Calculator The polar form of complex numbers

Rectangular form (standard form):
z = 5+8.660254i

Angle notation (phasor, module and argument):
z = 10 ∠ 60°

Polar form:
z = 10 × (cos 60° + i sin 60°)

Exponential form:
z = 10 × e^{i 1.0471976} = 10 × e^{i π/3}

Polar coordinates:
r = |z| = 10 ... magnitude (modulus, absolute value)
θ = arg z = 1.0471976 rad = 60° = 0.3333333π = π/3 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 5+8.660254i
Real part: x = Re z = 5
Imaginary part: y = Im z = 8.66025404

Calculation steps

Angle notation (phasor, module and argument): 10 L 60 = 5+8.660254i
10 ∠ 60° = 10 L 60° = 10 cis 60° = 10 * (cos 60° + i sin 60°) = 5+8.660254i
We assume trigonometric angle argument in degrees.

This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As an imaginary unit, use i or j (in electrical engineering), which satisfies the basic equation i² = −1 or j² = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

Basic operations with complex numbers

We hope that working with the complex number is quite easy because you can work with imaginary unit i as a variable. And use the definition i² = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

Addition

It is very simple: add up the real parts (without i) and add up the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i

(1+i) + (6-5i) = 7-4i
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5

Subtraction

Again it is very simple: subtract the real parts and subtract the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i

(1+i) - (3-5i) = -2+6i
-1/2 - (6-5i) = -6.5+5i
(10-5i) - (-5+5i) = 15-10i

Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and apply i² = -1.
This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i

(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i
-1/2 * (6-5i) = -3+2.5i
(10-5i) * (-5+5i) = -25+75i

Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the denominator's complex conjugate. This approach avoids imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it real), multiply with conjugate c-di:

(c+di)(c-di) = c²+d²

\frac{a + b i}{c + d i} = \frac{( a + b i ) ( c - d i )}{( c + d i ) ( c - d i )} = \frac{a c + b d + i ( b c - a d )}{c ^{2} + d ^{2}} = \frac{a c + b d}{c ^{2} + d ^{2}} + \frac{b c - a d}{c ^{2} + d ^{2}} i

(10-5i) / (1+i) = 2.5-7.5i
-3 / (2-i) = -1.2-0.6i
6i / (4+3i) = 0.72+0.96i

Absolute value or modulus

The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142136
|6i| = 6
abs(2+5i) = 5.3851648

Square root

The square root of a complex number (a+bi) is z, if z² = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is to use De Moivre's formula. Our calculator is on edge because the square root is not a well-defined function on a complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.1213203+2.1213203i
sqrt(10-6i) = 3.2910412-0.9115656i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225i
pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303i

Square, power, complex exponentiation

Our calculator can power any complex number to an integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:

i^{i} = e^{- π / 2}

i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224i
i^i = 0.2078795764
pow(1+i,3) = -2+2i

Functions

sqrt: Square Root of a value or expression.
sin: the sine of a value or expression. Autodetect radians/degrees.
cos: the cosine of a value or expression. Autodetect radians/degrees.
tan: tangent of a value or expression. Autodetect radians/degrees.
exp: e (the Euler Constant) raised to the power of a value or expression
pow: Power one complex number to another integer/real/complex number
ln: The natural logarithm of a value or expression
log: The base-10 logarithm of a value or expression
abs or |1+i|: The absolute value of a value or expression
phase: Phase (angle) of a complex number
cis: is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i
conj: the conjugate of a complex number - example: conj(4i+5) = 5-4i