## Calculation:

z = ((1+2i)^(1/3))*sqrt(4)

## Result:

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

#### z1 = ((1 + 2i)^(1/3))*sqrt(4) = 2.43923302+0.94342254i = 2.61532097 × ei 0.1174721Calculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = 1.21961651+0.47171127i
4. Square root: sqrt(4) = 4 = 2
5. Multiple: the result of step No. 3 * the result of step No. 4 = (1.21961651+0.47171127i) * 2 = 2.43923302+0.94342254i
The result z1
Rectangular form:
z = 2.43923302+0.94342254i

Angle notation (phasor):
z = 2.61532097 ∠ 21°8'42″

Polar form:
z = 2.61532097 × (cos 21°8'42″ + i sin 21°8'42″)

Exponential form:
z = 2.61532097 × ei 0.1174721

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

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.43923302+0.94342254i
Real part: x = Re z = 2.439
Imaginary part: y = Im z = 0.94342254

#### z2 = ((1 + 2i)^(1/3))*sqrt(4) = -2.03664439+1.64072649i = 2.61532097 × ei 0.7841388Calculation 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+0.82036324i
4. Square root: sqrt(4) = 4 = 2
5. Multiple: the result of step No. 3 * the result of step No. 4 = (-1.0183222+0.82036324i) * 2 = -2.03664439+1.64072649i
The result z2
Rectangular form:
z = -2.03664439+1.64072649i

Angle notation (phasor):
z = 2.61532097 ∠ 141°8'42″

Polar form:
z = 2.61532097 × (cos 141°8'42″ + i sin 141°8'42″)

Exponential form:
z = 2.61532097 × ei 0.7841388

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

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.03664439+1.64072649i
Real part: x = Re z = -2.037
Imaginary part: y = Im z = 1.64072649

#### z3 = ((1 + 2i)^(1/3))*sqrt(4) = -0.40258863-2.58414903i = 2.61532097 × ei (-0.5491945)Calculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -0.20129431-1.29207451i
4. Square root: sqrt(4) = 4 = 2
5. Multiple: the result of step No. 3 * the result of step No. 4 = (-0.20129431-1.29207451i) * 2 = -0.40258863-2.58414903i
The result z3
Rectangular form:
z = -0.40258863-2.58414903i

Angle notation (phasor):
z = 2.61532097 ∠ -98°51'18″

Polar form:
z = 2.61532097 × (cos (-98°51'18″) + i sin (-98°51'18″))

Exponential form:
z = 2.61532097 × ei (-0.5491945)

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

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.40258863-2.58414903i
Real part: x = Re z = -0.403
Imaginary part: y = Im z = -2.58414903

#### z4 = ((1 + 2i)^(1/3))*sqrt(4) = -2.43923302-0.94342254i = 2.61532097 × ei (-0.8825279)Calculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = 1.21961651+0.47171127i
4. Square root: sqrt(4) = 4 = -2
5. Multiple: the result of step No. 3 * the result of step No. 4 = (1.21961651+0.47171127i) * (-2) = -2.43923302-0.94342254i
The result z4
Rectangular form:
z = -2.43923302-0.94342254i

Angle notation (phasor):
z = 2.61532097 ∠ -158°51'18″

Polar form:
z = 2.61532097 × (cos (-158°51'18″) + i sin (-158°51'18″))

Exponential form:
z = 2.61532097 × ei (-0.8825279)

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

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.43923302-0.94342254i
Real part: x = Re z = -2.439
Imaginary part: y = Im z = -0.94342254

#### z5 = ((1 + 2i)^(1/3))*sqrt(4) = 2.03664439-1.64072649i = 2.61532097 × ei (-0.2158612)Calculation 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+0.82036324i
4. Square root: sqrt(4) = 4 = -2
5. Multiple: the result of step No. 3 * the result of step No. 4 = (-1.0183222+0.82036324i) * (-2) = 2.03664439-1.64072649i
The result z5
Rectangular form:
z = 2.03664439-1.64072649i

Angle notation (phasor):
z = 2.61532097 ∠ -38°51'18″

Polar form:
z = 2.61532097 × (cos (-38°51'18″) + i sin (-38°51'18″))

Exponential form:
z = 2.61532097 × ei (-0.2158612)

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

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.03664439-1.64072649i
Real part: x = Re z = 2.037
Imaginary part: y = Im z = -1.64072649

#### z6 = ((1 + 2i)^(1/3))*sqrt(4) = 0.40258863+2.58414903i = 2.61532097 × ei 0.4508055Calculation steps

1. Complex number: 1+2i
2. Divide: 1 / 3 = 0.33333333
3. Cube root: ∛(the result of step No. 1) = ∛(1+2i) = -0.20129431-1.29207451i
4. Square root: sqrt(4) = 4 = -2
5. Multiple: the result of step No. 3 * the result of step No. 4 = (-0.20129431-1.29207451i) * (-2) = 0.40258863+2.58414903i
The result z6
Rectangular form:
z = 0.40258863+2.58414903i

Angle notation (phasor):
z = 2.61532097 ∠ 81°8'42″

Polar form:
z = 2.61532097 × (cos 81°8'42″ + i sin 81°8'42″)

Exponential form:
z = 2.61532097 × ei 0.4508055

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

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.40258863+2.58414903i
Real part: x = Re z = 0.403
Imaginary part: y = Im z = 2.58414903

### Calculate next expression:

This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i2 = −1 or j2 = −1. The calculator also provides conversion of 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 angle (phase) in degrees, for example, 5L65 which is same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

Why the next complex numbers calculator when we have WolframAlpha? Because Wolfram tool is slow and some features such as step by step are charged premium service.
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 work 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 number 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

Division of two complex number is based on avoid imaginary unit i from denominator. This can be done only via i2 = -1. If denominator is c+di, to make it without i (or make it real), just multiply with conjugate c-di:

(c+di)(c-di) = c2+d2

$\dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{(c+di)(c-di)} = \dfrac{ac+bd+i(bc-ad)}{c^2+d^2} = \dfrac{ac+bd}{c^2+d^2}+\dfrac{bc-ad}{c^2+d^2} i$

(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

Absolute value or modulus is distance of image of complex number from origin in plane. That use Pythagorean theorem, just as case of 2D vector. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142135623731
|6i| = 6
abs(2+5i) = 5.3851648071345

### 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 probably to go with De Moivre's formula. Here our calculator is on edge, because square root is not a well defined function on complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.12132034+2.12132034i
sqrt(10-6i) = 3.29104116-0.91156563i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.43923302+0.94342254i
pow(-5i,1/8)*pow(8,1/3) = 2.39869586-0.47713027i

### Square, power, complex exponentiation

Yes, our calculator can power any complex number to any integer (positive, negative), real or even complex number. In another words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:
$i^i = e^{-\pi/2}$
i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.13779853-2284.55578905i
(6-5i)^(-3+32i) = 2929449.0670531-9022199.6661184i
i^i = 0.2078795764
pow(1+i,3) = -2+2i

### Functions

sqrt
Square Root of a value or expression.
sin
sine of a value or expression. Autodetect radians/degrees.
cos
cosine of a value or expression. Autodetect radians/degrees.
tan/tg
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/comple number
ln
The natural logarithm of a value or expression
log
The base-10 logarithm of a value or expression
abs or |1+i|
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

#### 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