Complex Number Calculator

A complex number calculator works with numbers of the form a + bi, where i is the imaginary unit and i² = −1. Enter two numbers and it adds, subtracts, multiplies or divides them, then reports the modulus, argument and polar form. Example: (3 + 4i) × (1 − 2i) = 11 − 2i, with modulus 11.1803.

Three neighboring tools cover related questions: the Quadratic Formula Calculator is the right page when a negative discriminant hands you complex roots and you want them solved from a, b and c, the Square Root Calculator handles ordinary real square roots, and the Scientific Calculator covers general real-number arithmetic with trig and logs. Use this page when the numbers themselves have an imaginary part and you need the arithmetic, the modulus and argument, or a conversion between rectangular and polar form.

The four arithmetic rules

All complex arithmetic follows from one substitution: wherever i² appears, write −1. That single rule produces the four standard formulas:

Addition and subtraction are componentwise, exactly like adding vectors: real with real, imaginary with imaginary. Multiplication is the binomial expansion with i² replaced by −1. Division is the only one that needs a trick, and that trick is the conjugate.

Worked example: all four operations on 3 + 4i and 1 − 2i

Take z₁ = 3 + 4i and z₂ = 1 − 2i, the values the calculator loads by default. Working each operation by hand:

• Sum: (3 + 1) + (4 − 2)i = 4 + 2i.
• Difference: (3 − 1) + (4 + 2)i = 2 + 6i.
• Product: real part 3·1 − 4·(−2) = 3 + 8 = 11; imaginary part 3·(−2) + 4·1 = −6 + 4 = −2. So the product is 11 − 2i.
• Quotient: multiply top and bottom by the conjugate 1 + 2i. The numerator becomes (3 + 4i)(1 + 2i) = −5 + 10i and the denominator becomes 1² + 2² = 5, so the quotient is −1 + 2i.

Now check the product and the quotient against the polar rules. The modulus of z₁ is √(9 + 16) = 5 and its argument is 53.13°; the modulus of z₂ is √(1 + 4) = 2.2361 and its argument is −63.4349°. Multiplying should multiply the moduli and add the angles: 5 × 2.2361 = 11.1803 and 53.1301° − 63.4349° = −10.3048°, which is exactly the polar form of 11 − 2i. Dividing should divide the moduli and subtract the angles: 5 ÷ 2.2361 = 2.2361 and 53.1301° + 63.4349° = 116.5651°, which is exactly the polar form of −1 + 2i. The two routes always agree, and that agreement is the fastest way to catch a sign error.

Second worked example: division step by step

Divide 7 + 2i by 3 − 4i. The conjugate of the divisor is 3 + 4i, so multiply both parts of the fraction by it. The numerator becomes (7 + 2i)(3 + 4i) = 21 + 28i + 6i + 8i² = 21 + 34i − 8 = 13 + 34i. The denominator becomes 3² + 4² = 25, a plain real number. Dividing each part gives 13/25 + (34/25)i = 0.52 + 1.36i. Check it in polar form: |7 + 2i| = √53 = 7.2801 and |3 − 4i| = 5, so the quotient should have modulus 7.2801 ÷ 5 = 1.456, and indeed √(0.52² + 1.36²) = 1.456. The arguments behave the same way: 15.9454° − (−53.1301°) = 69.0755°, which matches the argument of 0.52 + 1.36i.

Modulus, argument and polar form

Plot a + bi as the point (a, b) in the complex plane and two quantities describe it completely: how far it is from the origin, and which direction it points. Those are the modulus and the argument:

The modulus is just the Pythagorean theorem applied to the real and imaginary parts. The argument uses the two-argument arctangent so the quadrant comes out right: plain arctan(b/a) cannot tell 1 + i from −1 − i, because both give a ratio of 1. This calculator reports the principal argument between −180° and 180°, which is the standard convention. Add or subtract 360° and you get the same point, so an angle of 225° and an angle of −135° describe the identical number.

Rectangular to polar conversion table

Ten common values, converted with the formulas above. Angles are shown in degrees and in radians, since textbooks use both:

Notice how the four numbers with modulus 5 sit on the same circle and differ only by direction, and how a purely real negative number lands at exactly 180°. Going the other way is just as mechanical: 8∠120° becomes 8 cos 120° + i·8 sin 120° = −4 + 6.9282i.

Powers and De Moivre's theorem

Raising a complex number to a power by repeated multiplication gets ugly fast. Polar form makes it trivial, because multiplying two numbers multiplies their moduli and adds their angles. Apply that n times and you get De Moivre's theorem:

The table below tracks 1 + i, which has modulus 1.4142 and argument 45°, through the first eight powers. Each step multiplies the modulus by 1.4142 and rotates the point another 45°, so after eight steps it has gone all the way around and lands on the positive real axis:

The same theorem explains the powers of 3 + 4i, whose modulus is 5: the moduli of the successive powers are 25, 125, 625 and 3,125, and the rectangular values are −7 + 24i, −117 + 44i, −527 − 336i and −237 − 3,116i. Notice that the modulus grows as a plain geometric sequence while only the angle does anything interesting.

The powers of i cycle every four steps

Because i sits at modulus 1 and angle 90°, multiplying by i is a quarter turn. Four quarter turns bring you back to the start, which is why the powers of i repeat with period four:

So i²⁷ has remainder 3 and equals −i, and i¹⁰⁰ has remainder 0 and equals 1. No calculator needed once you see the pattern.

How to use this complex number calculator

1. Enter the first number: type the real part in the left box and the imaginary part in the right box. For 3 + 4i that is 3 and 4; for −2i it is 0 and −2.
2. Pick the operation: the five choice cards cover add, subtract, multiply, divide and integer power. The layout changes to match, so the power option asks for an exponent instead of a second number.
3. Enter the second number (or the exponent). For powers, the shortcut buttons jump straight to the common cases 2, 3, 4, 5 and −1, where −1 is the reciprocal.
4. Choose the angle unit: degrees or radians. This affects every angle shown on the page, including the polar-to-rectangular converter at the bottom.
5. Read the result card: the headline is the answer in rectangular form, and the tiles underneath give its modulus, argument, conjugate, polar form and trigonometric form.
6. Follow the steps: the numbered breakdown shows the algebra as you would write it on paper, which is what you need if the work has to be shown, not just the answer.

Two extra tables sit below the steps: one listing the modulus, argument and conjugate of both inputs, and one running the first five powers of the first number so you can see De Moivre's theorem in action on your own values.

Who this calculator is for

• Algebra 2 and precalculus students checking homework on complex arithmetic, conjugates and polar form.
• Engineering undergraduates converting between rectangular and phasor notation for AC circuits.
• Electrical and RF engineers who need a quick impedance or reflection-coefficient calculation without opening a full math package.
• Signal-processing and controls students working with poles, zeros and transfer functions.
• Anyone who hit a negative discriminant in a quadratic and wants to verify the pair of complex roots.
• Teachers and tutors who need worked examples with visible intermediate steps.

Key terms explained

• Imaginary unit (i): the number defined by i² = −1. Engineers usually write j instead, because i is already taken by current.
• Real part and imaginary part: in a + bi, a is the real part and b is the imaginary part. Both are ordinary real numbers; the i is a label, not part of b.
• Purely imaginary: a number whose real part is zero, such as 5i. It sits on the vertical axis of the complex plane.
• Conjugate: a − bi, the mirror image of a + bi across the real axis. A number times its conjugate is always the real value a² + b².
• Modulus: the distance from the origin, √(a² + b²). Also called the absolute value or magnitude.
• Argument: the direction angle measured from the positive real axis, counterclockwise positive.
• Principal value: the argument reduced to the range −180° to 180°, which is what this calculator reports.
• Polar form: r(cos θ + i sin θ), often abbreviated r∠θ. Engineers call this phasor notation.
• Complex plane: the two-dimensional picture with the real part on the horizontal axis and the imaginary part on the vertical axis.

What changes the result the most

• The sign of the imaginary part: a single flipped sign turns a number into its conjugate and mirrors the argument, so a stray minus is the most common source of a wrong answer.
• Which number is the divisor: division is not commutative. Swapping the two inputs inverts the modulus and negates the argument.
• The quadrant: −1 − i and 1 + i have the same modulus but arguments 180° apart, and only the two-argument arctangent tells them apart.
• The exponent: the modulus grows geometrically with n, so a modest base like 3 + 4i reaches a modulus of 3,125 by the fifth power.
• The angle unit: 45 degrees and 45 radians are wildly different directions. Radians are the default in most software; degrees are common in circuit work.

Tips for working with complex numbers

• Choose the form that fits the job. Add and subtract in rectangular form; multiply, divide and raise to powers in polar form. Converting first often saves more work than it costs.
• Check with the modulus. Because |z₁z₂| = |z₁| × |z₂| and |z₁/z₂| = |z₁| ÷ |z₂|, one quick multiplication verifies an entire product or quotient.
• Rationalize before you simplify. An answer is not in standard form while an i remains in the denominator. Multiply by the conjugate first, then reduce.
• Reduce powers of i early. Replace i²⁷ with −i at the start of a problem rather than carrying the exponent through every line.
• Sketch the point. Even a rough plot of (a, b) tells you which quadrant the argument belongs in and catches a sign error in seconds.
• Watch the units in engineering work. Phasor magnitudes may be peak or RMS values depending on the convention in use, and the two differ by a factor of √2.

Where complex numbers show up in practice

Complex numbers are the natural language for anything that rotates or oscillates, because multiplying by a unit-modulus complex number is exactly a rotation. In AC circuit analysis, impedance is written Z = R + jX, so a resistance of 30 ohms in series with a net reactance of 40 ohms gives an impedance of 30 + 40j, a magnitude of 50 ohms and a phase angle of 53.13° between voltage and current. Ohm's law then works unchanged, with complex numbers standing in for real ones. The same structure runs through the Fourier transform in signal processing, pole and zero locations in control systems, the reflection coefficient in RF work, and the wavefunctions of quantum mechanics. In every one of these, the modulus carries the size of the effect and the argument carries the phase, which is why this calculator always shows both.

Limitations and assumptions

• Results are computed in double-precision floating point and rounded for display, so a value shown as 0 can be a tiny rounding remainder rather than an exact zero.
• Output is decimal, not symbolic: an answer of √2/2 appears as 0.7071 rather than an exact radical.
• The argument is always the principal value between −180° and 180°; if your textbook wants an angle in the range 0° to 360°, add 360° to any negative result.
• Exponents are restricted to whole numbers from −20 to 20. Fractional exponents, complex roots, complex logarithms and complex exponentials are outside the scope of this tool.
• The calculator handles two numbers and one operation at a time. A longer expression has to be evaluated one step at a time, feeding each result back in as the next input.
• It does not parse typed expressions such as "(3+4i)*(1-2i)"; the real and imaginary parts go in separate boxes.

How it compares to related calculators

This page answers "what is the result of this complex-number operation, and what is it in polar form?" If your question is different, a sister tool fits better:

• To get complex roots from a quadratic equation's coefficients, use the Quadratic Formula Calculator.
• For real square roots and simplified radicals, use the Square Root Calculator.
• For general arithmetic with trig and logarithms on real numbers, use the Scientific Calculator.
• The modulus is a hypotenuse, so the Pythagorean Theorem Calculator covers the same right-triangle step on its own.
• To go from an angle to sine and cosine values when converting polar to rectangular by hand, use the Trigonometry Calculator.
• For real-number powers and roots, use the Exponent Calculator; for base-10 and natural logs, the Logarithm Calculator.

Sources

Every figure on this page comes from deterministic mathematics, so no external data source applies. Specifically:

• The four arithmetic identities, the conjugate rule, the modulus √(a² + b²), the argument atan2(b, a) and De Moivre's theorem are exact definitions and theorems of complex algebra, not measured values.
• Every number in the worked examples and the tables above was computed directly from those identities and verified independently.
• The degree and radian angle measures follow the International System of Units, in which the radian is the coherent SI unit for plane angle and the degree is the accepted non-SI unit equal to π/180 rad, as documented in NIST Special Publication 811, Guide for the Use of the International System of Units.
• The engineering convention of writing the imaginary unit as j, and of expressing AC impedance as Z = R + jX, is standard practice in electrical engineering; the ohm values used above are illustrative, not measured.