What is a complex number? — Complex number definition
Our complex number calculator (also known as an imaginary number calculator) is an excellent tool for solving basic operations with complex numbers. Read on to find the answer to the question "what is a complex number?", learn about the algebraic and polar forms of complex numbers, and master the skills of multiplying and dividing complex numbers.
To answer the question of what is a complex number, we first have to ask: "What is an imaginary number?" An imaginary number is the square root of a negative number. The basic imaginary number is denoted with the letter i (sometimes j, e.g. in electronics), and is defined by:
i = √−1
The definition of a complex number z is the combination of a real part a and an imaginary part b·i, so that:
z = a + bi
Here, both a and b are classically understood as real numbers. When b = 0, the number is purely real, and if a = 0, we have a purely imaginary number. You can use this complex number calculator as an imaginary number calculator — just set the real component equal to 0.
Another way to write the two parts of a complex number is Re and Im, so that Re(z) = a and Im(z) = b. In fact, there are also numbers with more imaginary parts (quaternions) — fortunately, we don't have to worry about them here.
The conjugate of a complex number is defined as:
z = a − bi
As we can see, the conjugate of a complex number doesn't affect the real part, while the imaginary part has the opposite sign to the original.
Polar form of complex numbers
Every complex number z = a + bi can also be written in polar form using its modulus (absolute value) r = |z| and its argument (angle) θ:
- r = √(a² + b²)
- θ = atan2(b, a)
The polar form is then:
z = r(cos θ + i sin θ) = r eiθ
Polar form makes multiplication, division, powers, and roots of complex numbers much easier to work with.
Basic operations with complex numbers — sum and difference
Adding and subtracting complex numbers is done component by component. For z1 = a + bi and z2 = c + di:
- Sum: z1 + z2 = (a + c) + (b + d)i
- Difference: z1 − z2 = (a − c) + (b − d)i
Multiplying and dividing complex numbers
Multiplication uses the distributive law together with i² = −1:
z1 × z2 = (ac − bd) + (ad + bc)i
Division is done by multiplying the numerator and denominator by the conjugate of the denominator:
z1 ÷ z2 = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
Division is only defined when z2 ≠ 0, that is, when c² + d² > 0.
Complex power and complex logarithm
Raising a complex number to a power is easiest in polar form, using De Moivre's theorem:
zn = rn(cos nθ + i sin nθ)
The complex (natural) logarithm of a non-zero number z uses the modulus and argument:
ln(z) = ln|z| + i arg(z) = ln r + iθ
The logarithm is undefined at z = 0, and the value shown here uses the principal argument in (−π, π].
How to use the complex number calculator
- Enter the real and imaginary parts of the first number, z1 = a + bi.
- Enter the real and imaginary parts of the second number, z2 = c + di.
- Choose the exponent n for the power z1n.
- Read off the sum, difference, product, quotient, polar form, power, and logarithm — all computed instantly.
To use it as an imaginary number calculator, just set the real part to 0.
Properties of complex numbers
Many properties of complex numbers are based on the conjugate or the absolute value:
- z · z = a² + b² = |z|²
- z + z = 2a = 2·Re(z)
- z − z = 2bi = 2i·Im(z)
- |z1 · z2| = |z1| · |z2| and arg(z1 · z2) = arg(z1) + arg(z2)
Complex numbers in real life
Although they may seem abstract, complex numbers have many practical applications. They are used in electrical engineering to describe alternating current and impedance, in signal processing and the Fourier transform, in control theory, in quantum mechanics, and in fluid dynamics and fractal geometry (for example, the Mandelbrot set). Whenever a problem involves rotations, oscillations, or waves, complex numbers offer an elegant and powerful way to describe it.