Complex Number Calculator

Rectangular form (standard form):
z = 1.25

Angle notation (phasor, modulus and argument):
z = 1.25 ∠ 0°

Polar form:
z = 1.25 × (cos 0° + i sin 0°)

Exponential form:
z = 1.25 × e^{i 0} = 1.25 × e^{i 0}

Polar coordinates:
r = |z| = 1.25 ... magnitude (modulus, absolute value)
θ = arg z = 0 rad = 0° = 0π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.25
Real part: x = Re z = 1.25
Imaginary part: y = Im z = 0

Calculation steps

Divide: 1 / 2 = 1/1 · 1/2 = 1 · 1/1 · 2 = 1/2 = 0.5
The first operand is an integer. It is equivalent to a fraction 1/1. Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 2/1 is 1/2) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, the fraction cannot be simplified further by cancelling.
In other words, one divided by two equals one half.
Divide: 3 / 4 = 3/1 · 1/4 = 3 · 1/1 · 4 = 3/4 = 0.75
The first operand is an integer. It is equivalent to a fraction 3/1. Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 4/1 is 1/4) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, the fraction cannot be simplified further by cancelling.
In other words, three divided by four equals three quarters.
Add: the result of step No. 1 + the result of step No. 2 = 0.5 + 0.75 = 1.25

This calculator supports all operations with complex numbers and evaluates expressions in the complex number system. You can use i (mathematics) or j (electrical engineering) as the imaginary unit, both satisfying the fundamental property i² = −1 or j² = −1.

Additionally, the calculator can convert complex numbers into:

Angle notation (phasor notation)
Exponential form
Polar coordinates (magnitude and angle)

Example input: 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may be written 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 complex numbers is quite easy because you can work with the 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 equivalent to using the 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 equivalent to using the 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 the distributive law, expand the binomials, and apply i² = -1.
This is equivalent to using the 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 eliminates the 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 can handle this because the square root is not a well-defined function on complex numbers. 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 raise 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: The tangent of a value or expression. Autodetect radians/degrees.
exp: e (the Euler's number) raised to the power of a value or expression
pow: Raise one complex number to another integer/real/complex power
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: a lesser-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

Examples:

• cube root: cuberoot(1 - 27i)
• roots of Complex Numbers: pow(1 + i,1/7)
• phase, complex number angle: phase(1 + i)
• cis form complex numbers: 5 * cis(45°)
• The polar form of complex numbers: 10L60
• complex conjugate calculator: conj(4 + 5i)
• equation with complex numbers: (z + i/2 )/(1 - i) = 4z + 5i
• system of equations with imaginary numbers: x - y = 4 + 6i; 3ix + 7y=x + iy
• De Moivre's theorem - equation: z ^ 4=1
• multiplication of three complex numbers: (1 + 3i)(3 + 4i)(−5 + 3i)
• Find the product of 3-4i and its conjugate.: (3 - 4i) * conj(3 - 4i)
• operations with complex numbers: (3 - i) ^ 3