Complex Number Calculator
a + bi arithmetic with every operation at once — sum, difference, product, quotient by the conjugate — plus moduli and conjugates, with the 3-4-5 triangle hiding in |3 + 4i| = 5.
Complex Number Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- One pair of numbers per run
- Rectangular (a + bi) form only — no polar conversion
In short: (3 + 4i) and (1 − 2i): the product is (3·1 − 4·(−2)) + (3·(−2) + 4·1)i = 11 − 2i — FOIL with i² = −1 folding the middle. The quotient divides by the conjugate: (3 + 4i)/(1 − 2i) = −1 + 2i, denominator a clean 5. The moduli are |3 + 4i| = 5 and |1 − 2i| = √5 ≈ 2.236068 — the 3-4-5 triangle wearing imaginary clothes.
Formula
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
(a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²)
i² = −1 is the only new rule; everything else is ordinary algebra.
Worked Example
- Add and subtract. real with real, imaginary with imaginary — the two parts never mix on a plus or minus.
- Multiply. FOIL, then fold: i² = −1 merges the outer products into the real part. (ac − bd) + (ad + bc)i.
- Divide. multiply top and bottom by the conjugate (c − di): the denominator becomes the real number c² + d² and the fraction lands in a + bi form.
The division trick is the page’s centerpiece: (3 + 4i)/(1 − 2i) multiplies by (1 + 2i)/(1 + 2i) — a disguised 1 — the denominator becomes 1² + 2² = 5, and the quotient reads −1 + 2i. Round-trip it on the product card: (1 − 2i)(−1 + 2i) returns exactly 3 + 4i.
Strengths & Limits Of This Model
Where this engine is strong
- Every operation at once — the five views are compared, not hidden
- The quotient’s conjugate trick is shown with the denominator it produces
Where it stops
- No polar/exponential form
- No powers or roots of i
Practical Use Cases
AC circuits
impedance adds and scales exactly this way
Roots of quadratics
the conjugate pair the Quadratic Formula page prints
Plane geometry
multiplication by i is a 90° turn; complex numbers ARE 2D directions
Methodology & Editorial Standards
Four real fields parsed as doubles. Sum, difference, product ((ac − bd) + (ad + bc)i), quotient by the conjugate ((ac + bd) + (bc − ad)i over c² + d², refused only when both c and d are zero), moduli as √(a² + b²), conjugates by sign flip. All operations printed simultaneously in a + bi form.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Complex Number Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does i² = − 1 change multiplication?
It only adds one folding rule. FOIL (3 + 4i)(1 − 2i) into 3 − 6i + 4i − 8i², then i² = −1 turns −8i² into +8, landing at 11 − 2i. Real parts and imaginary parts stay separate lists except through that single substitution — no other algebra changes.
Why does division multiply by the conjugate?
To make the bottom REAL. (c + di)(c − di) = c² + d² exactly — the imaginary parts cancel by construction — so multiplying by (c − di)/(c − di), a disguised 1, leaves the value alone while the denominator becomes the plain number c² + d². One trick, every quotient.
What does the modulus mean?
Distance from the origin: |a + bi| = √(a² + b²), the Pythagorean hypotenuse of the point (a, b). On the default that is √25 = 5 — literally the 3-4-5 triangle, which is why complex numbers and the vector page agree when you identify i-direction with the y-axis.
What is a conjugate, and what is it for?
The same real part with the sign of the imaginary part flipped: the conjugate of 1 − 2i is 1 + 2i. Its superpower: z · z̄ is always real — it is literally |z|². That makes it the division tool, and it is why polynomial roots with real coefficients always arrive in conjugate pairs: one root drags its mirror along.
How are addition and subtraction different from multiplication?
No cross-talk. Sum and difference pair real with real and imaginary with imaginary directly: (3 + 4i) + (1 − 2i) = 4 + 2i. Multiplication is the only operation where the parts interact — through i² = −1 — and that interaction is exactly what makes 3 × 4 triangles of modulus multiply into |5|·|√5| = 5√5.
What does multiplying by i do geometrically?
A quarter turn. i takes 1 to i, i to −1, −1 to −i: every application rotates the plane 90° counterclockwise. That is the deepest reason complex numbers are the natural language of 2D rotation — and why multiplication by (3 + 4i) both scales (by 5) and turns (by 53.130102°) at once.
Why are there no mode buttons on this page?
Because the operations are five lines of algebra and hiding three of them behind a dropdown would hide the point: they are all the same two numbers seen five ways. Sum, product, quotient, moduli — printed together, checked against each other, exactly like the percent forks page prints its three readings.
Can both numbers be real, or both imaginary?
Absolutely — the algebra never assumes otherwise. Two reals make the imaginary parts vanish (b = d = 0 gives an ordinary product); two pure imaginaries multiply to a real number (bi·di = −bd). Try a = 0 with d = 0 and watch the cards stay consistent all the way down.
Where do complex numbers show up outside class?
Everywhere oscillation lives: AC circuit analysis adds impedances as complex numbers, signal processing runs on complex exponentials, and the Quadratic Formula’s negative-discriminant case produces exactly the conjugate pairs this page manipulates. The algebra here is the working vocabulary of all of it.