a + bi Form Calculator
This a + bi form calculator works with a complex number written in rectangular form, a + bi, where a is the real part and b is the imaginary part. Enter a and b and it returns the number's magnitude (modulus) √(a² + b²) and its phase (argument) in degrees, found with atan2(b, a). Together these give the polar form of the complex number.
Formula
Magnitude = √(a² + b²), Phase = atan2(b, a)
- A complex number in a + bi form has a real part a and an imaginary part b.
- Its magnitude (modulus) is the distance from the origin in the complex plane: |z| = √(a² + b²).
- Its phase (argument) is the angle from the positive real axis, computed with atan2(b, a) so it falls in the correct quadrant, then converted to degrees.
- Magnitude and phase are the polar coordinates of the number: z = |z|(cos θ + i sin θ).
- A positive real number (b = 0) has phase 0°; a positive pure imaginary number (a = 0, b > 0) has phase 90°.
Example Calculation
Inputs
- Real Part (a): 3
- Imaginary Part (b): 4
For 3 + 4i, the magnitude is √(3² + 4²) = √25 = 5 and the phase is atan2(4, 3) ≈ 53.13°. So 3 + 4i ≈ 5 at an angle of 53.13°.
Frequently asked questions
What is a + bi form?
It is the rectangular form of a complex number, where a is the real part and b is the coefficient of the imaginary unit i. For example, 3 + 4i has a = 3 and b = 4.
What do magnitude and phase mean?
The magnitude (modulus) is how far the number sits from the origin in the complex plane, √(a² + b²). The phase (argument) is the angle that point makes with the positive real axis.
Why use atan2 instead of arctan(b/a)?
Plain arctan(b/a) loses quadrant information and is undefined when a = 0. atan2(b, a) returns the correct angle in all four quadrants.
What is the magnitude and phase of 3 + 4i?
The magnitude is √(3² + 4²) = 5 and the phase is atan2(4, 3) ≈ 53.13°.
How do magnitude and phase relate to polar form?
They are the polar coordinates of the complex number: z = r(cos θ + i sin θ), where r is the magnitude and θ is the phase.