# Complex number calculator

**Rectangular 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

### Calculation steps

- Exponentiation:
**i**^ 7 = 1^{7}**i**^{3}=**i**^{2}***i**= -**i**

This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers.
As an imaginary unit, use

Complex numbers in the angle notation or phasor (

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.

**i**or**j**(in electrical engineering), which satisfies the basic equation**i**or^{2}= −1**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 work with the complex number is quite easy because you can work with imaginary unit**i**as a variable. And use definition

**i**to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

^{2}= -1### 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+b

**i**)+(c+d

**i**) = (a+c) + (b+d)

**i**

(1+i) + (6-5i) = 7-4

**i**

12 + 6-5i = 18-5

**i**

(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+b

**i**)+(c+d

**i**) = (a-c) + (b-d)

**i**

(1+i) - (3-5i) = -2+6

**i**

-1/2 - (6-5i) = -6.5+5

**i**

(10-5i) - (-5+5i) = 15-10

**i**

### Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and apply**i**.

^{2}= -1This 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+8

**i**

-1/2 * (6-5i) = -3+2.5

**i**

(10-5i) * (-5+5i) = -25+75

**i**

### 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}

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

(10-5i) / (1+i) = 2.5-7.5

**i**

-3 / (2-i) = -1.2-0.6

**i**

6i / (4+3i) = 0.72+0.96

**i**

### 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 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 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+2.1213203

**i**

sqrt(10-6i) = 3.2910412-0.9115656

**i**

pow(-32,1/5)/5 = -0.4

pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225

**i**

pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303

**i**

### 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 = -1i^61 =

**i**

(6-2i)^6 = -22528-59904

**i**

(6-i)^4.5 = 2486.1377428-2284.5557378

**i**

(6-5i)^(-3+32i) = 2929449.03994-9022199.58262

**i**

i^i = 0.2078795764

pow(1+i,3) = -2+2

**i**

### 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) = 5-4
**i**

#### 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:

- 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? - Eq2 equations

For each of the following problems, determine the roots of the equation. Given the roots, sketch the graph and explain how your sketch matches the roots given and the form of the equation: g(x)=36x^{2}-12x+5 h(x)=x^{2}-4x+20 f(x)=4x^{2}-24x+45 p(x)=9x^{2}-36x+40 - Conjugate equation

Find the numbers of a and b, if (a - bi) (3 + 5i) is the Conjugate of (-6 - 24i) - 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. - Subtracting complex in polar

Given w =√2(cosine (p/4) + i sine (pi/4) ) and z = 2 (cosine (pi/2) + i sine (pi/2) ), what is w - z expressed in polar form? - 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. - Is complex

Are these numbers 2i, 4i, 2i + 1, 8i, 2i + 3, 4 + 7i, 8i, 8i + 4, 5i, 6i, 3i complex? - Distance two imaginary numbs

Find the distance between two complex number: z_{1}=(-8+i) and z_{2}=(-1+i). - Bearing

A plane flew 50 km on a bearing 63°20' and the flew on a bearing 153°20' for 140km. Find the distance between the starting point and the ending point. - Mappings of complex numbers

Find the images of the following points under mappings: z=3-2j w=2zj+j-1 - Mistake in expression

While attempting to multiply the expression (2 - 5i)(5 + 2i) a student made a mistake. (2 - 5i)(5 + 2i) = 10 + 4i - 25i - 10i2 = 10 + 4(-1) - 25(-1) - 10(1) = 10 - 4 + 25 - 10 = 21 Complete the explanation and correct the error. Hint: The student inc - Let z1=x1+y1i

Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)

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