Complex number calculator

There are 6 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains roots of complex numbers or powers to 1/n.

z₁ = ((1 + 2i)^(1/3))*sqrt(4) = 2.439233+0.9434225i = 2.615321 × e^{i 0.3690496} = 2.615321 × e^{i 0.1174721 π} Calculation steps principal root

Complex number: 1+2i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
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 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛(the result of step No. 1) = ∛(1+2i) = 1.2196165+0.4717113i
Square root: sqrt(4) = √ 4 = 2
Multiple: the result of step No. 3 * the result of step No. 4 = (1.2196165+0.4717113i) * 2 = 2.439233+0.9434225i

The result z₁

Rectangular form (standard form):
z = 2.439233+0.9434225i

Angle notation (phasor, module and argument):
z = 2.615321 ∠ 21°8'42″

Polar form:
z = 2.615321 × (cos 21°8'42″ + i sin 21°8'42″)

Exponential form:
z = 2.615321 × e^{i 0.3690496} = 2.615321 × e^{i 0.1174721 π}

Polar coordinates:
r = |z| = 2.615321 ... magnitude (modulus, absolute value)
θ = arg z = 0.3690496 rad = 21.14498° = 21°8'42″ = 0.1174721π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.439233+0.9434225i
Real part: x = Re z = 2.439
Imaginary part: y = Im z = 0.94342254

z₂ = ((1 + 2i)^(1/3))*sqrt(4) = -2.0366444+1.6407265i = 2.615321 × e^{i 2.4634447} = 2.615321 × e^{i 0.7841388 π} Calculation steps

Complex number: 1+2i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
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 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -1.0183222+0.8203632i
Square root: sqrt(4) = √ 4 = 2
Multiple: the result of step No. 3 * the result of step No. 4 = (-1.0183222+0.8203632i) * 2 = -2.0366444+1.6407265i

The result z₂

Rectangular form (standard form):
z = -2.0366444+1.6407265i

Angle notation (phasor, module and argument):
z = 2.615321 ∠ 141°8'42″

Polar form:
z = 2.615321 × (cos 141°8'42″ + i sin 141°8'42″)

Exponential form:
z = 2.615321 × e^{i 2.4634447} = 2.615321 × e^{i 0.7841388 π}

Polar coordinates:
r = |z| = 2.615321 ... magnitude (modulus, absolute value)
θ = arg z = 2.4634447 rad = 141.14498° = 141°8'42″ = 0.7841388π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.0366444+1.6407265i
Real part: x = Re z = -2.037
Imaginary part: y = Im z = 1.6407265

z₃ = ((1 + 2i)^(1/3))*sqrt(4) = -0.4025886-2.584149i = 2.615321 × e^{i -1.7253455} = 2.615321 × e^{i (-0.5491945) π} Calculation steps

Complex number: 1+2i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
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 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -0.2012943-1.2920745i
Square root: sqrt(4) = √ 4 = 2
Multiple: the result of step No. 3 * the result of step No. 4 = (-0.2012943-1.2920745i) * 2 = -0.4025886-2.584149i

The result z₃

Rectangular form (standard form):
z = -0.4025886-2.584149i

Angle notation (phasor, module and argument):
z = 2.615321 ∠ -98°51'18″

Polar form:
z = 2.615321 × (cos (-98°51'18″) + i sin (-98°51'18″))

Exponential form:
z = 2.615321 × e^{i -1.7253455} = 2.615321 × e^{i (-0.5491945) π}

Polar coordinates:
r = |z| = 2.615321 ... magnitude (modulus, absolute value)
θ = arg z = -1.7253455 rad = -98.85502° = -98°51'18″ = -0.5491945π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.4025886-2.584149i
Real part: x = Re z = -0.403
Imaginary part: y = Im z = -2.58414903

z₄ = ((1 + 2i)^(1/3))*sqrt(4) = -2.439233-0.9434225i = 2.615321 × e^{i -2.7725431} = 2.615321 × e^{i (-0.8825279) π} Calculation steps

Complex number: 1+2i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
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 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛(the result of step No. 1) = ∛(1+2i) = 1.2196165+0.4717113i
Square root: sqrt(4) = √ 4 = -2
Multiple: the result of step No. 3 * the result of step No. 4 = (1.2196165+0.4717113i) * (-2) = -2.439233-0.9434225i

The result z₄

Rectangular form (standard form):
z = -2.439233-0.9434225i

Angle notation (phasor, module and argument):
z = 2.615321 ∠ -158°51'18″

Polar form:
z = 2.615321 × (cos (-158°51'18″) + i sin (-158°51'18″))

Exponential form:
z = 2.615321 × e^{i -2.7725431} = 2.615321 × e^{i (-0.8825279) π}

Polar coordinates:
r = |z| = 2.615321 ... magnitude (modulus, absolute value)
θ = arg z = -2.7725431 rad = -158.85502° = -158°51'18″ = -0.8825279π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.439233-0.9434225i
Real part: x = Re z = -2.439
Imaginary part: y = Im z = -0.94342254

z₅ = ((1 + 2i)^(1/3))*sqrt(4) = 2.0366444-1.6407265i = 2.615321 × e^{i -0.678148} = 2.615321 × e^{i (-0.2158612) π} Calculation steps

Complex number: 1+2i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
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 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -1.0183222+0.8203632i
Square root: sqrt(4) = √ 4 = -2
Multiple: the result of step No. 3 * the result of step No. 4 = (-1.0183222+0.8203632i) * (-2) = 2.0366444-1.6407265i

The result z₅

Rectangular form (standard form):
z = 2.0366444-1.6407265i

Angle notation (phasor, module and argument):
z = 2.615321 ∠ -38°51'18″

Polar form:
z = 2.615321 × (cos (-38°51'18″) + i sin (-38°51'18″))

Exponential form:
z = 2.615321 × e^{i -0.678148} = 2.615321 × e^{i (-0.2158612) π}

Polar coordinates:
r = |z| = 2.615321 ... magnitude (modulus, absolute value)
θ = arg z = -0.678148 rad = -38.85502° = -38°51'18″ = -0.2158612π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.0366444-1.6407265i
Real part: x = Re z = 2.037
Imaginary part: y = Im z = -1.6407265

z₆ = ((1 + 2i)^(1/3))*sqrt(4) = 0.4025886+2.584149i = 2.615321 × e^{i 1.4162471} = 2.615321 × e^{i 0.4508055 π} Calculation steps

Complex number: 1+2i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
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 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -0.2012943-1.2920745i
Square root: sqrt(4) = √ 4 = -2
Multiple: the result of step No. 3 * the result of step No. 4 = (-0.2012943-1.2920745i) * (-2) = 0.4025886+2.584149i

The result z₆

Rectangular form (standard form):
z = 0.4025886+2.584149i

Angle notation (phasor, module and argument):
z = 2.615321 ∠ 81°8'42″

Polar form:
z = 2.615321 × (cos 81°8'42″ + i sin 81°8'42″)

Exponential form:
z = 2.615321 × e^{i 1.4162471} = 2.615321 × e^{i 0.4508055 π}

Polar coordinates:
r = |z| = 2.615321 ... magnitude (modulus, absolute value)
θ = arg z = 1.4162471 rad = 81.14498° = 81°8'42″ = 0.4508055π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.4025886+2.584149i
Real part: x = Re z = 0.403
Imaginary part: y = Im z = 2.58414903

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 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

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