Imaginary Number Calculator
Enter two complex numbers (a + bi form) and choose an operation to compute the result instantly. The calculator also shows the modulus, argument in degrees, conjugate, and polar form for every number, together with a step-by-step breakdown of the arithmetic. Switch the operation with the dropdown and all values update in real time.
What is an imaginary number?
An imaginary number is a real multiple of i, the imaginary unit, where i is defined as the square root of -1. That definition makes i^2 = -1, a property that ordinary real numbers cannot satisfy. The first systematic treatment of imaginary numbers was developed in the 16th century by Italian mathematicians working on cubic equations, and the term "imaginary" was coined by Rene Descartes in 1637 as a dismissive label that has stubbornly remained. Today imaginary numbers are fully accepted, and together with real numbers they form the complex number system that underlies electrical engineering, quantum mechanics, signal processing, and control theory.
Complex numbers in rectangular and polar form
A complex number z has two standard representations. Rectangular form writes it as z = a + bi, where a is the real part and b is the imaginary part. Polar form writes it as z = r(cos(theta) + i sin(theta)), or more compactly as r e^(i theta) using Eulers formula. The modulus r = sqrt(a^2 + b^2) is the distance from the origin in the complex plane, and the argument theta = atan2(b, a) is the angle from the positive real axis measured anticlockwise. Converting between forms is straightforward: a = r cos(theta) and b = r sin(theta). Polar form is especially convenient for multiplication and division, because you multiply moduli and add arguments rather than distributing terms.
How to multiply and divide complex numbers
Multiplication follows the standard FOIL rule applied to (a + bi)(c + di), giving (ac - bd) + (ad + bc)i, with the simplification i^2 = -1 turning -bd into a real term. Division requires eliminating i from the denominator by multiplying both numerator and denominator by the complex conjugate of the denominator, which is c - di. The denominator becomes c^2 + d^2, a real number, and the numerator expands to (ac + bd) + (bc - ad)i. You then divide each part by c^2 + d^2 to get the final rectangular form. A faster mental check: in polar form, multiplication gives r1*r2 at angle theta1 + theta2, and division gives r1/r2 at angle theta1 - theta2.
The complex conjugate and its uses
The conjugate of z = a + bi is written z* (or z-bar) and equals a - bi: you simply flip the sign of the imaginary part. Multiplying a complex number by its conjugate always produces a real number: (a + bi)(a - bi) = a^2 + b^2, which is the square of the modulus. This is precisely why conjugates appear in the denominator-clearing step of complex division. Conjugates also feature in the quadratic formula: when a polynomial with real coefficients has a complex root, its conjugate is always a root too, so complex roots of real polynomials come in conjugate pairs.
Quick reference: complex number operations
| Operation | Formula | Key rule |
|---|---|---|
| Addition | (a+c) + (b+d)i | Add component by component |
| Subtraction | (a-c) + (b-d)i | Subtract component by component |
| Multiplication | (ac-bd) + (ad+bc)i | FOIL and use i^2 = -1 |
| Division | [(ac+bd) + (bc-ad)i] / (c^2+d^2) | Multiply by conjugate of denominator |
| Modulus | sqrt(a^2 + b^2) | Pythagorean distance from origin |
| Argument | atan2(b, a) | Angle from positive real axis |
| Conjugate | a - bi | Negate the imaginary part |
| Polar form | r(cos theta + i sin theta) | r = modulus, theta = argument |
Formulas for the four basic operations on z1 = a + bi and z2 = c + di.
Frequently asked questions
What is i (the imaginary unit)?
i is defined as the square root of -1, so i^2 = -1. No real number satisfies this equation because squaring any real number gives a non-negative result. Defining i as a new quantity and extending the real number line into a plane gives the system of complex numbers. Higher powers cycle in a pattern: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, then the cycle repeats with period 4.
How do I add two complex numbers?
Add the real parts together and the imaginary parts together separately. For z1 = a + bi and z2 = c + di, the sum is (a + c) + (b + d)i. For example, (3 + 2i) + (1 - i) = (3+1) + (2-1)i = 4 + i.
How do I multiply two complex numbers?
Use the FOIL method: (a + bi)(c + di) = ac + adi + bci + bdi^2. Because i^2 = -1, the last term becomes -bd, giving a real contribution. Collecting terms: real part = ac - bd, imaginary part = ad + bc. For example, (3 + 2i)(1 - i) = (3x1 - 2x(-1)) + (3x(-1) + 2x1)i = (3 + 2) + (-3 + 2)i = 5 - i.
Why do we multiply by the conjugate when dividing?
The goal of division is to express the result in a + bi form with a real denominator. Multiplying both numerator and denominator by the conjugate of the denominator (c - di) turns the denominator into c^2 + d^2, which is a real number because (c + di)(c - di) = c^2 - (di)^2 = c^2 + d^2. The numerator is then a complex number divided by a real scalar, which gives the rectangular form directly.
What is the modulus of a complex number?
The modulus (also called the absolute value or magnitude) of z = a + bi is |z| = sqrt(a^2 + b^2). Geometrically it is the distance from the point (a, b) to the origin in the complex plane (the Argand diagram). The modulus is always a non-negative real number. For a purely real number b = 0 and |z| = |a|, the ordinary absolute value.
What is the argument of a complex number?
The argument of z = a + bi, written arg(z), is the angle theta = atan2(b, a) that the line from the origin to the point (a, b) makes with the positive real axis, measured anticlockwise. The result is usually given in radians in mathematics and in degrees in engineering. The principal value of the argument lies in the range (-180, 180] degrees (or (-pi, pi] radians).
What is the polar form of a complex number?
Polar form expresses z as r(cos(theta) + i sin(theta)) where r is the modulus and theta is the argument. By Eulers formula, this can also be written as r e^(i theta). Polar form is especially convenient for multiplication (multiply moduli, add arguments) and for finding powers and roots using De Moivre's theorem: z^n = r^n (cos(n theta) + i sin(n theta)).