Complex arithmetic with exact real and imaginary parts
This complex number calculator adds, subtracts, multiplies and divides two numbers written in the form a + bi, where i² = −1. It also calculates a conjugate, reciprocal, integer power or the properties of a single number. Every supported operation shows the entered values and the algebra used to obtain the result.
Integers, ordinary decimals and fractions are accepted as component inputs. Arithmetic results stay in exact reduced a + bi form; the modulus is shown as an exact rational or square root. The principal argument is a separate approximate angle, not an exact symbolic multiple of π. This distinction prevents an angle rounded for display from being mistaken for an exact identity.
How to enter complex numbers and choose an operation
- Choose an operation. Addition, subtraction, multiplication and division use two complex numbers; the other operations use only the first.
- Split the first number into its real and imaginary coefficients. For 3 − 4i, enter real part 3 and imaginary coefficient −4. Do not type the i symbol into either field.
- For a binary operation, enter both components of the second number. Enter zero when a component is absent: 5i has real part 0 and imaginary coefficient 5.
- For a power, enter an integer from −20 to 20. Calculate, then read the standard-form answer, modulus, angle and worked steps.
Subtraction means first number minus second number; division means first number divided by second number. A fraction such as 1/3 is entered directly, not as a rounded repeating decimal. Fields that do not apply to the selected mode are hidden and disabled. Editing an input clears its old result. Copy, Download and Print include the explanation, while Reset returns to blank addition inputs.
Affiliate disclosure: As an Amazon Associate I earn from qualifying purchases.
For an optional numerical check, browse scientific calculators (paid link). Confirm the supported functions; a numerical device does not necessarily reproduce this tool’s symbolic working. No purchase is needed.
Add, subtract and multiply complex numbers
Addition and subtraction combine real components with real components and imaginary coefficients with imaginary coefficients. For example, (3 + 4i) + (1 − 2i) = 4 + 2i, while subtraction gives (3 + 4i) − (1 − 2i) = 2 + 6i. Subtracting the second negative imaginary component changes its sign.
Multiplication requires all four products. The identity is (a + bi)(c + di) = (ac − bd) + (ad + bc)i, because bd·i² becomes −bd. With 3 + 4i and 1 − 2i, ac = 3, bd = −8, ad = −6 and bc = 4. The result is 11 − 2i.
A fraction component follows the same rules. Adding 1/2 + (1/3)i to 1/4 − (1/6)i gives 3/4 + (1/6)i. The calculator reduces each coefficient and omits a zero component from the main answer. A displayed real result, such as −1, can therefore be a valid complex-arithmetic answer with imaginary coefficient zero.
Dividing complex numbers using a conjugate
To divide by c + di, multiply both numerator and denominator by c − di. The denominator becomes c² + d², a positive real number unless both divisor components are zero. The resulting real component is (ac + bd)/(c² + d²), and the imaginary coefficient is (bc − ad)/(c² + d²).
For (3 + 4i)/(1 − 2i), multiply by the conjugate 1 + 2i. The numerator becomes −5 + 10i and the denominator is 5, giving −1 + 2i. The worked output displays the actual conjugate, numerator and denominator rather than a generic instruction alone.
The reciprocal uses the same method with numerator one. For 1/(3 + 4i), the result is 3/25 − (4/25)i. Division by 0 + 0i and the reciprocal of zero are rejected; returning infinity would not be a valid finite complex result.
Modulus, principal argument, conjugates and powers
The modulus of a + bi is the nonnegative value √(a² + b²). For 3 + 4i it equals 5. Its principal argument is computed using atan2(b, a), which preserves the quadrant; dividing b by a and applying arctangent alone can choose the wrong direction or fail when a is zero. This page reports radians and degrees with a principal range of (−π, π].
Zero has modulus zero, but its direction is undefined. The calculator states that explicitly instead of assigning a meaningful zero-degree angle. A negative real number has principal argument π radians, or 180 degrees. Angles and the numerical magnitude cross-check are rounded to 12 significant digits.
The conjugate changes a + bi to a − bi. A positive integer power is repeated exact multiplication; a negative integer power uses the nonzero reciprocal first. For example, i² = −1, i⁴ = 1 and (1 + i)⁻¹ = 1/2 − (1/2)i. A nonzero number raised to zero gives one; 0⁰ and negative powers of zero are rejected.
Supported inputs, precision and limitations
Each real or imaginary component requires a numeric value with 1–48 characters. Accepted forms are signed integers, ordinary decimals with up to six places, or a single fraction. Each number in the input and the resulting component must have magnitude at most 1,000,000. Denominators must be nonzero. Mixed numbers, scientific notation, commas and expressions such as 2+3 are not component inputs.
Integer powers are restricted to −20 through 20. Exact intermediate numerator and denominator lengths must remain within 256 digits. Large powers can exceed that safety limit even when every starting component is allowed. The tool then asks for smaller inputs instead of silently rounding an exact answer.
This version does not parse a full complex expression, calculate complex roots or logarithms, or convert a polar input to rectangular form. Its rectangular answer, modulus and argument are enough for many arithmetic checks, but they do not make it a complete circuit simulator. Use the Completing the Square Calculator to see where complex conjugate roots arise in quadratics, or the Fraction Calculator for separate real fraction arithmetic.
Complex Number Calculator FAQs
Sources and methodology
These educational references explain the underlying mathematics. This independently implemented tool is not affiliated with their publisher. Supported methods, numerical precision and restrictions are described above.