Complex numbers, rectangular form, and polar form explained
A complex number calculator helps you add, subtract, multiply, divide, inspect complex-number properties, and convert complex numbers between rectangular, polar, trigonometric, and exponential form. It is especially useful when you want to check real and imaginary parts, modulus, argument, conjugate, reciprocal, or polar form without redoing every step by hand.
What the calculator solves
A complex number is usually written in rectangular form as a + bi, where a is the real part, b is the imaginary part, and i² = -1. That notation makes it easy to combine numbers on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
This calculator focuses on the most common classroom and reference tasks: adding, subtracting, multiplying, dividing, finding the conjugate and inverse, and converting between rectangular and polar form. In practice that means a complex number calculator online can help with algebra homework, trigonometry review, phasor checks, and quick checks of work that would otherwise require several manual steps.
z = a + bi
Standard rectangular form for a complex number.
i² = -1
The defining relation for the imaginary unit.
Arithmetic rules and the conjugate
Addition and subtraction combine the real parts and the imaginary parts separately. Multiplication uses distribution, then simplifies i² back to -1. Division is handled by multiplying numerator and denominator by the conjugate of the denominator so the result returns to standard form a + bi.
That last step is the reason many searchers look for a complex number calculator or an imaginary number calculator when they are stuck on a fraction. The conjugate turns a complex denominator into a real denominator, which is what makes the quotient easy to read and compare.
(a + bi) + (c + di) = (a + c) + (b + d)i
Add the matching real and imaginary parts.
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Multiply by distribution, then combine the i² term with the real part.
Multiply by the conjugate so the denominator becomes a real number.
Real parts stay on the horizontal axis and imaginary parts stay on the vertical axis.
The conjugate of a + bi is a - bi.
Division is undefined when the denominator is 0 + 0i.
A purely real number is just the special case b = 0.
Conjugate, inverse, and exponential form
Many people do not need a two-number operation first. They need to inspect one complex number: find its conjugate, reciprocal, modulus, argument, polar form, or exponential form. That is why the calculator includes a properties mode in addition to arithmetic and conversion modes.
The conjugate of a + bi is a - bi, which reflects the point across the real axis. The reciprocal uses the same conjugate divided by |z|², so 1 / (a + bi) = (a - bi) / (a² + b²) as long as the number is not 0 + 0i. Exponential form writes the same point as re^(iθ), which is often the compact form used in Euler’s formula, rotations, waves, and electrical phasor work.
conjugate(a + bi) = a - bi
Reflects the complex number across the real axis.
1 / (a + bi) = (a - bi) / (a² + b²)
Uses the conjugate and modulus squared to find the reciprocal.
z = re^(iθ)
Exponential form of the same complex number when r and θ are known.
Rectangular, polar, and trigonometric form
Rectangular form and polar form are two ways to describe the same complex number. Rectangular form keeps the a + bi structure, while polar form uses the modulus r and argument θ to describe the same point as a distance and angle. Trigonometric form adds the cosine and sine components, and exponential form writes the same point as re^(iθ), which makes multiplication, division, and rotation easier to visualise.
The modulus is the distance from the origin, and the argument is the angle measured from the positive real axis. On many calculators and textbooks the angle is shown in degrees or radians, so the same complex number calculator can support both classroom conventions without changing the underlying maths.
r = √(a² + b²)
The modulus or absolute value of a complex number.
a = r cos θ
Convert polar form back to the real part.
b = r sin θ
Convert polar form back to the imaginary part.
z = r(cos θ + i sin θ)
The standard trigonometric form of a complex number.
Worked example
A simple worked example makes the calculator output easier to interpret. If z1 = 3 + 4i and z2 = 1 - 2i, then addition gives 4 + 2i. Multiplication gives 11 - 2i. If you divide 4 + 2i by 1 - i, the result is 1 + 3i. Converting 3 + 4i to polar form gives a modulus of 5 and an argument of about 53.13 degrees.
Those are the same values the calculator shows in the result block, but it also displays the working that gets there. That makes it easier to check whether you are using the correct sign, the correct conjugate, and the correct angle convention before you copy the answer into homework or notes.
Load the example inputs to see a realistic calculation without starting from zero.
Use properties mode when you only need the conjugate, inverse, modulus, argument, or exponential form of one complex number.
Use the result label to confirm whether the page is showing a rectangular answer or a polar answer.
Check the modulus and argument when the result matters more in geometry, trigonometry, or phasor work.
A common mistake is to treat the argument as interchangeable with the modulus. The modulus is a length, while the argument is an angle, so they answer different questions. Another common mistake is forgetting that the zero complex number does not have a unique polar angle, which means there is no single correct polar representation for 0 + 0i.
The page is intentionally a calculator, not a symbolic algebra system. It does not simplify arbitrary expressions, solve equations for unknown variables, or explore deeper topics such as complex exponentiation and logarithms. Those topics use the same ideas, but they need more advanced notation than a quick online calculator.
Check whether your angle should be in degrees or radians before comparing the result with a textbook.
Remember that polar form is undefined at the origin because the angle is not unique.
Use the conjugate to keep division in standard a + bi form.
For symbolic work, a full algebra package is better than a calculator.
Frequently asked questions
What is a complex number?
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with i² = -1. If b = 0, the number is real. If a = 0 and b is not zero, the number is purely imaginary. Complex numbers are usually plotted on the complex plane rather than on a one-dimensional number line.
What is the difference between rectangular form and polar form?
Rectangular form shows a complex number as a + bi, which makes it easy to read the real and imaginary parts directly. Polar form shows the same number as a distance from the origin and an angle, usually written with a modulus r and an argument θ. Rectangular form is often better for addition and subtraction, while polar form is often better for multiplication and division.
How do you add complex numbers?
Add the real parts together and add the imaginary parts together. For example, (3 + 4i) + (1 - 2i) = 4 + 2i. The calculator does the same thing automatically and then shows the result in standard a + bi form.
Why do you multiply by the conjugate when dividing complex numbers?
Multiplying by the conjugate turns the denominator into a real number because (c + di)(c - di) = c² + d². That removes the imaginary part from the denominator and leaves the quotient in standard rectangular form. Without that step, the result is harder to interpret and compare.
How do you find the conjugate of a complex number?
Keep the real part the same and change the sign of the imaginary part. The conjugate of a + bi is a - bi. For example, the conjugate of 3 + 4i is 3 - 4i.
How do you find the inverse of a complex number?
Use the conjugate divided by the modulus squared: 1 / (a + bi) = (a - bi) / (a² + b²). The reciprocal is undefined for 0 + 0i because the modulus squared is zero.
What is exponential form for a complex number?
Exponential form writes the same complex number as re^(iθ), where r is the modulus and θ is the argument in radians. It is closely related to polar and trigonometric form through Euler’s formula.
What is the modulus of a complex number?
The modulus is the distance from the origin to the point a + bi on the complex plane. It is calculated with r = √(a² + b²). The modulus is always zero or positive, and it is also called the absolute value of the complex number.
What is the argument or phase of a complex number?
The argument is the angle from the positive real axis to the complex number’s point on the complex plane. It is commonly measured in degrees or radians depending on the course or textbook. Many calculators report the principal argument, which is the main angle choice in a standard range rather than every possible coterminal angle.
Can the zero complex number be written in polar form?
The zero complex number has modulus 0, but it does not have a unique angle. That means polar form is not uniquely defined at the origin. A calculator should warn about that instead of pretending there is one exact polar answer.
Where are complex numbers used in real life?
Complex numbers appear in algebra, trigonometry, signal processing, electrical engineering, wave analysis, and rotation problems. They are especially useful whenever a problem is easier to model as a point on the complex plane rather than as a single real value. That is why complex number and polar-form calculators are common reference tools in math and engineering work.
