# Calculator complex conjugate calculator

**Rectangular form:**

*z*= 4-5

**i**

**Angle notation (phasor):**

*z*= 6.4031242 ∠ -51°20'25″

**Polar form:**

*z*= 6.4031242 × (cos (-51°20'25″) +

**i**sin (-51°20'25″))

**Exponential form:**

*z*= 6.4031242 ×

**e**

^{i -0.8960554}= 6.4031242 ×

**e**

^{i (-0.2852233) π}

**Polar coordinates:**

r = |

*z*| = 6.4031242 ... magnitude (modulus, absolute value)

θ = arg

*z*= -0.8960554 rad = -51.34019° = -51°20'25″ = -0.2852233π rad ... angle (argument or phase)

**Cartesian coordinates:**

Cartesian form of imaginary number:

*z*= 4-5

**i**

Real part: x = Re

*z*= 4

Imaginary part: y = Im

*z*= -5

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}

$c+dia+bi =(c+di)(c−di)(a+bi)(c−di) =c_{2}+d_{2}ac+bd+i(bc−ad) =c_{2}+d_{2}ac+bd +c_{2}+d_{2}bc−ad i$

(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:

$i_{i}=e_{−π/2}$

i^2 = -1i^61 =

**i**

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

**i**

(6-i)^4.5 = 2486.1377428-2284.5557378

**i**

(6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224

**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**

## Complex numbers in word problems:

- Zdeněk

Zdeněk picked up 15 l of water from a 100-liter full-water barrel. Write a fraction of what part of Zdeněk's water he picked. - Fruit basket

If there are 7 apples and 5 oranges in the basket then what fraction of oranges are there in the fruit basket? - Fraction and a decimal

Write as a fraction and a decimal. One and two plus three and five hundredths - Fraction to decimal

Write the fraction 3/22 as a decimal. - Evaluate expression

Calculate the value of the expression z/3 - 2 z/9 + 1/6, for z = 2 - Classroom 4

In a class of 36 pupils, 2/3 are girls. How many girls and boys are in the class? - The parsley

Milka's grandmother planted 12 rows of vegetables. 1/6 of the rows are carrots, the rest is parsley. How many rows are planted with parsley? - Which 5

Which mixed number is equivalent to 2.68? A:2 and 6 eighths B:2 and 68 tenths C:2 and 6 over 68 - Someone

Someone ate 1/10 of a cake, leaving only 9/10. If you eat 2/3 of the cake that is left, how much of a whole cake will you have eaten? - ReIm notation

Let z = 6 + 5i and w = 3 - i. Compute the following and express your answer in a + bi form. w + 3z - Addition of Roman numbers

Added together and write as decimal number: LXVII + MLXIV - Evaluate 18

Evaluate the expression (-4-7i)-(-6-9i) and write the result in the form a+bi (Real + i* Imaginary). - Lowest terms 2

The expression 4/12 can be expressed in its lowest term as 1/3. What is 3/15 expressed in its lowest term?

next math problems »