Maths › Complex numbers › Complex arithmetic and the Argand diagram
Complex arithmetic and the Argand diagram
One new number, i, with i squared equal to minus one, and every quadratic suddenly has its roots. Arithmetic follows the old rules; a diagram makes the numbers visible.
Builds on Quadratic functions and Vectors in two dimensions.
IN THIS TOPIC
- Solve any quadratic, reading complex roots from a negative discriminant.
- Add, multiply and divide complex numbers, using the conjugate to clear denominators.
- Plot numbers and their conjugates on the Argand diagram and read addition as vectors.
WHAT YOU PROBABLY THINK
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 the opening claim collapses at once: z2 = −9 has the solutions z = ±3i. A complex number is any z = x + yi with x and y real; x is the real part, 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 they arrive as a conjugate pair, x + yi with 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.
Arithmetic, and the conjugate's job
Addition and subtraction work part by part. Multiplication is a bracket expansion: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i2 = 1 + 5i. Division needs one idea: multiplying a number by its conjugate kills the imaginary part, since (x + yi)(x − yi) = x2 + y2. 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: (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.
The parallel with rationalising a denominator is worth noticing: 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: 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.
YOUR TURN
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.
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); z* = 2 − 3i is its reflection at (2, −3), the same distance below the real axis as z is above.
THE EXAM BIT
- Keep the ± with the i when reading roots from the formula: (6 ± 4i)/2 simplifies both parts.
- Complex roots of real quadratics come in conjugate pairs; quote the pair, not one root.
- In division, compute the real denominator x² + y² first and keep it as one fraction.
- 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
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i squared is minus one; everything else is ordinary algebra.
To divide, multiply top and bottom by the conjugate of the denominator.
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.
CHECK YOUR PROGRESS
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- Solve any quadratic, reading complex roots from a negative discriminant.
- Add, multiply and divide complex numbers, using the conjugate to clear denominators.
- 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.
No animated video for this topic yet; these notes stand alone.