Imaginary Calculator

Imaginary numbers stop being mysterious the moment you can push them through real arithmetic. This imaginary calculator adds, subtracts, multiplies and divides complex numbers, finds the conjugate, modulus and argument of a single value, and adds the two modes most tools leave out: square roots of negative numbers, so √(−9) comes back as 3i instead of an error, and powers of i resolved through the four-step cycle.

Quick Answer

How to use the imaginary number calculator

Pick an operation first, because the operation decides which fields matter, then enter the parts and press Calculate. Results stay hidden until you press it, and switching operations hides the previous result.

1. Add, subtract, multiply, divide: enter both numbers, a + bi and c + di. Blank part boxes are treated as 0 without an error message, so type only the parts you have.
2. Conjugate, or modulus and argument: only the first number is used. The second number stays on screen in these two modes and the single-value field appears labelled “Not used”; both are ignored, which the FAQ below states again so nobody trusts a stray value sitting in them.
3. Square root of a negative number: the second number hides and the single field takes one value, usually negative. A negative input returns an imaginary root with a check line; zero or a positive input returns the ordinary real square root with a note saying so.
4. Power of i: enter any whole-number exponent, including negatives. A decimal exponent is refused with “The exponent must be a whole number.”
5. Copy or share from the results box: the copied line names the operation and the result, for example “Imaginary number calculation (Add: (a + bi) + (c + di)): 4 + 2i”.

The eight operations and the formulas behind them

For z1 = a + bi and z2 = c + di, each operation is one short rule, and the calculator applies the same rules a textbook does. The formulas for modulus and argument follow the standard treatment in OpenStax Precalculus, section 8.5 on the polar form of complex numbers, where the modulus is defined as the distance from the origin.

Division is the one students get wrong, so it is worth seeing why the formula looks that way. Multiplying top and bottom by the conjugate of the denominator turns c + di into c² + d², a plain real number, and the division then splits into two ordinary divisions. The University of Wisconsin lecture notes on complex numbers and the complex exponential make the same point: remember the conjugate trick, not the finished formula. The argument uses atan2 rather than a plain arctangent because atan2 reads the signs of both parts and puts the angle in the right quadrant; it is the quadrant-aware relative of the function behind the arctan calculator.

Worked examples across five modes, verified against the tool

Every row below is a separate run of the calculator above with the inputs shown, and the outputs are copied from the tool at its own four-decimal display precision. Run any row yourself to check it.

Two display habits are worth knowing before you trust a screenshot of a result. First, the formatter drops empty parts and prints coefficients in full: a result of 1 + i appears as “1 + 1i”, a pure imaginary result such as the product (9999 + 9999i) × (9999 + 9999i) appears as “199960002i”, and 0 + 0i appears as plain “0”. Second, in modulus mode the Result row repeats the number you entered; the new information in that mode is the modulus and argument beside it, not the Result row.

Powers of i: the four-step cycle, and why i⁸ is 1

The powers of i cycle through four values and then restart: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Any exponent, however large or negative, lands on one of those four, and n mod 4 tells you which one.

This is the point where hurried write-ups slip: i⁸ is sometimes “worked” as i⁴ × i³, but those exponents add to 7, not 8. The clean split is i⁸ = (i⁴)² = 1² = 1, and the tool agrees: enter 8 and it returns 1 with the extra line “n mod 4 = 0”. The same check gives i⁷ = −i (7 mod 4 = 3), i¹⁰ = −1 (10 mod 4 = 2), i⁻¹ = −i, and i^123456789 = i, because 123456789 mod 4 = 1. General power work with ordinary bases belongs to the exponent calculator; this mode is only for the unit i itself.

Modulus and argument: reading the polar readout

The modulus is a distance and the argument is a direction: together they are the polar coordinates of the result. For arithmetic results the tool prints both automatically, in radians and degrees, at four-decimal precision.

Read 4 + 2i as a point: modulus 4.4721 says the point sits 4.4721 units from the origin, and argument 0.4636 rad (26.5651 deg) says it points up and to the right, in Quadrant I, which is also what the green badge in the results box reports. Negative arguments are normal, not errors: 11 − 2i sits in Quadrant IV, so its argument prints as −0.1799 rad (−10.3048 deg), measured clockwise from the positive real axis. The principal argument here always lands between −180 and 180 degrees because atan2 returns that range; add 360 degrees to a negative value if a course wants the 0 to 360 form. The Stan reference documentation for its complex special functions defines abs, arg and conj in exactly these terms, and the LibreTexts precalculus text treats the same conversion under the trigonometric form of a complex number. Two modes print “not applicable” for the argument instead: square roots and powers of i, where the result is defined directly rather than as a plotted a + bi value. In square root mode the modulus row shows the size of the root, and in powers mode it is fixed at 1, since every power of i has modulus 1.

Where imaginary numbers actually get used

Complex numbers earn their keep anywhere a quantity has both size and direction. In AC circuit work, impedance pairs resistance with reactance as a + bi, and the modulus and argument printed here are the magnitude and phase angle an engineer reads off. In signal processing, the same pair describes the amplitude and phase of each frequency component.

The conjugate pair pattern shows up in algebra too: a real 2×2 matrix with a rotation in it has eigenvalues that arrive as a conjugate pair such as 3 ± 4i, which is why the complex eigenvalue calculator reports values in the same a + bi form this tool manipulates. Quadratic equations with no real roots end here as well: x² + 9 = 0 has solutions 3i and −3i, exactly the output of the square root mode on −9 and its conjugate.

What competing calculators miss

The rivals for this keyword split into expression engines and single-job tools, and each group leaves a gap this page is built to close:

• HackMath complex number calculator: the deepest of the expression engines, with phasor, polar and exponential forms and worked steps. The cost is the input: you must type a well-formed expression such as 5*(1+i)(-2-5i)^2, and a syntax slip returns nothing useful. There is no guided a + bi form and no dedicated square-root-of-a-negative or powers-of-i workflow.
• Omni Calculator complex number calculator: broad operation coverage and good polar-form teaching, but it is organised around general complex arithmetic; the two school-level jobs students search for by name, √(−x) and iⁿ, have no dedicated modes.
• MiniWebTool complex number calculator: a wide menu including conversions, powers and roots of full complex numbers, which overshoots the homework intent: someone who only needs i³⁷ or the conjugate of 3 + 4i wades through De Moivre machinery to find it.
• Cuemath imaginary number calculator: the opposite extreme, a single-job tool for square roots of negative numbers with worked examples such as √−49 = 7i. It does that one job well and nothing else, so the follow-up multiplication needs a second site.
• ZeeSharp and similar form-based tools: clean a + bi entry and the four operations with modulus, argument and conjugate, which matches this tool on arithmetic, but no negative-root mode, no powers-of-i mode, and no copy or share of the result.
• The gap this page fills: guided part-by-part entry, all four operations plus conjugate and polar readout, dedicated modes for √(−x) and iⁿ with their check lines, and honest notes about the fields each mode ignores, on one page.

Frequently asked questions

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References