How this instrument works
A complex number can be written two ways: rectangular form, a + bi, giving its real and imaginary parts directly, or trigonometric (polar) form, r(cos θ + i sin θ), giving its distance from the origin and the angle it makes with the positive real axis. Converting from polar to rectangular applies the same relationship that places a point on a circle: real = r·cos θ and imaginary = r·sin θ, scaled by r rather than fixed to the unit circle.
This is the identical geometry behind reading coordinates off the unit circle, generalized to any radius r instead of exactly 1 — a complex number's modulus r plays the role of the circle's radius, and its argument θ plays the role of the angle, with the real and imaginary axes standing in for the ordinary x and y axes.
Rectangular form is usually easier for addition and subtraction (just combine the real parts and, separately, the imaginary parts), while polar form is usually easier for multiplication and division (multiply or divide the moduli, add or subtract the angles) — this conversion is what lets a calculation move freely between whichever form is more convenient at each step.
- Enter the complex number's modulus (its distance from the origin) into the Modulus (r) field.
- Enter its argument (the angle from the positive real axis) into the Argument (θ) field.
- Read Real part and Imaginary part: the sheet applies r·cos(θ) and r·sin(θ) respectively.
Worked example — modulus 5, argument 53.13°
A complex number has modulus 5 and argument 53.13°. Converting to rectangular form: real = 5 × cos(53.13°) ≈ 3, imaginary = 5 × sin(53.13°) ≈ 4 — landing on 3 + 4i, the same complex number whose modulus and argument were computed by the inverse conversion on this site's companion page.
A modulus of 1 with argument 0° converts to exactly 1 + 0i, the real number 1 with no rotation applied at all, while a modulus of 2 with argument 90° converts to exactly 0 + 2i, a purely imaginary result, since a quarter-turn argument sends every bit of the number's length straight up the imaginary axis.
Questions
How do you convert polar form to rectangular form?
Multiply the modulus r by the cosine of the argument θ to get the real part, and by the sine of θ to get the imaginary part: real = r·cos(θ), imaginary = r·sin(θ). This is the same relationship that locates a point on a circle of radius r.
What do modulus and argument mean?
The modulus is a complex number's distance from the origin on the complex plane, and the argument is the angle it makes with the positive real axis. Together, they describe the same number that rectangular form describes with a real part and an imaginary part.
Why convert to rectangular form at all?
Rectangular form makes addition and subtraction of complex numbers simple — just combine the real parts and, separately, the imaginary parts. Polar form is instead more convenient for multiplication, division, and powers, so calculations often convert back and forth between the two as needed.
What happens at an argument of 0°?
The imaginary part becomes exactly zero, since sin(0°) = 0, and the real part equals the full modulus, since cos(0°) = 1 — the complex number sits purely on the real axis, with no rotation applied.
Is this the same conversion used for polar coordinates in physics?
Yes — the underlying trigonometry is identical to converting polar coordinates (r, θ) to Cartesian coordinates (x, y) in any physics or engineering context; complex numbers simply relabel x and y as the real and imaginary axes.