Last updated: July 31, 2026
Imaginary Number Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
For z = a + bi, magnitude = √(a² + b²), argument = atan2(b, a) × 180/π, and conjugate(z) = a - bi.
Where:
- a=Real part
- b=Imaginary part
- |z|=Magnitude
- arg(z)=Argument in degrees(degrees)
- z̄=Complex conjugate
Worked Examples
Classic 3-4-5 complex number
Evaluate z = 3 + 4i.
- 1Write the complex number as z = 3 + 4i.
- 2Compute magnitude: √(3² + 4²) = √25 = 5.
- 3Compute argument: atan2(4, 3) × 180/π ≈ 53.130102°.
- 4Conjugate keeps the real part and flips the sign of the imaginary part.
Purely real number
Evaluate z = 1 + 0i.
- 1Magnitude is √(1² + 0²) = 1.
- 2The point lies on the positive real axis, so the argument is 0°.
- 3The conjugate of a purely real number is the same value.
Second-quadrant complex number
Evaluate z = -2 + 5i.
- 1Magnitude is √((-2)² + 5²) = √29 ≈ 5.385165.
- 2atan2(5, -2) returns an angle in quadrant II, about 111.801409°.
- 3Conjugate is found by changing +5i to -5i.
Negative imaginary component
Evaluate z = 6 - 8i.
- 1Magnitude is √(6² + (-8)²) = √100 = 10.
- 2atan2(-8, 6) gives a negative angle, about -53.130102°.
- 3Conjugate changes -8i to +8i.
Introduction
Imaginary and complex numbers extend the real number system so equations like x² + 1 = 0 have meaningful solutions. This calculator focuses on the geometric properties of a single complex number in rectangular form, making it easy to connect algebra, geometry, and polar interpretation.
What z = a + bi Means
A complex number has a real component a and an imaginary component b. Together they locate a point (a, b) on the complex plane, where the horizontal axis is real and the vertical axis is imaginary.
a controls left-right position.
b controls up-down position.
The ordered pair (a, b) behaves like a vector from the origin.
Rectangular form is often the easiest way to enter a complex number into a calculator.
If b = 0, the complex number is purely real. If a = 0, it is purely imaginary.
How Magnitude Is Calculated
Magnitude is the distance from the origin to the point representing the complex number. The Pythagorean theorem gives that distance directly.
Square the real part.
Square the imaginary part.
Add the squares.
Take the square root to obtain |z|.
Magnitude is always non-negative, even if one or both components are negative.
Why atan2 Is Used for the Argument
The argument is the direction angle of the complex number. Using atan2(b, a) instead of a plain arctangent keeps the angle in the correct quadrant and handles signs correctly.
Quadrant I angles are positive and acute.
Quadrant II angles are positive and obtuse.
Quadrant III and IV can produce negative principal angles.
The calculator returns the result in degrees for easy interpretation.
When the number is exactly 0 + 0i, the argument is mathematically undefined, but atan2(0, 0) returns 0 in JavaScript. This tool reports that numeric result for consistency.
Understanding the Conjugate
The conjugate reflects the point across the real axis. It keeps the same real part and reverses the sign of the imaginary part.
Conjugate of a + bi is a - bi.
A number and its conjugate have the same magnitude.
Multiplying a number by its conjugate removes the imaginary part.
Conjugates are used in rationalizing denominators and signal processing.
Reading the Standard Form Output
The form output is a display-friendly version of the same complex number. It uses a plus sign only when the imaginary coefficient is non-negative, so results read naturally as a+bi or a-bi.
3 and 4 become 3+4i.
6 and -8 become 6-8i.
0 and 5 become 0+5i.
The formatted string is useful for copy-paste into notes or homework.
How to Use This Calculator
Enter the real part and imaginary part, then review the magnitude first and the angle second. The conjugate and formatted form help you verify sign conventions quickly.
Enter a finite real number for a.
Enter a finite real number for b.
Read the magnitude as the complex-plane distance.
Use the argument to identify the direction or quadrant.
Where These Outputs Are Useful
Complex-number properties appear in electrical engineering, control systems, wave analysis, and advanced algebra courses.
Polar conversion checks.
Impedance and phasor calculations.
Roots of polynomials.
Visualization of rotations and oscillations.
If you later need multiplication or division of complex numbers, the magnitude and argument already give you a strong starting point.
FAQs
What does the magnitude represent?
It is the distance from the origin to the point (a, b) on the complex plane, computed as √(a² + b²).
Why is the argument given in degrees?
Degrees are easier for many students to interpret quickly. Internally the angle comes from atan2 and is converted from radians to degrees.
Can the argument be negative?
Yes. A principal argument in quadrant IV is often negative, such as -53.130102° for 6 - 8i.
What is the conjugate used for?
Conjugates help simplify complex fractions, remove imaginary denominators, and compute products like z·z̄ = |z|².
Does the calculator support decimals?
Yes. Any finite real-number inputs are accepted for both the real and imaginary parts.
What happens for 0 + 0i?
The magnitude is 0. JavaScript returns 0 for atan2(0, 0), so the calculator reports 0° for a stable output even though the argument is theoretically undefined.
Why is the form output shown as text?
The text output preserves the exact sign pattern of the entered complex number and makes the result easy to read or copy.
How precise are the numeric outputs?
Magnitude and argument are rounded to six decimal places to balance clarity with accuracy.