How this instrument works
A complex number's polar form describes it with two numbers instead of a real and imaginary part directly: a modulus (its distance from the origin) and an angle (the direction it points from the positive real axis). Converting back to the ordinary a+bi (rectangular) form uses the same right-triangle relationship trigonometry always relies on — the real part is the modulus times the cosine of the angle, and the imaginary part is the modulus times the sine of the angle.
This page runs the exact reverse of this site's Complex Number calculator, which starts from a+bi and finds the modulus and angle instead; together, the two pages cover both directions of the identical conversion, letting a complex number move freely between whichever description is more convenient for the task at hand.
Polar form is especially convenient for MULTIPLYING complex numbers together — moduli multiply and angles simply add — but eventually a result often needs converting back to a+bi form for further use, which is exactly the step this page performs.
- Enter the modulus into the r field.
- Enter the angle into the θ field.
- Read Real part: the modulus times the cosine of the angle.
- Read Imaginary part: the modulus times the sine of the angle.
Worked example — modulus 10, angle 30°
A complex number with modulus 10 and angle 30° converts to rectangular form as a=10×cos(30°)≈8.660 and b=10×sin(30°)=5, giving the complex number 8.660+5i.
A modulus of 5 at an angle of exactly 90° converts to a=5×cos(90°)=0 and b=5×sin(90°)=5 — landing purely on the imaginary axis, since 90° points straight up on the complex plane. A modulus of 1 at an angle of 0° converts to a=1 and b=0 — landing exactly on the positive real axis, the complex number 1 itself.
Questions
How do you convert from polar form back to a+bi?
Multiply the modulus by the cosine of the angle to get the real part, and by the sine of the angle to get the imaginary part — the same right-triangle relationship that connects any radius-and-angle description to horizontal and vertical coordinates.
How is this different from the Complex Number calculator on this site?
That page starts from a+bi and finds the modulus and angle; this page runs the identical relationship in reverse, starting from the modulus and angle and converting back to a+bi.
Why bother with polar form if it just converts back to a+bi anyway?
Polar form makes MULTIPLYING complex numbers together far simpler — moduli multiply and angles simply add — but after that multiplication, converting the result back to a+bi (which this page does) is often needed for further use.
What complex number does a modulus of 0 always give?
Exactly 0+0i, regardless of the angle entered — a modulus of 0 places the point right at the origin, where direction no longer has any meaning.
What does an angle of 180° give?
A purely negative real number: cos(180°)=−1 and sin(180°)=0, so the result lands directly on the negative real axis at distance r from the origin.