# 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 × ei 0.3690496 = 2.615321 × ei 0.1174721 πCalculation stepsprincipal root

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = 1.2196165+0.4717113i
4. Square root: sqrt(4) = 4 = 2
5. 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):
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 × ei 0.3690496 = 2.615321 × ei 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 × ei 2.4634447 = 2.615321 × ei 0.7841388 πCalculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -1.0183222+0.8203632i
4. Square root: sqrt(4) = 4 = 2
5. 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):
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 × ei 2.4634447 = 2.615321 × ei 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 × ei -1.7253455 = 2.615321 × ei (-0.5491945) πCalculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -0.2012943-1.2920745i
4. Square root: sqrt(4) = 4 = 2
5. 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):
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 × ei -1.7253455 = 2.615321 × ei (-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 × ei -2.7725431 = 2.615321 × ei (-0.8825279) πCalculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = 1.2196165+0.4717113i
4. Square root: sqrt(4) = 4 = -2
5. 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):
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 × ei -2.7725431 = 2.615321 × ei (-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 × ei -0.678148 = 2.615321 × ei (-0.2158612) πCalculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -1.0183222+0.8203632i
4. Square root: sqrt(4) = 4 = -2
5. 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):
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 × ei -0.678148 = 2.615321 × ei (-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 × ei 1.4162471 = 2.615321 × ei 0.4508055 πCalculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -0.2012943-1.2920745i
4. Square root: sqrt(4) = 4 = -2
5. 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):
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 × ei 1.4162471 = 2.615321 × ei 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 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 definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

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 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

(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

Square root of 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 go with 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^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
conjugate of complex number - example: conj(4i+5) = 5-4i