Gaussian integer multiplication calculator

Enter the four coefficients separately, and the calculator applies the distributive rule while accounting for i squared equaling negative one. The result contains a real component and an imaginary coefficient, ready for number theory exercises, algebra checks, program tests, or any workflow that needs a dependable Gaussian integer product.

Represent both Gaussian integers with four components

A Gaussian integer has the form a + bi, where a and b are ordinary integers and i is the imaginary unit. To multiply two of them, identify the real and imaginary coefficients before entering anything. The first value uses a_real for a and a_imag for b. The second uses b_real for c and b_imag for d. For example, to multiply 3 + 2i by 1 - 4i, enter 3, 2, 1, and -4 in those fields. Keep the sign attached to each coefficient: the coefficient of -4i is -4, not 4. A purely real Gaussian integer has an imaginary coefficient of zero, while a purely imaginary one has a real component of zero. Zero is therefore a valid value in every field. All four fields are required because an omitted coefficient is ambiguous in an API request. Components must be actual JSON integers within the published bounds; decimal fractions, numeric text, infinity, and other non-integer values are rejected instead of being rounded or silently converted. This strict contract keeps exact arithmetic distinct from approximate complex-number calculation.

Understand how the product is calculated

The calculation expands (a + bi)(c + di) using the distributive law. Multiplying each term gives ac + adi + bci + bdi squared. Since i squared equals -1, the last term becomes -bd. Combining the terms without i produces the real component ac - bd, and combining the coefficients of i produces the imaginary component ad + bc. The calculator returns those two integers as real and imaginary. For 3 + 2i and 1 - 4i, the real component is 3 times 1 minus 2 times -4, while the imaginary coefficient is 3 times -4 plus 2 times 1. This representation avoids formatting uncertainty around plus and minus signs and is convenient for further machine processing. The algorithm uses only bounded integer multiplication, addition, and subtraction. The component limit guarantees that every intermediate value and final result remains within JavaScript's exact safe-integer range, so there is no floating-point rounding hidden in a successful response. The operation is deterministic: identical inputs always produce identical output, with no network service, model inference, clock, or randomness involved.

Use the result in mathematics and software

Gaussian integers form a ring, so their product is always another Gaussian integer. That closure makes multiplication useful in elementary number theory, including reasoning about sums of two squares, Gaussian primes, norms, and factorization in Z[i]. You can verify a manual expansion by comparing its two coefficients with the response, or feed the returned real and imaginary values into the next exact step of a calculation. A useful independent check is the multiplicative norm N(a + bi) = a squared + b squared: the norm of the returned product should equal the product of the two input norms. In software tests, include sign changes, zero components, purely real factors, purely imaginary factors, and multiplication by the units 1, -1, i, and -i. The endpoint costs $0.002 per API request, while the browser implementation can run locally for interactive checks. Invalid components fail with an invalid-input response and are not approximated. If your data contains fractional real or imaginary parts, it describes general complex numbers rather than Gaussian integers, so use a complex-number operation designed for non-integer coefficients instead of changing the values to make this calculator accept them.