.

  6. (a + bi)2 = -15 - 8i, where a and b are real and a > 0. Find b.
    Show answer

    -4. a2 - b2 = -15 and 2ab = -8, so b = -4/a. Then a2 - 16/a2 = -15 gives a4 + 15a2 - 16 = 0, (a2 + 16)(a2 - 1) = 0, a = 1 and b = -4. Check: (1 - 4i)2 = 1 - 8i - 16 = -15 - 8i.

  7. z = 1 + i. What is z/z*?
    • -i
    • i
    • 1
    • 2i
    Show answer

    i. (1 + i)/(1 - i) = (1 + i)2/((1 - i)(1 + i)) = 2i/2 = i.

Practise with instant feedback

Sources

Last updated 8 November 2026. Spotted a mistake? Tell us.