Math & Conversions

Complex Number Calculator

Add and multiply two complex numbers z1 = a + bi and z2 = c + di, returning the sum and the product in a + bi form.

What this calculator does

The Complex Number Calculator estimates your result instantly from the values you enter, using standard math & conversions formulas. Add and multiply two complex numbers z1 = a + bi and z2 = c + di, returning the sum and the product in a + bi form. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.

Complex Number Calculator

Result

—
z1 + z2
—
z1 x z2

About Complex Number Calculator

This calculator works with two complex numbers, z1 = a + bi and z2 = c + di. You supply the real and imaginary parts of each, and it returns two results: the sum z1 + z2 and the product z1 × z2, both written in standard a + bi form.

It is handy for students checking algebra homework or anyone who wants to skip the by-hand steps of combining like terms and expanding (a + bi)(c + di). Note that this tool covers addition and multiplication only.

Formula & Mathematical Calculation

z1 + z2 = f(Real part of z1 (a), Imag part of z1 (b), Real part of z2 (c), Imag part of z2 (d))

Calculated according to standard math formulas implemented in the calculator core.

Calculation Example

Take z1 = 3 + 2i (a = 3, b = 2) and z2 = 1 − 4i (c = 1, d = −4).

  • Sum: (3 + 1) + (2 + (−4))i = 4 − 2i
  • Product: (ac − bd) + (ad + bc)i = (3×1 − 2×(−4)) + (3×(−4) + 2×1)i = 11 − 10i

The calculator reports z1 + z2 = 4 - 2i and z1 x z2 = 11 - 10i.

Key Benefits

  • Combines both the sum and the product of two complex numbers in one step
  • Shows results in standard a + bi form so they are easy to read and check
  • Handles negative and decimal parts for real and imaginary components

How to Use this Calculator

  1. Enter z1
    Type the real part of z1 in the 'Real part of z1 (a)' field and the imaginary part in 'Imag part of z1 (b)'.
  2. Enter z2
    Type the real part of z2 in the 'Real part of z2 (c)' field and the imaginary part in 'Imag part of z2 (d)'.
  3. Calculate
    Press Calculate to see z1 + z2 and z1 x z2 displayed in a + bi form.

Assumptions & Limitations

What the calculation assumes

  • Every value you enter is treated as an exact, known amount.
  • Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
  • Results use standard published formulas and account only for the values entered above.
  • Rounding is applied for display; the underlying calculation uses your full precision.
  • Results are returned in the same units you enter, unless the tool states the output unit.

What it does not cover

  • Complex Number Calculator produces a calculated result for reference and learning, not a certified or official measurement.
  • Real-world outcomes differ when tolerances, rounding rules or local standards apply.
  • Definitions and standards vary by region and change over time; verify the convention that applies to your case.
  • Figures are only as accurate as the inputs you supply — check them before acting on the result.
  • For work where the result carries real consequences, confirm it against an authoritative source.

Frequently Asked Questions

It returns the sum z1 + z2 and the product z1 x z2. It does not compute subtraction, division, modulus, or argument.

Using (a + bi)(c + di) = (ac - bd) + (ad + bc)i, so both the real and imaginary parts are computed at once.

Yes. Each part accepts negative numbers and decimals, and the results reflect those inputs directly.

Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.