This calculator adds, subtracts, multiplies, and divides complex numbers, and converts them between rectangular and polar form. Enter your numbers below and get the result instantly, along with the steps behind it, whether you are checking a homework problem or working out impedance for a circuit.
What Is a Complex Number and How Is It Written?
A complex number is written as a + bi. The letter a is the real part, and b is the imaginary part. The letter i stands for the imaginary unit, and it has one property that makes it useful: i squared equals negative one. No real number behaves that way, since squaring a real number always gives you something positive or zero. That gap is exactly what i was invented to fill.
Take 3 + 4i as an example. Here 3 is the real part and 4 is the imaginary part. If b happens to be zero, you are just left with an ordinary real number, which is why every real number technically counts as a complex number too, just one with no imaginary part attached.
How Does the Complex Number Calculator Work?
Using the tool is straightforward. Type in your first complex number, and if the operation needs a second one, enter that as well. Choose the operation you want, from addition to polar conversion, and the calculator returns the answer in both rectangular form (a + bi) and polar form, so you do not need to switch tools partway through a problem.
What Input Format Does the Calculator Accept?
You can type your numbers the standard way, such as 3+4i, and the calculator reads it correctly. If you come from an electrical engineering background, you can also use j instead of i, so 3+4j works exactly the same way as 3+4i and returns the same result. Use whichever symbol matches your textbook or your workplace.
Input Format | Example | Accepted |
Standard form (i) | 3+4i | Yes |
Engineering form (j) | 3+4j | Yes |
Decimal values | 2.5-1.75i | Yes |
Negative real or imaginary part | -3+4i | Yes |
Which Operations Can the Calculator Perform?
Every function on this page is available directly in the calculator above. Here is the full list at a glance:
• Addition, combines two complex numbers
• Subtraction, finds the difference between two complex numbers
• Multiplication, multiplies two complex numbers together
• Division, divides one complex number by another
• Modulus, finds the distance from the origin
• Conjugate, flips the sign of the imaginary part
• Powers, raises a complex number to a power
• Roots, finds the roots of a complex number
• Rectangular to polar conversion, and back again
How Do You Add and Subtract Complex Numbers?
Adding and subtracting complex numbers is the simplest part of working with them. You add or subtract the real parts together, then do the same with the imaginary parts, keeping the two separate the entire way through.
(3 + 4i) + (1 - 2i) = (3 + 1) + (4i - 2i) = 4 + 2i
(3 + 4i) - (1 - 2i) = (3 - 1) + (4i + 2i) = 2 + 6i
This is the one case where rectangular form is actually easier to work with than polar form. Multiplication, division, powers, and roots all go the other way, as covered below.
How Do You Multiply and Divide Complex Numbers?
Multiplication uses the same distributive approach you would use on any two binomials, with one extra step: replacing i squared with negative one wherever it shows up.
(2 + 3i)(4 + i) = 8 + 2i + 12i + 3i² = 8 + 14i - 3 = 5 + 14i
Division works differently, since you cannot leave i sitting in a denominator. Instead, you multiply the top and bottom by the conjugate of the denominator to clear it out.
(3 + 2i) / (1 - i) = (3 + 2i)(1 + i) / (1 - i)(1 + i) = (1 + 5i) / 2 = 0.5 + 2.5i
One mistake shows up constantly here: treating every operation like it is the same binomial multiplication. Addition and subtraction only combine matching parts, they do not multiply anything, so applying the multiplication method to an addition problem gives the wrong answer every time.
What Is the Modulus of a Complex Number?
The modulus of a complex number tells you how far it sits from zero on the complex plane. For a + bi, it is calculated as the square root of a squared plus b squared, which is really the Pythagorean theorem applied to two dimensions instead of one.
|6 + 3i| = √(6² + 3²) = √45 ≈ 6.71
The result is always a non-negative real number, since it represents a distance, and distances do not go negative. The calculator works this out automatically alongside every result, so there is no need for a separate step.
What Is the Conjugate of a Complex Number?
The conjugate of a + bi is a - bi. All that changes is the sign on the imaginary part, which mirrors the number across the real axis on the complex plane.
The conjugate of 5 + 2i is 5 - 2i
Its main job is clearing i out of a denominator during division, as shown above. Multiply any complex number by its own conjugate and the imaginary parts cancel out completely, leaving you with a plain real number.
How Do You Convert a Complex Number Between Rectangular and Polar Form?
Rectangular form (a + bi) tells you how far to move left or right and up or down. Polar form describes the same point a different way, using distance (r) and angle (θ) instead.
To go from rectangular to polar:
r = √(a² + b²)
θ = atan2(b, a)
A plain arctangent function gives the wrong angle whenever a is negative, since it cannot tell which quadrant the number actually sits in. That is why this calculator uses atan2 instead, which corrects for the quadrant automatically.
To go the other way, from polar back to rectangular:
a = r cos(θ)
b = r sin(θ)
For example, 1 + i converts to r = √2 and θ = 45 degrees (or π/4 radians), so 1 + i = √2(cos 45° + i sin 45°). You can enter the angle in either degrees or radians, whichever your problem calls for.
How Do You Find Powers and Roots of a Complex Number?
Raising a complex number to a power, or finding its roots, is far easier once you switch to polar form. De Moivre's theorem handles it directly: raise r to the power, and multiply the angle by that same power.
(1 + i)² in polar form: r = √2, θ = 45° → r² = 2, θ = 90° → 0 + 2i
Roots work the same way in reverse, but with a catch that trips a lot of people up. Every root beyond the first has more than one valid answer. A square root has two, a cube root has three, and so on. The calculator returns the principal root by default, which is the standard convention, though it is worth knowing the others exist if your problem specifically calls for them.
What Mistakes Do Users Commonly Make With Complex Numbers?
A few small errors account for most of the wrong answers people get with complex numbers. Here are the ones worth watching for:
Mistake | Why It Happens | Fix |
Forgetting i² = -1 | Easy to skip during simplification | Always replace i² with -1 before finishing |
Mixing real and imaginary parts | Treating a+bi as one number instead of two | Keep the two parts separate until the final step |
Misplaced parentheses in multiplication | Skipping the full distributive step | Write out every term before combining |
Wrong conjugate in division | Only flipping part of the denominator | Take the conjugate of the entire denominator |
Dropping i from the final answer | Simplifying too quickly | Check that the imaginary part survived to the end |
Where Are Complex Numbers Used Outside the Classroom?
Complex numbers do not disappear once the test is over. Electrical engineers use them to describe how current and voltage behave in AC circuits, since resistance and reactance combine naturally into a real and an imaginary part. Audio engineers rely on them to break sound into waveforms during recording and mixing. GPS systems use complex number math to account for timing shifts as signals travel from satellite to receiver, and MRI machines lean on them to turn raw radio wave data into an image. A few tools on this site build directly on the ideas covered above.
How Do You Calculate Impedance in an AC Circuit With Complex Numbers?
In an AC circuit, impedance combines resistance and reactance into a single complex number, written as Z = R + jX (electrical engineers use j here instead of i). Phasors representing voltage and current add and subtract easily in rectangular form, but multiplying or dividing them is far simpler once converted to polar form. If you are working through a real circuit, the impedance calculator handles both forms and the parallel combination of two impedances in one step, so you are not converting back and forth by hand.
How Does a Quadratic Equation Calculator Handle Complex Roots?
When a quadratic equation's discriminant, b² - 4ac, comes out negative, the two solutions are not real numbers at all, they are complex conjugates. Take x² + 1 = 0: there is no real number you can square to get negative one, so the two solutions are i and -i. This is exactly why the fundamental theorem of algebra guarantees every polynomial equation has a solution somewhere in the complex numbers, even when the real number line comes up empty. If you are solving an equation of your own, the quadratic equation calculator works through the discriminant and returns whichever type of solution applies, real or complex.
How Do You Convert Between Polar and Rectangular Coordinates for Any Point?
The r and θ math used to convert a complex number into polar form is the same trigonometry used to convert any ordinary point on a graph. A complex number and a coordinate point are really two different ways of describing the same idea, a location described either by horizontal and vertical distance or by distance and angle. If you only need to convert a coordinate pair and do not need the rest of the complex number toolkit, the polar to rectangular coordinate converter does that single job directly.
How Does a Scientific Calculator Handle Trigonometric and Exponential Functions?
Complex logarithms, complex trig functions, and complex hyperbolic functions all lean on the same exponential and trigonometric functions found on any scientific calculator, just extended into the complex plane through Euler's formula. If your work involves standard scientific calculations that do not involve i at all, such as plain logarithms or trigonometric ratios, the scientific calculator covers those functions directly.
How Does a Matrix Calculator Handle Complex Number Entries?
Matrices used in electrical engineering and quantum mechanics often contain complex numbers as entries rather than plain real numbers, and operations like multiplication or finding a determinant follow the same complex arithmetic rules covered earlier on this page. If you are solving a system of equations where the coefficients are complex, the complex matrix calculator applies those rules across an entire matrix at once instead of one number at a time.