# Calculator roots of Complex Numbers

There are 7 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains roots of complex numbers or powers to 1/n.

#### z1 = ((1 + i)^(1/7)) = 1.0441497+0.1176474i = 1.0507566 × ei π/28Calculation stepsprincipal root

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei π/28 =
The result z1
Rectangular form:
z = 1.0441497+0.1176474i

Angle notation (phasor):
z = 1.0507566 ∠ 6°25'43″

Polar form:
z = 1.0507566 × (cos 6°25'43″ + i sin 6°25'43″)

Exponential form:
z = 1.0507566 × ei 0.1121997 = 1.0507566 × ei π/28

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = 0.1121997 rad = 6.42857° = 6°25'43″ = 0.0357143π = π/28 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.0441497+0.1176474i
Real part: x = Re z = 1.044
Imaginary part: y = Im z = 0.11764742

#### z2 = ((1 + i)^(1/7)) = 0.5590362+0.8897011i = 1.0507566 × ei 9π/28Calculation steps

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei 9π/28 =
The result z2
Rectangular form:
z = 0.5590362+0.8897011i

Angle notation (phasor):
z = 1.0507566 ∠ 57°51'26″

Polar form:
z = 1.0507566 × (cos 57°51'26″ + i sin 57°51'26″)

Exponential form:
z = 1.0507566 × ei 1.0097976 = 1.0507566 × ei 9π/28

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = 1.0097976 rad = 57.85714° = 57°51'26″ = 0.3214286π = 9π/28 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.5590362+0.8897011i
Real part: x = Re z = 0.559
Imaginary part: y = Im z = 0.88970108

#### z3 = ((1 + i)^(1/7)) = -0.3470429+0.9917917i = 1.0507566 × ei 17π/28Calculation steps

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei 17π/28 =
The result z3
Rectangular form:
z = -0.3470429+0.9917917i

Angle notation (phasor):
z = 1.0507566 ∠ 109°17'9″

Polar form:
z = 1.0507566 × (cos 109°17'9″ + i sin 109°17'9″)

Exponential form:
z = 1.0507566 × ei 1.9073955 = 1.0507566 × ei 17π/28

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = 1.9073955 rad = 109.28571° = 109°17'9″ = 0.6071429π = 17π/28 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.3470429+0.9917917i
Real part: x = Re z = -0.347
Imaginary part: y = Im z = 0.99179168

#### z4 = ((1 + i)^(1/7)) = -0.9917917+0.3470429i = 1.0507566 × ei 25π/28Calculation steps

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei 25π/28 =
The result z4
Rectangular form:
z = -0.9917917+0.3470429i

Angle notation (phasor):
z = 1.0507566 ∠ 160°42'51″

Polar form:
z = 1.0507566 × (cos 160°42'51″ + i sin 160°42'51″)

Exponential form:
z = 1.0507566 × ei 2.8049934 = 1.0507566 × ei 25π/28

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = 2.8049934 rad = 160.71429° = 160°42'51″ = 0.8928571π = 25π/28 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.9917917+0.3470429i
Real part: x = Re z = -0.992
Imaginary part: y = Im z = 0.34704292

#### z5 = ((1 + i)^(1/7)) = -0.8897011-0.5590362i = 1.0507566 × ei (-23π/28)Calculation steps

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei (-23π/28) =
The result z5
Rectangular form:
z = -0.8897011-0.5590362i

Angle notation (phasor):
z = 1.0507566 ∠ -147°51'26″

Polar form:
z = 1.0507566 × (cos (-147°51'26″) + i sin (-147°51'26″))

Exponential form:
z = 1.0507566 × ei -2.580594 = 1.0507566 × ei (-23π/28)

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = -2.580594 rad = -147.85714° = -147°51'26″ = -0.8214286π = -23π/28 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.8897011-0.5590362i
Real part: x = Re z = -0.89
Imaginary part: y = Im z = -0.55903624

#### z6 = ((1 + i)^(1/7)) = -0.1176474-1.0441497i = 1.0507566 × ei (-15π/28)Calculation steps

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei (-15π/28) =
The result z6
Rectangular form:
z = -0.1176474-1.0441497i

Angle notation (phasor):
z = 1.0507566 ∠ -96°25'43″

Polar form:
z = 1.0507566 × (cos (-96°25'43″) + i sin (-96°25'43″))

Exponential form:
z = 1.0507566 × ei -1.6829961 = 1.0507566 × ei (-15π/28)

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = -1.6829961 rad = -96.42857° = -96°25'43″ = -0.5357143π = -15π/28 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.1176474-1.0441497i
Real part: x = Re z = -0.118
Imaginary part: y = Im z = -1.04414971

#### z7 = ((1 + i)^(1/7)) = 0.7429971-0.7429971i = 1.0507566 × ei (-π/4)Calculation steps

1. Complex number: 1+i
2. Divide: 1 / 7 = 0.14285714
3. Exponentiation: the result of step No. 1 ^ the result of step No. 2 = (1+i) ^ 0.14285714 = (1.4142136 × ei π/4)0.142857142857 = 1.41421360.142857142857 × ei 0.142857142857 × π/4 = 1.0507566 × ei (-π/4) =
The result z7
Rectangular form:
z = 0.7429971-0.7429971i

Angle notation (phasor):
z = 1.0507566 ∠ -45°

Polar form:
z = 1.0507566 × (cos (-45°) + i sin (-45°))

Exponential form:
z = 1.0507566 × ei -0.7853982 = 1.0507566 × ei (-π/4)

Polar coordinates:
r = |z| = 1.0507566 ... magnitude (modulus, absolute value)
θ = arg z = -0.7853982 rad = -45° = -0.25π = -π/4 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.7429971-0.7429971i
Real part: x = Re z = 0.743
Imaginary part: y = Im z = -0.74299715
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 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 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

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

(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

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:
${i}^{i}={e}^{-\pi \mathrm{/}2}$
i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.03994-9022199.58262i
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

## Complex numbers in word problems:

• Complex number coordinates Which coordinates show the location of -2+3i
• De Moivre's formula There are two distinct complex numbers z, such that z3 is equal to 1 and z is not equal to 1. Calculate the sum of these two numbers.
• Log Calculate value of expression log |3 +7i +5i2| .
• Let 4 Let z = 6 + 5i and w = 3 - i. Compute the following and express your answer in a + bi form. w + 3z
• Evaluate 18 Evaluate the expression (-4-7i)-(-6-9i) and write the result in the form a+bi (Real + i* Imaginary).
• Reciprocal Calculate reciprocal of z=0.8-1.8i:
• Imaginary numbers Find two imaginary numbers whose sum is a real number. How are the two imaginary numbers related? What is its sum?
• Conjugate equation Find the numbers of a and b, if (a - bi) (3 + 5i) is the Conjugate of (-6 - 24i)
• Cis notation Evaluate multiplication of two complex numbers in cis notation: (6 cis 120°)(4 cis 30°) Write result in cis and Re-Im notation.
• ABS CN Calculate the absolute value of complex number -15-29i.
• Turtles 2 A box turtle hibernates in the sand at 11 5/8. A spotted turtle hibernates at 11 16/25 feet. Which turtle is deeper? Write answer as number 1 or 2.
• Modulus and argument Find the mod z and argument z if z=i
• Moivre 2 Find the cube roots of 125(cos 288° + i sin 288°).