The complex number calculator above performs the four arithmetic operations on two complex numbers and then reports everything you normally need alongside the answer: modulus, argument in degrees and radians, conjugate, and both polar and exponential forms. The working panel shows the algebra rather than just the result, including the conjugate multiplication that makes division possible.
Arb Digital publishes this as part of a free algebra set. Complex numbers are where a lot of students first meet a system that obeys familiar rules with exactly one new one — i2 = −1 — and seeing that single substitution do its work in each operation is most of the battle.
What This Complex Number Calculator Does
Enter two complex numbers as their real and imaginary parts, pick an operation, and the calculator returns the result in standard a + bi form. It also gives the modulus (the distance from the origin), the argument (the angle from the positive real axis), the conjugate, and the polar form r(cosθ + i sinθ), which is the form multiplication and division are easiest to reason about.
Division by zero is handled explicitly: dividing by 0 + 0i is undefined and the calculator says so rather than returning infinity or NaN.
How to Use It
- Enter the first number as its real part a and imaginary part b. For 3 + 4i, enter 3 and 4.
- Enter the second number the same way. For 1 − 2i, enter 1 and −2.
- Choose the operation from the dropdown. Order matters for subtraction and division.
- Press Calculate to see the result and all the derived quantities.
- Read the working panel to follow the expansion, including where i2 was replaced by −1.
The Four Operations
Addition and subtraction work componentwise: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. Real parts combine with real parts, imaginary with imaginary, exactly like adding vectors.
Multiplication is ordinary expansion followed by one substitution: (a + bi)(c + di) = ac + adi + bci + bdi2, and since i2 = −1 the last term becomes −bd. Collecting gives (ac − bd) + (ad + bc)i. The minus sign in the real part is the entire content of complex multiplication, and it is where nearly every hand-worked error occurs.
Division needs a trick, because you cannot divide by an expression containing i directly. Multiply top and bottom by the conjugate of the denominator, c − di. The denominator becomes (c + di)(c − di) = c2 + d2, a real number, and the division reduces to scaling a complex numerator. The identities behind this are collected in the Wolfram MathWorld entry on complex numbers.
A Fully Worked Division Example
Compute (3 + 4i) ÷ (1 − 2i).
Step 1 — multiply by the conjugate. The conjugate of 1 − 2i is 1 + 2i. Multiply both numerator and denominator by it, which is multiplying by 1 and therefore changes nothing.
Step 2 — expand the denominator. (1 − 2i)(1 + 2i) = 1 + 2i − 2i − 4i2 = 1 + 4 = 5. The imaginary terms always cancel here — that is the whole point of using the conjugate.
Step 3 — expand the numerator. (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i2 = 3 − 8 + 10i = −5 + 10i.
Step 4 — divide through. (−5 + 10i)/5 = −1 + 2i.
Check by multiplying back: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i2 = −1 + 4i + 4 = 3 + 4i. Correct. The modulus of the answer is √(1 + 4) = √5 ≈ 2.2361, and its argument is arctan(2 / −1) adjusted into the second quadrant, giving about 116.57°.
Modulus and Argument: The Geometry Behind the Algebra
Plot a + bi as the point (a, b) and every complex operation acquires a picture. The modulus |z| = √(a2 + b2) is the distance from the origin — the Pythagorean theorem applied to the real and imaginary parts, which is why the Pythagorean theorem calculator computes exactly the same quantity for a right triangle with those legs.
The argument arg(z) is the angle that line makes with the positive real axis. Computing it needs care: plain arctan(b/a) cannot distinguish 1 + i from −1 − i, because both give a ratio of 1. The correct calculation uses the two-argument arctangent, which looks at the signs of a and b separately to place the angle in the right quadrant. This calculator uses that version, so a result in the second or third quadrant reports a genuinely different angle rather than an incorrect one folded back into the first.
By convention the principal argument is reported in the range −180° to 180°. Any multiple of 360° could be added and still describe the same point, which is precisely why complex logarithms and fractional powers are multi-valued.
Why Polar Form Makes Multiplication Trivial
In polar form z = r(cosθ + i sinθ), multiplication becomes almost effortless: multiply the moduli and add the arguments. Division divides the moduli and subtracts the arguments. Compare that to the four-term expansion in rectangular form, and you can see why polar form dominates in engineering and physics.
Geometrically, multiplying by a complex number is a rotation combined with a scaling. Multiplying by i has modulus 1 and argument 90°, so it rotates a point a quarter turn anticlockwise and changes nothing else. That is why i2 = −1 stops being mysterious: two quarter turns is a half turn, and a half turn takes 1 to −1.
The same rule gives De Moivre's theorem: raising to the n-th power raises the modulus to the n-th power and multiplies the argument by n. Finding the fifth root of a complex number, which is unpleasant in rectangular form, becomes a matter of taking the fifth root of r and dividing θ by 5 — with five distinct answers spaced evenly around a circle. The exponential form underlying De Moivre's theorem is defined in the NIST Digital Library of Mathematical Functions, section 4.14.
The Conjugate and What It Is For
The conjugate of a + bi is a − bi: the same point reflected across the real axis. Three properties make it useful. Multiplying a number by its conjugate always gives the real value a2 + b2, which equals |z|2 — this is what makes division work. Adding a number to its conjugate gives 2a, twice the real part, with the imaginary parts cancelling. And the conjugate of a sum or product equals the sum or product of the conjugates, so conjugation passes cleanly through any polynomial with real coefficients.
That last property has a direct consequence you have already met: complex roots of real polynomials always come in conjugate pairs. If p(z) = 0 for a polynomial with real coefficients, then conjugating both sides gives p(z̄) = 0 as well. This is why the quadratic equation solver never returns a single complex root on its own, and why a negative discriminant always produces two.
Where Complex Numbers Are Actually Used
Electrical engineering represents alternating current with complex impedance, where the real part is resistance and the imaginary part is reactance — the argument of that number is the phase shift between voltage and current, and the modulus is the total opposition to flow. Signal processing runs on the Fourier transform, which is complex-valued throughout. Control theory places system poles in the complex plane, and their positions determine whether a system is stable or oscillates. In each case the complex number is not a mathematical curiosity but the natural way to carry two related quantities — a size and a phase — in one object.
Arb Digital builds fast, dependency-free interactive calculators designed to rank and to be genuinely useful. Browse the free library or tell us what you need built.
Browse Free Tools Talk To UsCommon Mistakes to Avoid
- Forgetting that i2 = −1. In multiplication the bd term flips sign, giving ac − bd, not ac + bd.
- Trying to divide without the conjugate. You cannot leave i in a denominator; multiplying top and bottom by the conjugate clears it.
- Using plain arctan for the argument. It collapses opposite quadrants together, so the two-argument version is required.
- Mixing degrees and radians. Polar form is usually written in radians in analysis and in degrees in engineering; state which you are using.
- Comparing complex numbers with less than or greater than. There is no ordering on the complex numbers, though their moduli can certainly be compared.
Related Free Tools From Arb Digital
The Pythagorean theorem calculator computes the same square root the modulus uses, and the quadratic equation solver produces the conjugate pairs discussed above. Use the scientific calculator for the trigonometry in polar form, the square root calculator for moduli, and the exponent calculator when applying De Moivre's theorem. The full free online tools hub lists the rest.
Frequently Asked Questions
Multiply the numerator and denominator by the conjugate of the denominator. That makes the denominator the real number c squared plus d squared, after which you divide each part of the numerator by it.
It is the distance from the origin to the point in the complex plane, calculated as the square root of the real part squared plus the imaginary part squared.
It is the angle measured from the positive real axis to the line joining the origin to the point. It must be computed with a two-argument arctangent so the quadrant is correct.
It is the same number with the sign of the imaginary part reversed. Multiplying a complex number by its conjugate always produces a real number equal to the modulus squared.
That is the defining property of i. Geometrically, multiplying by i rotates a point ninety degrees anticlockwise, so multiplying by it twice is a half turn, which sends one to minus one.
It makes multiplication, division and powers far simpler. Multiplying multiplies the moduli and adds the arguments, and De Moivre's theorem extends that directly to powers and roots.
No. There is no ordering of the complex numbers consistent with arithmetic, so saying one is greater than another is meaningless. Their moduli, being real, can be compared.
This calculator is provided for study and reference. Angles are reported as principal values, so add or subtract full turns as your context requires.
