Short answer
Rectangular form (a + bi) is easier for addition and subtraction. Polar form (r∠θ) is easier for multiplication, division, powers and roots, because moduli multiply and angles add.
1. Which form is easier
| Operation | Easier in | Why |
|---|---|---|
| Add, subtract | Rectangular | Combine real and imaginary parts |
| Multiply, divide | Polar | Multiply or divide moduli, add or subtract angles |
| Powers, roots | Polar | De Moivre: rⁿ∠nθ |
| Reading real and imaginary parts | Rectangular | They are the numbers a and b |
2. The same product both ways
Rectangular: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i.
Polar: 3 + 4i = 5∠53.13° and 1 − 2i = √5∠−63.43°. Multiply the moduli: 5 × √5 = 11.18. Add the angles: 53.13° − 63.43° = −10.3°. So the product is 11.18∠−10.3°, and 11.18(cos 10.3° − i sin 10.3°) = 11 − 2i.
3. Engineering notation
Electrical engineers write j instead of i and call a polar complex number a phasor, such as 5∠53.13°. The maths is identical because j² = −1.
4. Converting
Rectangular to polar: r = √(a² + b²), θ = atan2(b, a). Polar to rectangular: a = r cos θ, b = r sin θ. Set your calculator to degrees or radians to match the angle you enter.
5. Frequently asked questions
When should I use polar form?expand_more
For multiplication, division, powers and roots.
When should I use rectangular form?expand_more
For addition and subtraction, and when you need the real and imaginary parts.
What is a phasor?expand_more
A complex number in polar form, used in AC circuit analysis.
Is j the same as i?expand_more
Yes. Engineers use j so that i can mean current.
6. Sources
- menu_bookWolfram MathWorld: Complex Number
- menu_bookWolfram MathWorld: De Moivre's Identity