Complex number calculator - calculation: z^4=1 (2024)

There are 4 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains roots of complex numbers or powers to 1/n.

z₁ = (1^(1/4)) = 1 Calculation steps

Divide: 1 / 4 = 0.25
Exponentiation: 1 ^ the result of step No. 1 = 1 ^ 0.25 = 1

z₂ = (1^(1/4)) = i = e^{i π/2} Calculation steps principal root

Divide: 1 / 4 = 0.25
Exponentiation: 1 ^ the result of step No. 1 = 1 ^ 0.25 = i

The result z₂

Rectangular form (standard form):
z = i

Angle notation (phasor):
z = 1 ∠ 90°

Polar form:
z = cos 90° + i sin 90°

Exponential form:
z = e^{i 1.5707963} = e^{i π/2}

Polar coordinates:
r = |z| = 1 ... magnitude (modulus, absolute value)
θ = arg z = 1.5707963 rad = 90° = 0.5π = π/2 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = i
Real part: x = Re z = 0
Imaginary part: y = Im z = 1

z₃ = (1^(1/4)) = -1 Calculation steps

Divide: 1 / 4 = 0.25
Exponentiation: 1 ^ the result of step No. 1 = 1 ^ 0.25 = -1

The result z₂

Rectangular form (standard form):
z = i

Angle notation (phasor):
z = 1 ∠ 90°

Polar form:
z = cos 90° + i sin 90°

Exponential form:
z = e^{i 1.5707963} = e^{i π/2}

Polar coordinates:
r = |z| = 1 ... magnitude (modulus, absolute value)
θ = arg z = 1.5707963 rad = 90° = 0.5π = π/2 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = i
Real part: x = Re z = 0
Imaginary part: y = Im z = 1

z₄ = (1^(1/4)) = -i = e^{i (-π/2)} Calculation steps

Divide: 1 / 4 = 0.25
Exponentiation: 1 ^ the result of step No. 1 = 1 ^ 0.25 = -i

The result z₄

Rectangular form (standard form):
z = -i

Angle notation (phasor):
z = 1 ∠ -90°

Polar form:
z = cos (-90°) + i sin (-90°)

Exponential form:
z = e^{i -1.5707963} = e^{i (-π/2)}

Polar coordinates:
r = |z| = 1 ... magnitude (modulus, absolute value)
θ = arg z = -1.5707963 rad = -90° = -0.5π = -π/2 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -i
Real part: x = Re z = 0
Imaginary part: y = Im z = -1

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

(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:

ii=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

Complex numbers in word problems:

Complex number coordinates Which coordinates show the location of -2+3i
Alternating circuit In an alternating circuit, the total voltage V is given by V=V1 +V2 If V=(12.2+6.8i) V and V1=(7.8-2.5i) V, find the voltage V2.
Suppose 5 Suppose z5=2+3i and z6=6+9i are complex numbers and 3 z5 + 7 z6= m+in. What is the value of m and n?
Calculate 3884 Calculate the ratio of the two fifth roots of the number 32.

Log Calculate the value of expression log |-7 -i +i²| .
Suppose 10 Suppose 4+7i is a solution of 5z2+Az+B=0, where A, B∈R. Find A and B.
Determine 3888 Determine the sum of the three-third roots of the number 64.