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Complex arithmetic and the Argand diagram

One new number, i, whose square is minus one, and every quadratic suddenly has its roots. Arithmetic follows the old rules, and a diagram makes the numbers visible.

Year FMEDEXCEL 9FM0 CP1 2.1, 2.2, 2.3, 2.4

Builds on Quadratic functions and Vectors in two dimensions.

IN THIS TOPIC

  • Solve any quadratic, reading complex roots from a negative discriminant.
  • Add, subtract and multiply complex numbers without special rules.
  • Divide by using the conjugate to clear the denominator.
  • Plot numbers and their conjugates on the Argand diagram, and read addition as vectors.

COMMON MISCONCEPTION

A negative number has no square root, so z² = −9 has no solutions.

A number whose square is negative

Define i by i2 = −1 and z2 = −9 acquires the solutions z = ±3i. A complex number is any z = x + yi with x and y real, where x is the real part and y the imaginary part. Nothing else in algebra changes. Brackets expand, like terms collect, and every i2 that appears becomes −1.

The quadratic formula now works unconditionally. A negative discriminant no longer means 'no roots'; it means the roots are complex, and for a quadratic with real coefficients they arrive as a conjugate pair, x + yi alongside x − yi, written z and z*.

WORKED EXAMPLE

A quadratic with no real roots

Solve z2 − 6z + 13 = 0.

The discriminant is 36 − 52 = −16, so the roots are complex.

z = (6 ± √(−16))/2 = (6 ± 4i)/2 = 3 ± 2i.

The two roots are conjugates, mirror images across the real axis, and their sum 6 and product 13 recover the original coefficients.

The complex number 3 + 2i plotted on the Argand diagram, with its conjugate mirrored below the real axisz = 3 + 2iz* = 3 − 2iReIm
FIG. 1z = 3 + 2i and its conjugate z* = 3 − 2i on the Argand diagram, reflections of each other in the real axis.

Arithmetic, and the conjugate's job

Addition and subtraction work part by part. Multiplication is a bracket expansion, so (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i2 = 1 + 5i. Division needs one idea. Multiplying a number by its conjugate removes the imaginary part, since (x + yi)(x − yi) = x2 + y2, a real number. So to divide, multiply top and bottom by the conjugate of the bottom.

WORKED EXAMPLE

Division via the conjugate

Write (3 + 2i)/(1 − i) in the form x + yi.

Multiply top and bottom by 1 + i. The denominator becomes (1 − i)(1 + i) = 1 + 1 = 2.

The numerator is (3 + 2i)(1 + i) = 3 + 3i + 2i − 2 = 1 + 5i.

So the quotient is 1/2 + (5/2)i. The conjugate is chosen so the cross terms cancel, exactly as with surd denominators.

You have met this before. 1/(3 − √2) and 1/(3 − 2i) are cleared by the same trick, a conjugate chosen to produce a difference of squares.

The Argand diagram

Plot x + yi at the point (x, y) and complex numbers become geometry. That picture is the Argand diagram. Real numbers live on the horizontal axis, purely imaginary ones on the vertical, and conjugation is reflection in the real axis. Addition is vector addition, nose to tail or by parallelogram, component by component.

Adding complex numbers is vector addition: 3 + i and 1 + 2i complete a parallelogram at 4 + 3i3 + i1 + 2i4 + 3i
FIG. 2Adding 3 + i and 1 + 2i on the Argand diagram: a parallelogram whose far corner sits at the sum, 4 + 3i.

GUIDED PRACTICE

Arithmetic, then a picture

For z = 2 + 3i and w = 4 − i, find z + w, zw and z/w, and state where z and z* sit on the Argand diagram.

Show the working

z + w = 6 + 2i. zw = (2 + 3i)(4 − i) = 8 − 2i + 12i + 3 = 11 + 10i.

For z/w, multiply by the conjugate of w. (2 + 3i)(4 + i)/((4 − i)(4 + i)) = (8 + 2i + 12i − 3)/17 = (5 + 14i)/17.

z sits at (2, 3), and z* = 2 − 3i is its reflection at (2, −3), the same distance below the real axis as z is above.

ASSESSMENT FOCUS

  • Keep the ± with the i when reading roots from the formula, then simplify both parts of (6 ± 4i)/2.
  • Complex roots of real quadratics come in conjugate pairs, so quote the pair, never one root alone.
  • In division, compute the real denominator x² + y² first and keep the answer as one fraction until the end.
  • State points on the Argand diagram as coordinates. 3 + 2i sits at (3, 2).

CHECK YOURSELF

Solve z2 + 4z + 29 = 0, and describe how the two roots sit on the Argand diagram.

Show a hint

Complete the square or use the formula; the discriminant is −100.

Show the answer

The discriminant is 16 − 116 = −100, so z = (−4 ± 10i)/2 = −2 ± 5i. The roots are a conjugate pair at (−2, 5) and (−2, −5), reflections of each other in the real axis.

i squared is minus one. Everything else is ordinary algebra.

To divide, multiply top and bottom by the conjugate of the denominator.

Conjugation reflects in the real axis, and real quadratics have conjugate root pairs.

WORKBOOK

Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.

Question workbook

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Question workbook

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Answer book

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7 questions on this topicAnswer them one at a time and mark yourself against the worked answer.Practise this topic

Or read them with their worked answers on the complex arithmetic and the argand diagram questions page.

CHECK YOUR PROGRESS

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  • Solve any quadratic, reading complex roots from a negative discriminant.
  • Add, subtract and multiply complex numbers without special rules.
  • Divide by using the conjugate to clear the denominator.
  • Plot numbers and their conjugates on the Argand diagram, and read addition as vectors.

Open the full revision checklist to see every objective in the course in one place.