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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.

Complex numbers representation

z = a + bi
z1 = + i

z2 = + i

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

In order to add two complex numbers represented in algebraic (rectangular) form, you need to add their real and imaginary parts:
(a + bi) + (c + di) = (a + c) + (b + d)i

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 − 6i

(3 + 4i) + (8 − 6i) = (3 + 8) + (4 − 6i) = 11 − 2i


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

To add a complex number a + bi and a real number c, you need to add a real number to the real part of the complex number:
(a + bi) + c = (a + c) + bi

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 −6 + 3i and the real number -23

(−6 + 3i) + (−23) = (−6 + (−23)) + 3i = −29 + 3i


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

In order to subtract two complex numbers represented in algebraic (rectangular) form, you need to subtract their real and imaginary parts:
(a + bi) − (c + di) = (a − c) + (b − d)i

Let's give examples:

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

(3 + 9i) − (5 + 6i) = (3 − 5) + (9 − 6)i = −2 + 3i


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

(6 + 23i) − (57 + 68i) = (6 − 57) + (23 − 68)i = −51 − 45i


Subtracting a Complex Number From a Real Number

To subtract the complex number c + di from a real number a, use the formula below:
a − (c + di) = (a − c) − di

Let's give examples:

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

6 − (1 + 7i) = (6 − 1) + 7i = 5 − 7i


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

−15 − (1 + (−7)i) = (−15 − 1) − (−7)i = −16 + 7i


Subtracting a Real Number From a Complex Number

To subtract the real number c from the complex number a + bi, you need to subtract the real number from the real part of the complex number:
(a + bi) − c = (a − c) + bi

Let's give examples:

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

(5 + 12i) − 8 = (5 − 8) + 12i = −3 + 12i


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

(−1 + (−5)i) − (−3) = (−1 − (−3)) + (−5)i = 2 − 5i


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

In order to multiply two complex numbers represented in algebraic (rectangular) form, you must use the formula below:
(a + bi) × (c + di) = ac + adi + bci + bdi2 = (ac + bdi2) + (bc + ad)i = (ac − 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) × (3 + 7i) = ((2 × 3) − (5 × 7)) + ((5 × 3) + (2 × 7))i = (6 − 35) + (15 + 14)i = −29 + 29i
Solution 2
(2 + 5i) × (3 + 7i) = (2 × 3) + (2 × 7i) + (5i × 3) + (5i × 7i) = 6 + (14i) + (15i) + (35i2) = 6 + (29i) + (35 × (−1)) = −29 + 29i


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

Solution 1
(0.4 + (−2)i) × (3.023 + 0.25i) = ((0.4 × 3.023) − (−2 × 0.25)) + (((−2) × 3.023) + (0.4 × 0.25))i = (1.2092 − (−0.5)) + (−6.046 + 0.1)i = 1.7092−5.946i
Solution 2
(0.4 + (−2)i) × (3.023 + 0.25i) = (0.4 × 3.023) + (0.4 × 0.25i) + ((−2)i × 3.023) + ((−2)i × 0.25i) = 1.2092 + (0.1i) + (−6.046i) + (−0.5i2) = 1.2092 + (−5.946i) + ((−0.5 × (−1))) = 1.7092 − 5.946i


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

In order to multiply a real number a by a complex number c + di, it is necessary to multiply the real and imaginary parts of the number c + di by this number:
a × (c + di) = ac + adi

Let's give examples:

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

1 × (3 + 4i) = (1 × 3) + (1 × 4)i = 3 + 4i


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

−74 × (−5 + 4i) = (−74 × (−5)) + (−74 × 4)i = 370 − 296i


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

In order to divide two complex numbers presented in algebraic (rectangular) form, you must use the formula below:
a + bi =
c + di
(a + bi) × (c − di) =
(c + di) × (c − di)
ac + bd +
c2 + d2
bc − ad
c2 + d2



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) × (5 − 8i) =
(5 + 8i) × (5 − 8i)
(4 × 5) + (3 × 8) +
(52 + 82)
(3 × 5) − (4 × 8)
(52 + 82)



i =
(20 + 24) +
(25 + 64)
15 − 32
25 + 64



i =
44 +
89
−17
89



i
= 0.49438202247191−0.191011235955056i


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

(6 + (−2)i) =
(−4 + 7i)
(6 + (−2)i) × (−4 − 7i) =
(−4 + 7i) × (−4 − 7i)
(6 × (−4)) + (−2 × 7) +
(−42 + 72)
(−2 × (−4)) − (6 × 7)
(−42 + 72)



i =
(−24 + (−14)) +
(16 + 49)
8 − 42
16 + 49



i =
−38 +
65
−34
65



i
= −0.584615384615385−0.523076923076923i


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

In order to divide the complex number a + bi by the real number c, it is necessary to divide the real part of the complex number by the real number and divide the imaginary part of the complex number by the real number:
a + bi =
c
a +
c
bi
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

To divide a real number a by a complex number c + di, use the formula below:
a =
c + di
a × (c − di) =
(c + di) × (c − di)
ac −
c2 + d2
ad
c2 + d2



i

Here's an example:

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

5 =
(2 + 9i)
5 × (2 − 9i) =
(2 + 9i) × (2 − 9i)
5 × −
(22 + 92)
5 ×
(22 + 92)



i =
10 −
(4 + 81)
45
4 + 81



i =
10 −
85
45
85



i
= 0.117647058823529−0.529411764705882i


Addition of Complex Numbers Represented in Polar Form

To add two complex numbers in polar form, use the formula below:
z1 + z2 = (|z1| × (cos α + i sin α)) + (|z2| × (cos β + i sin β)) = ((|z1| × cos α) + (|z2| × cos β)) + i((|z1| × sin α) + (|z2| × sin β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

Let's add two complex numbers √13 (cos 48° + i sin 48°) and √25 (cos 69° + i sin 69°)

√13 (cos 48° + i sin 48°) + √25 (cos 69° + i sin 69°) = ((√13 × cos(48°)) + (√25 × cos(69°))) + i((√13 × sin(48°)) + (√25 × sin(69°))) = (2.41258471120918 + 1.7918397477265) + (2.67944677335447 + 4.667902132486)i = 4.20442445893568 + 7.34734890584047i


Subtraction of Complex Numbers Represented in Polar Form

To subtract two complex numbers in polar form, use the formula below:
z1 − z2 = (|z1| × (cos α + i sin α)) − (|z2| × (cos β + i sin β)) = ((|z1| × cos α) − (|z2| × cos β)) + i((|z1| × sin α) − (|z2| × sin β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

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

1/2 (cos π/2 + i sin π/2) − 1/3 (cos π/3 + i sin π/3) = ((1/2 × cos((π/2))) − (1/3 × cos((π/3)))) + i((1/2 × sin((π/2))) − (1/3 × sin((π/3)))) = (0 − 0.166666666666666) + (0.5 − 0.288675134594813)i = −0.166666666666666 + 0.211324865405187i


Multiplication of Complex Numbers Represented in Polar Form

To multiply two complex numbers in polar form, use the formula below:
z1 × z2 = (|z1| × |z2|) × (cos(α + β) + i × sin(α + β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

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

2 (cos π/2 + i sin π/2) × 2 (cos π/3 + i sin π/3) = (2 × 2) × (cos(π/2 + (π/3)) + i × sin(π/2 + (π/3))) =

16
× (cos(5π/6) + i × sin(5π/6)) =
16
× (cos(150°) + i × sin(150°))


Division of Complex Numbers Represented in Polar Form

To divide two complex numbers represented in polar form, use the formula below:
z1 ÷ z2 = (|z1| ÷ |z2|) × (cos(α − β) + i × sin(α − β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

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

3 (cos 45° + i sin 45°) ÷ 2 (cos 37° + i sin 37°) = (3 ÷ 2) × (cos(45° − 37°) + i × sin(45° − 37°)) =

2.25
× (cos(8°) + i × sin(8°))=
2.25
× (cos(2π/45) + i × sin(2π/45))


Addition of Complex Numbers Represented in Exponential Form

To add two complex numbers in exponential form, use the formula below:
|z1| eiα + |z2| eiβ = (|z1| × (cos α + i sin α)) + (|z2| × (cos β + i sin β)) = ((|z1| × cos α) + (|z2| × cos β)) + i((|z1| × sin α) + (|z2| × sin β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

Let's add two complex numbers 3 × e(π/2)i and 2 × e(3π/2)i

3 × e(π/2)i + 2 × e(3π/2)i = ((3 × cos((π/2))) + (2 × cos((3π/2)))) + i((3 × sin((π/2))) + (2 × sin((3π/2)))) = (0 + 0) + (3 + (−2))i = 0 + 1i


Subtraction of Complex Numbers Represented in Exponential Form

To subtract two complex numbers in exponential form, use the formula below:
|z1| eiα − |z2| eiβ = (|z1| × (cos α + i sin α)) − (|z2| × (cos β + i sin β)) = ((|z1| × cos α) − (|z2| × cos β)) + i((|z1| × sin α) − (|z2| × sin β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

Subtract from the number 3 × e(π/2)i the number 2 × e(3π/2)i

3 × e(π/2)i - 2 × e(3π/2)i = ((3 × cos((π/2))) − (2 × cos((3π/2)))) + i((3 × sin((π/2))) − (2 × sin((3π/2)))) = (0 − 0) + (3 − (−2))i = 0 + 5i


Multiplication of Complex Numbers Represented in Exponential Form

To multiply two complex numbers in exponential form, use the formula below:
|z1| eiα × |z2| eiβ = (|z1| × |z2|) × (cos(α + β) + i × sin(α + β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

Multiply two complex numbers 3 × e(π/2)i and 2 × e(3π/2)i

(3 × e(π/2)i) × (2 × e(3π/2)i) = (3 × 2) × (cos(π/2 + (3π/2)) + i × sin(π/2 + (3π/2))) =

36
× (cos(360°) + i × sin(360°)) = 6 + 0i


Division of Complex Numbers Represented in Exponential Form

In order to divide two complex numbers presented in exponential form, use the formula below:
|z1| eiα ÷ |z2| eiβ = (|z1| ÷ |z2|) × (cos(α − β) + i × sin(α − β)), where

|z1| − modulus of the complex number z1
α − argument of the complex number z1
|z2| − modulus of the complex number z2
β − argument of the complex number z2

Here's an example:

Divide two complex numbers 3 × e(π/2)i and 2 × e(3π/2)i

(3 × e(π/2)i) ÷ (2 × e(3π/2)i) = (3 ÷ 2) × (cos(π/2 − (3π/2)) + i × sin(π/2 − (3π/2))) =

2.25
× (cos(−180°) + i × sin(−180°)) =
2.25
× (cos(−π) + i × sin(−π)) = −1.5 + 0i


Modulus of a Complex Number

The modulus of a complex number is equal to the distance from a point on the complex plane to the origin and is denoted by |z|.
The modulus of a complex number z is calculated by the formula:
|z| =
a2 + b2

Here's an example:

Calculate the modulus of a complex number 1 + 3i

|1 + 3i| =
12 + 32
=
1 + 9
=
10
= 3.16227766016838

Argument (Phase) Of a Complex Number

The argument of a nonzero complex number is the angle φ between the radius vector of the corresponding point and the positive real semiaxis. The argument of the number z is denoted by Arg(z).
The argument of the complex number z is calculated by the formula:
Arg(z) = arctg(b/a), a > 0

Arg(z) = arctg(b/a) + π, a < 0

Arg(z) = π/2, b > 0 and a = 0

Arg(z) = −π/2, b < 0 and a = 0
The principal value of the argument must be such that −π < arctg(b/a) ⩽ π. The calculator calculates exactly the main value of the argument.

Here's an example:

Calculate the argument of a complex number −4 + 7i

Arg(−4 + 7i) = arctg(7/(−4)) + π = 2.08994244104142 radians
Arg(−4 + 7i) = 119.744881296942° degrees

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

In order to convert a complex number represented in algebraic (rectangular) form to polar form, it is necessary to calculate the modulus and argument of the given complex number.
z = a + bi = |z| × (cos φ + i sin φ), where

|z| − modulus of a complex number z
φ − argument of a complex number z
The modulus of a complex number is equal to the distance from a point on the complex plane to the origin and is denoted by |z|.
The modulus of a complex number z is calculated by the formula:
|z| =
a2 + b2

Here's an example:

Converting the number 2 + 3i to the polar form

|2 + 3i| =
22 + 32
=
4 + 9
=
13
= 3.60555127546399

The argument of a nonzero complex number is the angle φ between the radius vector of the corresponding point and the positive real semiaxis. The argument of the number z is denoted by Arg(z).
The argument of the complex number z is calculated by the formula:
Arg(z) = arctg(b/a), a > 0

Arg(z) = arctg(b/a) + π, a < 0

Arg(z) = π/2, b > 0 and a = 0

Arg(z) = −π/2, b < 0 and a = 0
The principal value of the argument must be such that −π < arctg(b/a) ⩽ π. The calculator calculates exactly the main value of the argument.

Arg(2 + 3i) = arctg(3/2) = 0.982793723247329 radians
Arg(2 + 3i) = arctg(3/2) = 56.3099324740202° degrees


Now we can write the complex number z in polar form:

z = 2 + 3i =
13
× (cos(arctg(3/2)) + sin(arctg(3/2))i) = 3.60555127546399 × (cos(56.3099324740202°) + sin(56.3099324740202°)i)

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

In order to convert a complex number represented in algebraic (rectangular) form to indicative form, it is necessary to calculate the modulus and argument of the given complex number.
z = a + bi = |z| × (cos φ + i sin φ), where

|z| − modulus of a complex number z
φ − argument of a complex number z
The modulus of a complex number is equal to the distance from a point on the complex plane to the origin and is denoted by |z|.
The modulus of a complex number z is calculated by the formula:
|z| =
a2 + b2

Here's an example:

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

|2 + 2i| =
22 + 22
=
4 + 4
=
8
= 2.82842712474619

The argument of a nonzero complex number is the angle φ between the radius vector of the corresponding point and the positive real semiaxis. The argument of the number z is denoted by Arg(z).
The argument of the complex number z is calculated by the formula:
Arg(z) = arctg(b/a), a > 0

Arg(z) = arctg(b/a) + π, a < 0

Arg(z) = π/2, b > 0 and a = 0

Arg(z) = −π/2, b < 0 and a = 0
The principal value of the argument must be such that −π < arctg(b/a) ⩽ π. The calculator calculates exactly the main value of the argument.

Arg(2 + 2i) = arctg(2/2) = 0.785398163397448 radians
Arg(2 + 2i) = arctg(2/2) = 45° degrees


Now we can write the complex number z in exponential form:

z = 2 + 2i =
8
× e0.785398163397448i =
8
× e45°i =
8
× e(π/4)i

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

To convert a complex number in polar form to an algebraic (rectangular) form z = a + bi, it is necessary to represent the real part of a as cos φ times the modulus |z|, and the imaginary part of b as sin φ times the modulus |z| complex number z.
|z| (cos φ + i sin φ) = (|z| × (cos φ)) + (|z| × (sin φ)i) = a + bi, where

a = |z| × cos φ
b = |z| × sin φ

Here's an example:

Convert 2 × (cos(60°) + sin(60°)i) to algebraic (rectangular) form


2 × (cos(60°) + sin(60°)i) = 2 × (0.5 + 0.86602540378444i) = (2 × 0.5) + (2 × 0.86602540378444i) = 1 + 1.73205080756888i

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

To convert a complex number in polar form to exponential form, use the formula below:
|z| × (cos φ + i sin φ) = |z| eφi, where

|z| − modulus of complex number z
φ − argument of complex number z

Here's an example:

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

2 × (cos(60°) + sin(60°)i) = 2 × e60°i = 2 × e(π/3)i

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

To convert a complex number in exponential form to polar form, use the formula below:
|z| eφi = |z| × (cos φ + i sin φ), where

|z| − modulus of complex number z
φ − argument of complex number z

Here's an example:

Convert 2 × e(π/3)i to polar form

2 × e(π/3)i = 2 × (cos((π/3)) + sin((π/3))i)

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

In order to convert a complex number in exponential form to algebraic (rectangular) form, the number must first be converted to polar form.
|z| eφi = |z| × (cos φ + i sin φ), where
|z| − modulus of complex number z
φ − argument of complex number z
To convert a complex number from a polar form to an algebraic (rectangular) form z = a + bi, it is necessary to represent the real part of a as cos φ times the modulus |z|, and the imaginary part of b as sin φ times the modulus |z| complex number z.
|z| (cos φ + i sin φ) = (|z| × (cos φ)) + (|z| × (sin φ)i) = a + bi, where
a = |z| × cos φ
b = |z| × sin φ

Here's an example:

Convert the number 2 × e(π/3)i to the algebraic (rectangular) form

2 × e(π/3)i = 2 × (cos((π/3)) + sin((π/3))i) = 2 × (0.5 + 0.86602540378444i) = (2 × 0.5) + (2 × 0.86602540378444i) = 1 + 1.73205080756888i

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

Two complex conjugate numbers have the same real parts and opposite imaginary parts.
The conjugate of z is denoted as z.
For z = a + bi, the conjugate is z = a − bi.

Here's an example:

Calculate the conjugate number for the number 2 + 3i

2 + 3i = 2 − 3i

The Conjugate of a Complex Number Expressed in Polar Form

Two complex conjugate numbers have the same real parts and opposite imaginary parts.
The conjugate of z is denoted as z.
For z = |z| (cos φ + i sin φ) the conjugate number is z = |z| (cos φ − i sin φ).

Here's an example:

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

2(cos((π/3)) + sin((π/3))i) = 2(cos((π/3)) − sin((π/3))i)

The Conjugate of a Complex Number Expressed in Exponential Form

Two complex conjugate numbers have the same real parts and opposite imaginary parts.
The conjugate of z is denoted as z.
For z = |z| eiφ the conjugate is z = |z| e−iφ.

Here's an example:

Calculate the conjugate number for the number 2 × e(π/3)i

2 × e(π/3)i = 2 × e − (π/3)i

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

For every non-zero complex number, there is a reciprocal. To find the reciprocal of the number a + bi, you need to divide one by this number. Since the unit acts as a dividend, the formula for division is simplified and takes the form:
1 =
a + bi
1 × (a − bi) =
(a + bi) × (a − bi)
a −
a2 + b2
b
a2 + b2



i

Here's an example:

Calculate the reciprocal of the number 2 + 3i

1 =
(2 + 3i)
2 −
(22 + 32)
3
(22 + 32)



i =
2 −
(4 + 9)
3
4 + 9



i =
2 −
13
3
13



i
= 0.153846153846154−0.230769230769231i

The Reciprocal of a Complex Number Expressed in Polar Form

To calculate the reciprocal of a number in polar form, use the formula below:
1/z = 1/|z| × (cos φ − i sin φ), where

|z| − modulus of complex number z
φ − argument of complex number z

Here's an example:

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

1/z = 1/2 × (cos (π/2) − i sin π/2) =
0.25
× (cos(π/2) − i × sin(π/2))

Reciprocal of a Complex Number Represented in Exponential Form

To calculate the reciprocal of a number in exponential form, use the formula below:
1/z = (1/|z|)e−iφ, where

|z| − modulus of complex number z
φ − argument of complex number z

Here's an example:

Calculate the reciprocal of the number 2 × ei(π/2)

1/z = 1/2 × e−i (π/2) =
0.25
× e−i (π/2)

Additive Inversion of a Complex Number

The additive inverse of a complex number is one that, when added to the original number, produces zero. The additive inverse of a complex number is a number in which the real and imaginary parts are multiplied by −1. To multiply the number −1 by the complex number a + bi. it is necessary to multiply the real and complex parts of the number a + bi by this number:
−1 × (a + bi) = (−1 × a) + (−1 × bi)

Here's an example:

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

−z = (−1 × 2) + (−1 × 3)i = −2−3i

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

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

To extract the nth root of a non-zero complex number. first, the given number must be converted to polar form.

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 = a + bi = |z| × (cos φ + i sin φ), where
|z| − modulus of a complex number z
φ − argument of a complex number z
The modulus of a complex number is equal to the distance from a point on the complex plane to the origin and is denoted by |z|.
The modulus of a complex number z is calculated by the formula:
|z| =
a2 + b2
|3 + 2i| =
32 + 22
=
9 + 4
=
13
= 3.60555127546399

The argument of a nonzero complex number is the angle φ between the radius vector of the corresponding point and the positive real semiaxis. The argument of the number z is denoted by Arg(z).
The complex number argument z is calculated by the formula:
Arg(z) = arctg(b/a), a > 0

Arg(z) = arctg(b/a) + π, a < 0

Arg(z) = π/2, b > 0 and a = 0

Arg(z) = −π/2, b < 0 and a = 0
The principal value of the argument must be such that −π < arctg(b/a) ⩽ π. The calculator calculates exactly the main value of the argument.

Arg(3 + 2i) = arctg(2/3) = 0.588002603547568 radians
Arg(3 + 2i) = arctg(2/3) = 33.6900675259798° degrees


Now we can write the complex number z in polar form:

z = 3 + 2i =
13
× (cos(arctg(2/3)) + sin(arctg(2/3))i) = 3.60555127546399 × (cos(33.6900675259798°) + sin(33.6900675259798°)i)

Let's apply De Moivre's formula to extract the nth root:
n√z = n√|z| × (cos (φ + 2πk)/n + i sin (φ + 2πk)/n)), where
|z| − modulus of complex number z
φ − argument of complex number z
n – degree of root (number of roots)
k – takes values 0, 1, 2, … n−1


z1 = 3
3.60555127546399
× (cos((33.6900675259798° + (2 × 180 × 0))/3) + i sin ((33.6900675259798° + (2 × 180 × 0))/3)) = 1.53340623701639 × (cos((33.6900675259798° + 0)/3) + i sin ((33.6900675259798° + 0)/3)) = (1.53340623701639 × (cos 11.2300225086599°)) + (1.53340623701639 × (sin 11.2300225086599°))i = 1.50404648119798 + 0.298628314325244i

z2 = 3
3.60555127546399
× (cos((33.6900675259798° + (2 × 180 × 1))/3) + i sin ((33.6900675259798° + (2 × 180 × 1))/3)) = 1.53340623701639 × (cos((33.6900675259798° + 360)/3) + i sin ((33.6900675259798° + 360)/3)) = (1.53340623701639 × (cos 131.23002250866°)) + (1.53340623701639 × (sin 131.23002250866°))i = −1.01064294709398 + 1.15322830402742i

z3 = 3
3.60555127546399
× (cos((33.6900675259798° + (2 × 180 × 2))/3) + i sin ((33.6900675259798° + (2 × 180 × 2))/3)) = 1.53340623701639 × (cos((33.6900675259798° + 720)/3) + i sin ((33.6900675259798° + 720)/3)) = (1.53340623701639 × (cos 251.23002250866°)) + (1.53340623701639 × (sin 251.23002250866°))i = −0.493403534104007 − 1.45185661835266i


z1 = 1.50404648119798 + 0.298628314325244i
z2 = −1.01064294709398 + 1.15322830402742i
z3 = −0.493403534104007 − 1.45185661835266i


z1 = 1.53340623701639 × (cos(11.2300225086599°) + i sin(11.2300225086599°))
z2 = 1.53340623701639 × (cos(131.23002250866°) + i sin(131.23002250866°))
z3 = 1.53340623701639 × (cos(251.23002250866°) + i sin(251.23002250866°))

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

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

Let's apply De Moivre's formula to calculate the root of the nth degree:
n√z = n√|z| × (cos (φ + 2πk)/n + i sin (φ + 2πk)/n)), where
|z| − modulus of complex number z
φ − argument of complex number z
n – degree of root (number of roots)
k – takes values 0, 1, 2, … n−1


z1 = 3
3
× (cos((36° + (2 × 180 × 0))/3) + i sin ((36° + (2 × 180 × 0))/3)) = 1.44224957030741 × (cos((36° + 0)/3) + i sin ((36° + 0)/3)) = (1.44224957030741 × (cos 12°)) + (1.44224957030741 × (sin 12°))i = 1.41073295685556 + 0.299860546743801i

z2 = 3
3
× (cos((36° + (2 × 180 × 1))/3) + i sin ((36° + (2 × 180 × 1))/3)) = 1.44224957030741 × (cos((36° + 360)/3) + i sin ((36° + 360)/3)) = (1.44224957030741 × (cos 132°)) + (1.44224957030741 × (sin 132°))i = −0.965053329500603 + 1.07180030522094i

z3 = 3
3
× (cos((36° + (2 × 180 × 2))/3) + i sin ((36° + (2 × 180 × 2))/3)) = 1.44224957030741 × (cos((36° + 720)/3) + i sin ((36° + 720)/3)) = (1.44224957030741 × (cos 252°)) + (1.44224957030741 × (sin 252°))i = −0.445679627354959 − 1.37166085196474i


z1 = 1.41073295685556 + 0.299860546743801i
z2 = −0.965053329500603 + 1.07180030522094i
z3 = −0.445679627354959 − 1.37166085196474i


z1 = 1.44224957030741 × (cos(12°) + i sin(12°))
z2 = 1.44224957030741 × (cos(132°) + i sin(132°))
z3 = 1.44224957030741 × (cos(252°) + i sin(252°))

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

Extract the 3rd root of the number 2 × e45°i

To extract the nth root of a non-zero complex number. you must first represent this number in polar form,

In order to represent a complex number in exponential form in polar form, use the formula below:
|z| eφi = |z| × (cos φ + i sin φ), where
|z| − modulus of complex number z
φ − argument of complex number z
2 × e45°i = 2 × (cos(45°) + sin(45°)i)

Let's apply De Moivre's formula to extract the nth root:
n√z = n√|z| × (cos (φ + 2πk)/n + i sin (φ + 2πk)/n)), where
|z| − modulus of complex number z
φ − argument of complex number z
n – degree of root (number of roots)
k – takes values 0, 1, 2, … n−1


z1 = 3
2
× (cos((45° + (2 × 180 × 0))/3) + i sin ((45° + (2 × 180 × 0))/3)) = 1.25992104989487 × (cos((45° + 0)/3) + i sin ((45° + 0)/3)) = (1.25992104989487 × (cos 15°)) + (1.25992104989487 × (sin 15°))i = 1.2169902811787 + 0.326091563038355i

z2 = 3
2
× (cos((45° + (2 × 180 × 1))/3) + i sin ((45° + (2 × 180 × 1))/3)) = 1.25992104989487 × (cos((45° + 360)/3) + i sin ((45° + 360)/3)) = (1.25992104989487 × (cos 135°)) + (1.25992104989487 × (sin 135°))i = −0.890898718140328 + 0.89089871814034i

z3 = 3
2
× (cos((45° + (2 × 180 × 2))/3) + i sin ((45° + (2 × 180 × 2))/3)) = 1.25992104989487 × (cos((45° + 720)/3) + i sin ((45° + 720)/3)) = (1.25992104989487 × (cos 255°)) + (1.25992104989487 × (sin 255°))i = −0.326091563038355 − 1.2169902811787i


z1 = 1.2169902811787 + 0.326091563038355i
z2 = −0.890898718140328 + 0.89089871814034i
z3 = −0.326091563038355 − 1.2169902811787i


z1 = 1.25992104989487 × (cos(15°) + i sin(15°))
z2 = 1.25992104989487 × (cos(135°) + i sin(135°))
z3 = 1.25992104989487 × (cos(255°) + i sin(255°))



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