# Complex number calculator

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

#### z1 = ((-1)^(1/4)) = 0.7071068+0.7071068i = ei π/4Calculation steps

1. Divide: 1 / 4 = 0.25
2. Exponentiation: (-1) ^ the result of step No. 1 = (-1) ^ 0.25 = -1 ^ 0.25 = 0.7071068+0.7071068i
The result z1
Rectangular form:
z = 0.7071068+0.7071068i

Angle notation (phasor):
z = 1 ∠ 45°

Polar form:
z = cos 45° + i sin 45°

Exponential form:
z = ei 0.25 = ei π/4

Polar coordinates:
r = |z| = 1 ... 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 = 0.7071068+0.7071068i
Real part: x = Re z = 0.707
Imaginary part: y = Im z = 0.70710678

#### z2 = ((-1)^(1/4)) = -0.7071068+0.7071068i = ei 3π/4Calculation steps

1. Divide: 1 / 4 = 0.25
2. Exponentiation: (-1) ^ the result of step No. 1 = (-1) ^ 0.25 = -1 ^ 0.25 = -0.7071068+0.7071068i
The result z2
Rectangular form:
z = -0.7071068+0.7071068i

Angle notation (phasor):
z = 1 ∠ 135°

Polar form:
z = cos 135° + i sin 135°

Exponential form:
z = ei 0.75 = ei 3π/4

Polar coordinates:
r = |z| = 1 ... magnitude (modulus, absolute value)
θ = arg z = 2.3561945 rad = 135° = 0.75π = 3π/4 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.7071068+0.7071068i
Real part: x = Re z = -0.707
Imaginary part: y = Im z = 0.70710678

#### z3 = ((-1)^(1/4)) = -0.7071068-0.7071068i = ei (-3π/4)Calculation steps

1. Divide: 1 / 4 = 0.25
2. Exponentiation: (-1) ^ the result of step No. 1 = (-1) ^ 0.25 = -1 ^ 0.25 = -0.7071068-0.7071068i
The result z3
Rectangular form:
z = -0.7071068-0.7071068i

Angle notation (phasor):
z = 1 ∠ -135°

Polar form:
z = cos (-135°) + i sin (-135°)

Exponential form:
z = ei (-0.75) = ei (-3π/4)

Polar coordinates:
r = |z| = 1 ... magnitude (modulus, absolute value)
θ = arg z = -2.3561945 rad = -135° = -0.75π = -3π/4 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.7071068-0.7071068i
Real part: x = Re z = -0.707
Imaginary part: y = Im z = -0.70710678

#### z4 = ((-1)^(1/4)) = 0.7071068-0.7071068i = ei (-π/4)Calculation steps

1. Divide: 1 / 4 = 0.25
2. Exponentiation: (-1) ^ the result of step No. 1 = (-1) ^ 0.25 = -1 ^ 0.25 = 0.7071068-0.7071068i
The result z4
Rectangular form:
z = 0.7071068-0.7071068i

Angle notation (phasor):
z = 1 ∠ -45°

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

Exponential form:
z = ei (-0.25) = ei (-π/4)

Polar coordinates:
r = |z| = 1 ... 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 = 0.7071068-0.7071068i
Real part: x = Re z = 0.707
Imaginary part: y = Im z = -0.70710678
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:
${i}^{i}={e}^{-\pi \mathrm{/}2}$
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

#### 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

## Complex numbers in word problems:

• Reciprocal Calculate reciprocal of z=0.8-1.8i:
• Linear imaginary equation Given that 2(z+i)=i(z+i) "this is z star" Find the value of the complex number z.
• Linear combination of complex If z1=5+3i and z2=4-2i, write the following in the form a+bi a) 4z1+6z2 b) z1*z2
• Imaginary numbers Find two imaginary numbers whose sum is a real number. How are the two imaginary numbers related? What is its sum?
• Stadium A domed stadium is in the shape of spherical segment with a base radius of 150 m. The dome must contain a volume of 3500000 m³. Determine the height of the dome at its centre to the nearest tenth of a meter.
• 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.
• Is complex Are these numbers 2i, 4i, 2i + 1, 8i, 2i + 3, 4 + 7i, 8i, 8i + 4, 5i, 6i, 3i complex?
• Let z1=x1+y1i Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)
• Modulus and argument Find the mod z and argument z if z=i
• Mappings of complex numbers Find the images of the following points under mappings: z=3-2j w=2zj+j-1
• The modulus Find the modulus of the complex number 2 + 5i
• 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?
• Two grandmothers Two grandmothers went to sell eggs at the market, and they had a total of 100. When they sold all the eggs, they made the same money. The first grandmother says to the second: "If I sold my eggs for your price, I would earn 15 crowns. " The other grandmot