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 **i ^{2} = −1** or

**j**. 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

^{2}= −1Complex 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 working with the complex number is quite easy because you can work with imaginary unit **i** as a variable. And use the definition **i ^{2} = -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+b**i**)+(c+d**i**) = (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+b**i**)+(c+d**i**) = (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 **i ^{2} = -1**.

This is equal to use rule: (a+b

**i**)(c+d

**i**) = (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+d**i**, to make it without i (or make it real), multiply with conjugate c-d**i**:

(c+d**i**)(c-d**i**) = c^{2}+d^{2}

c+dia+bi=(c+di)(c−di)(a+bi)(c−di)=c2+d2ac+bd+i(bc−ad)=c2+d2ac+bd+c2+d2bc−adi

(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 z^{2} = (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

#### 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:

- Complex number coordinates Which coordinates show the location of -2+3i
- De Moivre's formula There are two distinct complex numbers, such that z³ is equal to 1 and z is not equal to 1. Calculate the sum of these two numbers.
- Evaluate 18 Evaluate the expression (-4-7i)-(-6-9i) and write the result in the form a+bi (Real + i* Imaginary).
- ReIm notation Let z = 6 + 5i and w = 3 - i. Compute the following and express your answer in a + bi form. w + 3z

- Complex plane mapping Show that the mapping w = z +c/z, where z = x+iy, w = u+iv, and c is a real number, maps the circle |z| = 1 in the z-plane into an ellipse in the (u, v) plane.
- Reciprocal Calculate the reciprocal of z=0.8-1.8i:
- Cis notation Evaluate multiplication of two complex numbers in cis notation: (6 cis 120°)(4 cis 30°) Write the result in cis and Re-Im notation.

more math problems »