Imaginary Number
Analyze any complex number a+bi by computing its modulus |z| = √(a²+b²) and phase angle θ in degrees. Enter real and imaginary parts to get the cartesian form, magnitude, and angle instantly. Free online math tool.
About This Calculator
Imaginary Number Calculator
This free online imaginary number calculator helps you analyze any complex number z = a + bi by computing its key properties. Enter the real part (a) and imaginary part (b) to instantly see the cartesian form, modulus (magnitude), and phase angle in degrees.
Formulas Used
- Cartesian Form: z = a + bi — standard representation with real part a and imaginary coefficient b.
- Modulus: |z| = √(a² + b²) — the distance from the origin on the complex plane.
- Angle (Argument): θ = arctan(b / a) — the direction angle from the positive real axis, measured in degrees.
Who Is This For?
Students learning complex numbers in algebra or precalculus classes, engineers working with AC circuit analysis and signal processing, and anyone needing quick analysis of imaginary or complex numbers. The calculator handles pure real numbers (b = 0), pure imaginary numbers (a = 0), and general complex numbers in all four quadrants.
Frequently Asked Questions
What is an imaginary number?
An imaginary number is a multiple of the imaginary unit i, where i = √-1. A purely imaginary number has the form bi (no real part), while a complex number has both a real and imaginary part a + bi. This calculator handles both cases, showing the cartesian form, modulus, and angle of any complex number you enter.
How does this Imaginary Number Calculator work?
Enter the real part (a) and imaginary part (b) of your complex number a+bi. The calculator computes the modulus |z| = √(a²+b²), which represents the distance from the origin on the complex plane, and the angle θ = arctan(b/a) in degrees, which is the direction from the positive real axis. For example, entering a=3 and b=4 gives modulus 5 and angle 53.13°.
What is the difference between an imaginary number and a complex number?
A complex number has both a real part and an imaginary part, written as a + bi. An imaginary number is a complex number where the real part is zero, such as 5i or -3i. In other words, all imaginary numbers are complex numbers, but not all complex numbers are imaginary numbers.
What is the modulus of a complex number used for?
The modulus (magnitude) |z| = √(a²+b²) represents the distance of the complex number from the origin on the complex plane. It is used extensively in electrical engineering (impedance magnitude), signal processing (amplitude of signals), quantum mechanics (wave function normalization), and control theory (system stability analysis).
Can I calculate the angle of a real number or a pure imaginary number?
Yes. For a pure real number (b=0), the angle is 0° if a > 0, or 180° if a < 0. For a pure imaginary number like 5i (a=0, b=5), the angle is 90°. For -5i, the angle is -90°. The calculator uses the atan2 function to handle all cases correctly across all four quadrants.
Is this imaginary number calculator free?
Yes, all calculators on Calculy are completely free to use with no registration or usage limits. You can bookmark or share the URL with your input values pre-filled.