# Complex Number Calculator

## The complex number calculator calculates: addition, subtraction, multiplication and division of two complex numbers. Converts a complex number to the rectangular (algebraic form), polar, and exponential forms of a complex number. Raises to real and complex powers, complex and real numbers. Calculates: modulus, argument, conjugate, reciprocal, additive inversion, nth root, logarithm, complex exponent, sine, cosine, tangent, hyperbolic sine, hyperbolic cosine, hyperbolic tangent, arcsine, arccosine, and arctangent. The calculator displays the detailed progress of solving the example.

z = a + bi

z1 =

z2 =

Theory

Addition of Two Complex Numbers Represented in Algebraic (Rectangular) Form.Addition of Real and Complex Numbers Represented in Algebraic (Rectangular) Form.

Subtraction of Two Complex Numbers Represented in Algebraic (Rectangular) Form.

Subtracting a Complex Number From a Real Number.

Subtracting a Real Number From a Complex Number.

Multiplication of Two Complex Numbers Represented in Algebraic (Rectangular) Form..

Multiplication of a Real and Complex Number Represented in Algebraic (Rectangular) Form.

Division of Two Complex Numbers, Represented in Algebraic (Rectangular) Form.

Division of a Complex Number Represented in Algebraic (Rectangular) Form by a Real Number.

Division of a Real Number by a Complex Number Represented in Algebraic (Rectangular) Form.

Addition of Complex Numbers Represented in Polar Form.

Subtraction of Complex Numbers Represented in Polar Form.

Multiplication of Complex Numbers Represented in Polar Form.

Division of Complex Numbers Represented in Polar Form.

Addition of Complex Numbers Represented in Exponential Form.

Subtraction of Complex Numbers Represented in Exponential Form.

Multiplication of Complex Numbers Represented in Exponential Form.

Division of Complex Numbers Represented in Exponential Form.

Modulus of a Complex Number.

Argument (Phase) Of a Complex Number.

Representation of a Complex Number Represented in Algebraic (Rectangular) Form in Polar Form.

Representation of a Complex Number Represented in Algebraic (Rectangular) Form in Exponential Form.

Representation of a Complex Number Represented in Polar Form in Algebraic (Rectangular) Form.

Representation of a Complex Number Represented in Polar Form in Exponential Form.

Representation of a Complex Number Represented in Exponential Form in Polar Form.

Representation of a Complex Number Written in Exponential Form in Algebraic (Rectangular) Form.

The Conjugate of a Complex Number Represented in Algebraic (Rectangular) Form.

The Conjugate of a Complex Number Expressed in Polar Form.

The Conjugate of a Complex Number Expressed in Exponential Form.

Reciprocal of a Complex Number Represented in Algebraic (Rectangular) Form.

Reciprocal of a Complex Number Represented in Polar Form.

Reciprocal of a Complex Number Represented in Exponential Form.

Additive Inversion of a Complex Number.

Extracting the Nth Root of a Complex Number Represented in Algebraic (Rectangular) Form.

Extracting the Nth Root of a Complex Number Represented in Polar Form.

Extracting the Nth Root of a Complex Number Represented in Exponential Form.

Addition of Two Complex Numbers Represented in Algebraic (Rectangular) Form

Let's give examples:

Example 1. Add two complex numbers 2 + 3i and 1.6 + 7i

(2 + 3i) + (1.6 + 7i) = (2 + 1.6) + (3 + 7)i = 3.6 + 10i

Example 2. Let us add two complex numbers 3 + 4i and 8 âˆ’ 6i

(3 + 4i) + (8 âˆ’ 6i) = (3 + 8) + (4 âˆ’ 6i) = 11 âˆ’ 2i

Addition of Real and Complex Numbers Represented in Algebraic (Rectangular) Form

Let's give examples:

Example 1. Add the complex number 2 + 3i and the real number 10

(2 + 3i) + 10 = (2 + 10) + 3i = 12 + 3i

Example 2. Add the complex number âˆ’6 + 3i and the real number -23

(âˆ’6 + 3i) + (âˆ’23) = (âˆ’6 + (âˆ’23)) + 3i = âˆ’29 + 3i

Subtraction of Two Complex Numbers Represented in Algebraic (Rectangular) Form

Let's give examples:

Example 1. Subtract two complex numbers 3 + 9i and 5 + 6i

(3 + 9i) âˆ’ (5 + 6i) = (3 âˆ’ 5) + (9 âˆ’ 6)i = âˆ’2 + 3i

Example 2. Subtract two complex numbers 6 + 23i and 57 + 68i

(6 + 23i) âˆ’ (57 + 68i) = (6 âˆ’ 57) + (23 âˆ’ 68)i = âˆ’51 âˆ’ 45i

Subtracting a Complex Number From a Real Number

Let's give examples:

Example 1. Subtract from the real number 6 the complex number 1 + 7i

6 âˆ’ (1 + 7i) = (6 âˆ’ 1) + 7i = 5 âˆ’ 7i

Example 2. Subtract from the real number -15 the complex number 1 + (âˆ’7)i

âˆ’15 âˆ’ (1 + (âˆ’7)i) = (âˆ’15 âˆ’ 1) âˆ’ (âˆ’7)i = âˆ’16 + 7i

Subtracting a Real Number From a Complex Number

Let's give examples:

Example 1. Subtract from the complex number 5 + 12i the real number 8

(5 + 12i) âˆ’ 8 = (5 âˆ’ 8) + 12i = âˆ’3 + 12i

Example 2. Subtract from the complex number âˆ’1 + (âˆ’5)i the real number âˆ’3

(âˆ’1 + (âˆ’5)i) âˆ’ (âˆ’3) = (âˆ’1 âˆ’ (âˆ’3)) + (âˆ’5)i = 2 âˆ’ 5i

Multiplication of Two Complex Numbers Represented in Algebraic (Rectangular) Form.

^{2}= (ac + bdi

^{2}) + (bc + ad)i = (ac âˆ’ bd) + (bc + ad)i

Let's give examples:

Example 1. Let's multiply two complex numbers 2 + 5i and 3 + 7i

Solution 1

(2 + 5i) Ã— (3 + 7i) = ((2 Ã— 3) âˆ’ (5 Ã— 7)) + ((5 Ã— 3) + (2 Ã— 7))i = (6 âˆ’ 35) + (15 + 14)i = âˆ’29 + 29i

Solution 2

(2 + 5i) Ã— (3 + 7i) = (2 Ã— 3) + (2 Ã— 7i) + (5i Ã— 3) + (5i Ã— 7i) = 6 + (14i) + (15i) + (35i^{2}) = 6 + (29i) + (35 Ã— (âˆ’1)) = âˆ’29 + 29i

Example 2. Let's multiply two complex numbers 0.4 + (âˆ’2)i and 3.023 + 0.25i

Solution 1

(0.4 + (âˆ’2)i) Ã— (3.023 + 0.25i) = ((0.4 Ã— 3.023) âˆ’ (âˆ’2 Ã— 0.25)) + (((âˆ’2) Ã— 3.023) + (0.4 Ã— 0.25))i = (1.2092 âˆ’ (âˆ’0.5)) + (âˆ’6.046 + 0.1)i = 1.7092âˆ’5.946i

Solution 2

(0.4 + (âˆ’2)i) Ã— (3.023 + 0.25i) = (0.4 Ã— 3.023) + (0.4 Ã— 0.25i) + ((âˆ’2)i Ã— 3.023) + ((âˆ’2)i Ã— 0.25i) = 1.2092 + (0.1i) + (âˆ’6.046i) + (âˆ’0.5i^{2}) = 1.2092 + (âˆ’5.946i) + ((âˆ’0.5 Ã— (âˆ’1))) = 1.7092 âˆ’ 5.946i

Multiplication of a Real and Complex Number Represented in Algebraic (Rectangular) Form

Let's give examples:

Example 1. Multiply the complex number 3 + 4i and the real number 1

1 Ã— (3 + 4i) = (1 Ã— 3) + (1 Ã— 4)i = 3 + 4i

Example 2. Multiply the complex number âˆ’5 + 4i and the real number âˆ’74

âˆ’74 Ã— (âˆ’5 + 4i) = (âˆ’74 Ã— (âˆ’5)) + (âˆ’74 Ã— 4)i = 370 âˆ’ 296i

Division of Two Complex Numbers, Represented in Algebraic (Rectangular) Form

a + bi | = | |

c + di |

(a + bi) Ã— (c âˆ’ di) | = | |

(c + di) Ã— (c âˆ’ di) |

ac + bd | + | |

c^{2} + d^{2} |

bc âˆ’ ad | |

c^{2} + d^{2} |

i | |

Let's give examples:

Example 1. Divide the complex number 4 + 3i by the complex number 5 + 8i

(4 + 3i) | = | |

(5 + 8i) |

(4 + 3i) Ã— (5 âˆ’ 8i) | = | |

(5 + 8i) Ã— (5 âˆ’ 8i) |

(4 Ã— 5) + (3 Ã— 8) | + | |

(5^{2} + 8^{2}) |

(3 Ã— 5) âˆ’ (4 Ã— 8) | |

(5^{2} + 8^{2}) |

i = | |

(20 + 24) | + | |

(25 + 64) |

15 âˆ’ 32 | |

25 + 64 |

i = | |

44 | + | |

89 |

âˆ’17 | |

89 |

i | |

= 0.49438202247191âˆ’0.191011235955056i | ||

Example 2. Divide the complex number 6 + (âˆ’2)i by the complex number âˆ’4 + 7i

(6 + (âˆ’2)i) | = | |

(âˆ’4 + 7i) |

(6 + (âˆ’2)i) Ã— (âˆ’4 âˆ’ 7i) | = | |

(âˆ’4 + 7i) Ã— (âˆ’4 âˆ’ 7i) |

(6 Ã— (âˆ’4)) + (âˆ’2 Ã— 7) | + | |

(âˆ’4^{2} + 7^{2}) |

(âˆ’2 Ã— (âˆ’4)) âˆ’ (6 Ã— 7) | |

(âˆ’4^{2} + 7^{2}) |

i = | |

(âˆ’24 + (âˆ’14)) | + | |

(16 + 49) |

8 âˆ’ 42 | |

16 + 49 |

i = | |

âˆ’38 | + | |

65 |

âˆ’34 | |

65 |

i | |

= âˆ’0.584615384615385âˆ’0.523076923076923i | ||

Division of a Complex Number Represented in Algebraic (Rectangular) Form by a Real Number

a + bi | = | |

c |

a | + | |

c |

b | i | |

c |

Here's an example:

Divide the complex number 3 + 6i by the real number 7

(3 + 6i) | = | |

7 |

3 | + | |

7 |

6 | |

7 |

i | |

= 0.428571428571429 + 0.857142857142857i | ||

Division of a Real Number by a Complex Number Represented in Algebraic (Rectangular) Form

a | = | |

c + di |

a Ã— (c âˆ’ di) | = | |

(c + di) Ã— (c âˆ’ di) |

ac | âˆ’ | |

c^{2} + d^{2} |

ad | |

c^{2} + d^{2} |

i | |

Here's an example:

Divide the real number 5 by the complex number 2 + 9i

5 | = | |

(2 + 9i) |

5 Ã— (2 âˆ’ 9i) | = | |

(2 + 9i) Ã— (2 âˆ’ 9i) |

5 Ã— | âˆ’ | |

(2^{2} + 9^{2}) |

5 Ã— | |

(2^{2} + 9^{2}) |

i = | |

10 | âˆ’ | |

(4 + 81) |

45 | |

4 + 81 |

i = | |

10 | âˆ’ | |

85 |

45 | |

85 |

i | |

= 0.117647058823529âˆ’0.529411764705882i | ||

Addition of Complex Numbers Represented in Polar Form

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Let's add two complex numbers âˆš13 (cos 48Â° + i sin 48Â°) and âˆš25 (cos 69Â° + i sin 69Â°)

âˆš13 (cos 48Â° + i sin 48Â°) + âˆš25 (cos 69Â° + i sin 69Â°) = ((âˆš13 Ã— cos(48Â°)) + (âˆš25 Ã— cos(69Â°))) + i((âˆš13 Ã— sin(48Â°)) + (âˆš25 Ã— sin(69Â°))) = (2.41258471120918 + 1.7918397477265) + (2.67944677335447 + 4.667902132486)i = 4.20442445893568 + 7.34734890584047i

Subtraction of Complex Numbers Represented in Polar Form

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Subtract from two complex number 1/2 (cos Ï€/2 + i sin Ï€/2) the number 1/3 (cos Ï€/3 + i sin Ï€/3)

1/2 (cos Ï€/2 + i sin Ï€/2) âˆ’ 1/3 (cos Ï€/3 + i sin Ï€/3) = ((1/2 Ã— cos((Ï€/2))) âˆ’ (1/3 Ã— cos((Ï€/3)))) + i((1/2 Ã— sin((Ï€/2))) âˆ’ (1/3 Ã— sin((Ï€/3)))) = (0 âˆ’ 0.166666666666666) + (0.5 âˆ’ 0.288675134594813)i = âˆ’0.166666666666666 + 0.211324865405187i

Multiplication of Complex Numbers Represented in Polar Form

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Multiply two complex numbers 2 (cos Ï€/2 + i sin Ï€/2) and 2 (cos Ï€/3 + i sin Ï€/3)

2 (cos Ï€/2 + i sin Ï€/2) Ã— 2 (cos Ï€/3 + i sin Ï€/3) = (2 Ã— 2) Ã— (cos(Ï€/2 + (Ï€/3)) + i Ã— sin(Ï€/2 + (Ï€/3))) =

Division of Complex Numbers Represented in Polar Form

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Divide two complex numbers 3 (cos 45Â° + i sin 45Â°) and 2 (cos 37Â° + i sin 37Â°)

3 (cos 45Â° + i sin 45Â°) Ã· 2 (cos 37Â° + i sin 37Â°) = (3 Ã· 2) Ã— (cos(45Â° âˆ’ 37Â°) + i Ã— sin(45Â° âˆ’ 37Â°)) =

Addition of Complex Numbers Represented in Exponential Form

^{iÎ±}+ |z2| e

^{iÎ²}= (|z1| Ã— (cos Î± + i sin Î±)) + (|z2| Ã— (cos Î² + i sin Î²)) = ((|z1| Ã— cos Î±) + (|z2| Ã— cos Î²)) + i((|z1| Ã— sin Î±) + (|z2| Ã— sin Î²)), where

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Let's add two complex numbers 3 Ã— e^{(Ï€/2)i} and 2 Ã— e^{(3Ï€/2)i}

3 Ã— e^{(Ï€/2)i} + 2 Ã— e^{(3Ï€/2)i} = ((3 Ã— cos((Ï€/2))) + (2 Ã— cos((3Ï€/2)))) + i((3 Ã— sin((Ï€/2))) + (2 Ã— sin((3Ï€/2)))) = (0 + 0) + (3 + (âˆ’2))i = 0 + 1i

Subtraction of Complex Numbers Represented in Exponential Form

^{iÎ±}âˆ’ |z2| e

^{iÎ²}= (|z1| Ã— (cos Î± + i sin Î±)) âˆ’ (|z2| Ã— (cos Î² + i sin Î²)) = ((|z1| Ã— cos Î±) âˆ’ (|z2| Ã— cos Î²)) + i((|z1| Ã— sin Î±) âˆ’ (|z2| Ã— sin Î²)), where

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Subtract from the number 3 Ã— e^{(Ï€/2)i} the number 2 Ã— e^{(3Ï€/2)i}

3 Ã— e^{(Ï€/2)i} - 2 Ã— e^{(3Ï€/2)i} = ((3 Ã— cos((Ï€/2))) âˆ’ (2 Ã— cos((3Ï€/2)))) + i((3 Ã— sin((Ï€/2))) âˆ’ (2 Ã— sin((3Ï€/2)))) = (0 âˆ’ 0) + (3 âˆ’ (âˆ’2))i = 0 + 5i

Multiplication of Complex Numbers Represented in Exponential Form

^{iÎ±}Ã— |z2| e

^{iÎ²}= (|z1| Ã— |z2|) Ã— (cos(Î± + Î²) + i Ã— sin(Î± + Î²)), where

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Multiply two complex numbers 3 Ã— e^{(Ï€/2)i} and 2 Ã— e^{(3Ï€/2)i}

(3 Ã— e^{(Ï€/2)i}) Ã— (2 Ã— e^{(3Ï€/2)i}) = (3 Ã— 2) Ã— (cos(Ï€/2 + (3Ï€/2)) + i Ã— sin(Ï€/2 + (3Ï€/2))) =

Division of Complex Numbers Represented in Exponential Form

^{iÎ±}Ã· |z2| e

^{iÎ²}= (|z1| Ã· |z2|) Ã— (cos(Î± âˆ’ Î²) + i Ã— sin(Î± âˆ’ Î²)), where

|z1| âˆ’ modulus of the complex number z1

Î± âˆ’ argument of the complex number z1

|z2| âˆ’ modulus of the complex number z2

Î² âˆ’ argument of the complex number z2

Here's an example:

Divide two complex numbers 3 Ã— e^{(Ï€/2)i} and 2 Ã— e^{(3Ï€/2)i}

(3 Ã— e^{(Ï€/2)i}) Ã· (2 Ã— e^{(3Ï€/2)i}) = (3 Ã· 2) Ã— (cos(Ï€/2 âˆ’ (3Ï€/2)) + i Ã— sin(Ï€/2 âˆ’ (3Ï€/2))) =

Modulus of a Complex Number

The modulus of a complex number z is calculated by the formula:

^{2}+ b

^{2}

Here's an example:

Calculate the modulus of a complex number 1 + 3i

|1 + 3i| =^{2}+ 3

^{2}

Argument (Phase) Of a Complex Number

The argument of the complex number z is calculated by the formula:

Arg(z) = arctg(b/a) + Ï€, a < 0

Arg(z) = Ï€/2, b > 0 and a = 0

Arg(z) = âˆ’Ï€/2, b < 0 and a = 0

Here's an example:

Calculate the argument of a complex number âˆ’4 + 7i

Arg(âˆ’4 + 7i) = arctg(7/(âˆ’4)) + Ï€ = 2.08994244104142 radiansArg(âˆ’4 + 7i) = 119.744881296942Â° degrees

Representation of a Complex Number Represented in Algebraic (Rectangular) Form in Polar Form

|z| âˆ’ modulus of a complex number z

Ï† âˆ’ argument of a complex number z

The modulus of a complex number z is calculated by the formula:

^{2}+ b

^{2}

Here's an example:

Converting the number 2 + 3i to the polar form

|2 + 3i| =^{2}+ 3

^{2}

The argument of the complex number z is calculated by the formula:

Arg(z) = arctg(b/a) + Ï€, a < 0

Arg(z) = Ï€/2, b > 0 and a = 0

Arg(z) = âˆ’Ï€/2, b < 0 and a = 0

Arg(2 + 3i) = arctg(3/2) = 56.3099324740202Â° degrees

z = 2 + 3i =

Representation of a Complex Number Represented in Algebraic (Rectangular) Form in Exponential Form

|z| âˆ’ modulus of a complex number z

Ï† âˆ’ argument of a complex number z

The modulus of a complex number z is calculated by the formula:

^{2}+ b

^{2}

Here's an example:

Let's represent the number 2 + 2i in exponential form

|2 + 2i| =^{2}+ 2

^{2}

The argument of the complex number z is calculated by the formula:

Arg(z) = arctg(b/a) + Ï€, a < 0

Arg(z) = Ï€/2, b > 0 and a = 0

Arg(z) = âˆ’Ï€/2, b < 0 and a = 0

Arg(2 + 2i) = arctg(2/2) = 45Â° degrees

z = 2 + 2i =

^{0.785398163397448i}=

^{45Â°i}=

^{(Ï€/4)i}

Representation of a Complex Number Represented in Polar Form in Algebraic (Rectangular) Form

a = |z| Ã— cos Ï†

b = |z| Ã— sin Ï†

Here's an example:

Convert 2 Ã— (cos(60Â°) + sin(60Â°)i) to algebraic (rectangular) form

2 Ã— (cos(60Â°) + sin(60Â°)i) = 2 Ã— (0.5 + 0.86602540378444i) = (2 Ã— 0.5) + (2 Ã— 0.86602540378444i) = 1 + 1.73205080756888i

Representation of a Complex Number Represented in Polar Form in Exponential Form

^{Ï†i}, where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

Here's an example:

Let's represent the number 2 Ã— (cos(60Â°) + sin(60Â°)i) in exponential form

2 Ã— (cos(60Â°) + sin(60Â°)i) = 2 Ã— e^{60Â°i}= 2 Ã— e

^{(Ï€/3)i}

Representation of a Complex Number Represented in Exponential Form in Polar Form

^{Ï†i}= |z| Ã— (cos Ï† + i sin Ï†), where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

Here's an example:

Convert 2 Ã— e^{(Ï€/3)i} to polar form

^{(Ï€/3)i}= 2 Ã— (cos((Ï€/3)) + sin((Ï€/3))i)

Representation of a Complex Number Written in Exponential Form in Algebraic (Rectangular) Form

^{Ï†i}= |z| Ã— (cos Ï† + i sin Ï†), where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

a = |z| Ã— cos Ï†

b = |z| Ã— sin Ï†

Here's an example:

Convert the number 2 Ã— e^{(Ï€/3)i} to the algebraic (rectangular) form

^{(Ï€/3)i}= 2 Ã— (cos((Ï€/3)) + sin((Ï€/3))i) = 2 Ã— (0.5 + 0.86602540378444i) = (2 Ã— 0.5) + (2 Ã— 0.86602540378444i) = 1 + 1.73205080756888i

The Conjugate of a Complex Number Represented in Algebraic (Rectangular) Form

The conjugate of z is denoted as

**z**.

For z = a + bi, the conjugate is

**z**= a âˆ’ bi.

Here's an example:

Calculate the conjugate number for the number 2 + 3i

**2 + 3i**= 2 âˆ’ 3i

The Conjugate of a Complex Number Expressed in Polar Form

The conjugate of z is denoted as

**z**.

For z = |z| (cos Ï† + i sin Ï†) the conjugate number is

**z**= |z| (cos Ï† âˆ’ i sin Ï†).

Here's an example:

Calculate the conjugate number for the number 2(cos((Ï€/3)) + sin((Ï€/3))i)

**2(cos((Ï€/3)) + sin((Ï€/3))i)**= 2(cos((Ï€/3)) âˆ’ sin((Ï€/3))i)

The Conjugate of a Complex Number Expressed in Exponential Form

The conjugate of z is denoted as

**z**.

For z = |z| e

^{iÏ†}the conjugate is

**z**= |z| e

^{âˆ’iÏ†}.

Here's an example:

Calculate the conjugate number for the number 2 Ã— e^{(Ï€/3)i}

**2 Ã— e**= 2 Ã— e

^{(Ï€/3)i}^{ âˆ’ (Ï€/3)i}

Reciprocal of a Complex Number Represented in Algebraic (Rectangular) Form

1 | = | |

a + bi |

1 Ã— (a âˆ’ bi) | = | |

(a + bi) Ã— (a âˆ’ bi) |

a | âˆ’ | |

a^{2} + b^{2} |

b | |

a^{2} + b^{2} |

i | |

Here's an example:

Calculate the reciprocal of the number 2 + 3i

1 | = | |

(2 + 3i) |

2 | âˆ’ | |

(2^{2} + 3^{2}) |

3 | |

(2^{2} + 3^{2}) |

i = | |

2 | âˆ’ | |

(4 + 9) |

3 | |

4 + 9 |

i = | |

2 | âˆ’ | |

13 |

3 | |

13 |

i | |

= 0.153846153846154âˆ’0.230769230769231i | ||

The Reciprocal of a Complex Number Expressed in Polar Form

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

Here's an example:

Calculate the reciprocal of the number 2(cos((Ï€/2)) + sin((Ï€/2))i)

1/z = 1/2 Ã— (cos (Ï€/2) âˆ’ i sin Ï€/2) =Reciprocal of a Complex Number Represented in Exponential Form

^{âˆ’iÏ†}, where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

Here's an example:

Calculate the reciprocal of the number 2 Ã— e^{i(Ï€/2)}

^{âˆ’i (Ï€/2)}=

^{âˆ’i (Ï€/2)}

Additive Inversion of a Complex Number

Here's an example:

Compute the additive inverse for the number z = 2 + 3i

âˆ’z = (âˆ’1 Ã— 2) + (âˆ’1 Ã— 3)**i**= âˆ’2âˆ’3

**i**

Extracting the Nth Root of a Complex Number Represented in Algebraic (Rectangular) Form

Extract the 3rd root of the number 3 + 2i

In order to represent a complex number written in algebraic (rectangular) form in polar form, it is necessary to find the modulus and argument of the given complex number.

|z| âˆ’ modulus of a complex number z

Ï† âˆ’ argument of a complex number z

The modulus of a complex number z is calculated by the formula:

^{2}+ b

^{2}

^{2}+ 2

^{2}

The complex number argument z is calculated by the formula:

Arg(z) = arctg(b/a) + Ï€, a < 0

Arg(z) = Ï€/2, b > 0 and a = 0

Arg(z) = âˆ’Ï€/2, b < 0 and a = 0

Arg(3 + 2i) = arctg(2/3) = 33.6900675259798Â° degrees

z = 3 + 2i =

^{n}âˆšz =

^{n}âˆš|z| Ã— (cos (Ï† + 2Ï€k)/n + i sin (Ï† + 2Ï€k)/n)), where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

n â€“ degree of root (number of roots)

k â€“ takes values 0, 1, 2, â€¦ nâˆ’1

z

_{1}=

^{3}

z

_{2}=

^{3}

z

_{3}=

^{3}

z

_{1}= 1.50404648119798 + 0.298628314325244i

z

_{2}= âˆ’1.01064294709398 + 1.15322830402742i

z

_{3}= âˆ’0.493403534104007 âˆ’ 1.45185661835266i

z

_{1}= 1.53340623701639 Ã— (cos(11.2300225086599Â°) + i sin(11.2300225086599Â°))

z

_{2}= 1.53340623701639 Ã— (cos(131.23002250866Â°) + i sin(131.23002250866Â°))

z

_{3}= 1.53340623701639 Ã— (cos(251.23002250866Â°) + i sin(251.23002250866Â°))

Extracting the Nth Root of a Complex Number Represented in Polar Form

Calculate the root of the 3rd degree from the number 3(cos((Ï€/5)) + sin((Ï€/5))i)

^{n}âˆšz =

^{n}âˆš|z| Ã— (cos (Ï† + 2Ï€k)/n + i sin (Ï† + 2Ï€k)/n)), where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

n â€“ degree of root (number of roots)

k â€“ takes values 0, 1, 2, â€¦ nâˆ’1

z

_{1}=

^{3}

z

_{2}=

^{3}

z

_{3}=

^{3}

z

_{1}= 1.41073295685556 + 0.299860546743801i

z

_{2}= âˆ’0.965053329500603 + 1.07180030522094i

z

_{3}= âˆ’0.445679627354959 âˆ’ 1.37166085196474i

z

_{1}= 1.44224957030741 Ã— (cos(12Â°) + i sin(12Â°))

z

_{2}= 1.44224957030741 Ã— (cos(132Â°) + i sin(132Â°))

z

_{3}= 1.44224957030741 Ã— (cos(252Â°) + i sin(252Â°))

Extracting the Nth Root of a Complex Number Represented in Exponential Form

Extract the 3rd root of the number 2 Ã— e^{45Â°i}

In order to represent a complex number in exponential form in polar form, use the formula below:

^{Ï†i}= |z| Ã— (cos Ï† + i sin Ï†), where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

^{45Â°i}= 2 Ã— (cos(45Â°) + sin(45Â°)i)

^{n}âˆšz =

^{n}âˆš|z| Ã— (cos (Ï† + 2Ï€k)/n + i sin (Ï† + 2Ï€k)/n)), where

|z| âˆ’ modulus of complex number z

Ï† âˆ’ argument of complex number z

n â€“ degree of root (number of roots)

k â€“ takes values 0, 1, 2, â€¦ nâˆ’1

z

_{1}=

^{3}

z

_{2}=

^{3}

z

_{3}=

^{3}

z

_{1}= 1.2169902811787 + 0.326091563038355i

z

_{2}= âˆ’0.890898718140328 + 0.89089871814034i

z

_{3}= âˆ’0.326091563038355 âˆ’ 1.2169902811787i

z

_{1}= 1.25992104989487 Ã— (cos(15Â°) + i sin(15Â°))

z

_{2}= 1.25992104989487 Ã— (cos(135Â°) + i sin(135Â°))

z

_{3}= 1.25992104989487 Ã— (cos(255Â°) + i sin(255Â°))