Calculator roots of Complex Numbers


   



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

z1 = ((1 + i)^(1/7)) = 1.0441497+0.1176474i = 1.0507566 × ei π/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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei π/28 = 1.0441497+0.1176474i
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 0.0357143 = 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/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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei 9π/28 = 0.5590362+0.8897011i
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 0.3214286 = 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/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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei 17π/28 = -0.3470429+0.9917917i
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 0.6071429 = 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/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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei 25π/28 = -0.9917917+0.3470429i
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 0.8928571 = 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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei (-23π/28) = -0.8897011-0.5590362i
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 (-0.8214286) = 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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei (-15π/28) = -0.1176474-1.0441497i
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 (-0.5357143) = 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)0.14285714285714 = 1.41421360.14285714285714 × ei 0.14285714285714 × π/4 = 1.0507566 × ei (-π/4) = 0.7429971-0.7429971i
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 (-0.25) = 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 imaginary unit use i or j (in electrical engineering), which satisfies 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.

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.

Addition

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 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 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) = 2.1213203+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 any 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:
ii=eπ/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
conjugate of complex number - example: conj(4i+5) = 5-4i

Complex numbers in word problems:

  • Is complex
    cplx Are these numbers 2i, 4i, 2i + 1, 8i, 2i + 3, 4 + 7i, 8i, 8i + 4, 5i, 6i, 3i complex?
  • De Moivre's formula
    sqrt3_complex 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.
  • ABS CN
    complex_num Calculate the absolute value of complex number -15-29i.
  • Mappings of complex numbers
    complex_numbers Find the images of the following points under mappings: z=3-2j w=2zj+j-1
  • The modulus
    abs_value Find the modulus of the complex number 2 + 5i
  • Linear imaginary equation
    cplx_function Given that 2(z+i)=i(z+i) "this is z star" Find the value of the complex number z.
  • Complex number coordinates
    complex_numbers Which coordinates show the location of -2+3i
  • Goniometric form
    complex_fnc Determine the goniometric form of a complex number z = √ 110 +4 i.
  • Linear combination of complex
    cplx_function If z1=5+3i and z2=4-2i, write the following in the form a+bi a) 4z1+6z2 b) z1*z2
  • Let z1=x1+y1i
    cplx_function Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)
  • Im>0?
    ReIm Is -10i a positive number?
  • Reciprocal
    complex_numbers Calculate reciprocal of z=0.8-1.8i:
  • Imaginary numbers
    Complex_conjugate_picture.svg Find two imaginary numbers whose sum is a real number. How are the two imaginary numbers related? What is its sum?


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