Multiply
Find (2 + 3i)(1 − i).
- Expand: 2 − 2i + 3i − 3i².
- Replace −3i² by +3, then collect parts.
Answer: 5 + i
Multiplying and dividing complex numbers
Multiply using i² = −1 and divide with a conjugate.
Before you start
You’ll need Expanding and factorising. Follow a link if you want to revise it first.
Working with i² = −1
A complex number has the form a + bi, where i² = −1. The imaginary coefficient b is a real number.
Expand products as ordinary brackets, then replace i² by −1 before collecting terms.
To divide, multiply numerator and denominator by the denominator’s conjugate. The denominator becomes a² + b², a positive real number unless both a and b are zero.
The conjugate changes the sign of the imaginary part. It does not negate the whole number.
A product and a quotient
Divide
Find (2 + 3i)/(1 − i).
- Multiply top and bottom by 1 + i.
- The numerator is −1 + 5i; the denominator is 2.
Answer: −1/2 + (5/2)i
Treating i² as +1 gives the wrong real part.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
Enable JavaScript for a fresh three-question check, or use the practice links above.
Why is the product of a nonzero complex number and its conjugate real and positive?