From 6eb9269d5acf2995bfabd4352255dffa062c8aae Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 12 Nov 2024 20:07:07 +0530
Subject: [PATCH 01/18] Updating description.md

---
 exercises/complex-numbers/description.md | 92 +++++++++++++++++++-----
 1 file changed, 74 insertions(+), 18 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index c1aca4a582..cfaef68ec0 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -1,29 +1,85 @@
 # Description
 
-A complex number is a number in the form `a + b * i` where `a` and `b` are real and `i` satisfies `i^2 = -1`.
+A **complex number** is expressed in the form:
 
-`a` is called the real part and `b` is called the imaginary part of `z`.
-The conjugate of the number `a + b * i` is the number `a - b * i`.
-The absolute value of a complex number `z = a + b * i` is a real number `|z| = sqrt(a^2 + b^2)`. The square of the absolute value `|z|^2` is the result of multiplication of `z` by its complex conjugate.
+```
+z = a + b * i
+```
 
-The sum/difference of two complex numbers involves adding/subtracting their real and imaginary parts separately:
-`(a + i * b) + (c + i * d) = (a + c) + (b + d) * i`,
-`(a + i * b) - (c + i * d) = (a - c) + (b - d) * i`.
+where:
 
-Multiplication result is by definition
-`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`.
+- `a` is the **real part** (a real number),
 
-The reciprocal of a non-zero complex number is
-`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`.
+- `b` is the **imaginary part** (also a real number), and
 
-Dividing a complex number `a + i * b` by another `c + i * d` gives:
-`(a + i * b) / (c + i * d) = (a * c + b * d)/(c^2 + d^2) + (b * c - a * d)/(c^2 + d^2) * i`.
+- `i` is the imaginary unit satisfying `i^2 = -1`.
 
-Raising e to a complex exponent can be expressed as `e^(a + i * b) = e^a * e^(i * b)`, the last term of which is given by Euler's formula `e^(i * b) = cos(b) + i * sin(b)`.
+### Key Properties
 
-Implement the following operations:
+1. **Conjugate**: The conjugate of the complex number `z = a + b * i` is given by:
 
-- addition, subtraction, multiplication and division of two complex numbers,
-- conjugate, absolute value, exponent of a given complex number.
+```
+z̅ = a - b * i
+```
 
-Assume the programming language you are using does not have an implementation of complex numbers.
+2. **Absolute Value**: The absolute value (or modulus) of `z` is defined as:
+
+```
+|z| = sqrt(a^2 + b^2)
+```
+
+The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
+
+```
+|z|² = z * z̅ = a² + b²
+```
+
+### Operations on Complex Numbers
+
+1. **Addition**: The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
+
+```
+z₁ + z₂ = (a + c) + (b + d) * i
+```
+
+2. **Subtraction**: The difference of two complex numbers is obtained by subtracting their respective parts:
+
+```
+z₁ - z₂ = (a - c) + (b - d) * i
+```
+
+3. **Multiplication**: The product of two complex numbers is defined as:
+
+```
+z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
+```
+
+4. **Division**: The division of one complex number by another is given by:
+
+```
+z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
+```
+
+5. **Reciprocal**: The reciprocal of a non-zero complex number is given by:
+
+```
+1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
+```
+
+6. **Exponentiation**: Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
+
+```
+e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))
+```
+
+### Implementation Requirements
+
+Given that you should not use built-in support for complex numbers, implement the following operations:
+
+- **Addition** of two complex numbers.
+- **Subtraction** of two complex numbers.
+- **Multiplication** of two complex numbers.
+- **Division** of two complex numbers.
+- Calculation of the **conjugate** of a complex number.
+- Calculation of the **absolute value** of a complex number.
+- Calculation of the **exponent** of a given complex number.

From f0e6fbdcd8d9c12821fed2c18d7f605cf1e51dc3 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 12 Nov 2024 22:25:35 +0530
Subject: [PATCH 02/18] Updating description.md

---
 exercises/complex-numbers/description.md | 26 ++++++++++++------------
 1 file changed, 13 insertions(+), 13 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index cfaef68ec0..7a965c90ae 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -2,7 +2,7 @@
 
 A **complex number** is expressed in the form:
 
-```
+```math
 z = a + b * i
 ```
 
@@ -14,65 +14,65 @@ where:
 
 - `i` is the imaginary unit satisfying `i^2 = -1`.
 
-### Key Properties
+## Key Properties
 
 1. **Conjugate**: The conjugate of the complex number `z = a + b * i` is given by:
 
-```
+```math
 z̅ = a - b * i
 ```
 
 2. **Absolute Value**: The absolute value (or modulus) of `z` is defined as:
 
-```
+```math
 |z| = sqrt(a^2 + b^2)
 ```
 
 The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
 
-```
+```math
 |z|² = z * z̅ = a² + b²
 ```
 
-### Operations on Complex Numbers
+## Operations on Complex Numbers
 
 1. **Addition**: The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
 
-```
+```math
 z₁ + z₂ = (a + c) + (b + d) * i
 ```
 
 2. **Subtraction**: The difference of two complex numbers is obtained by subtracting their respective parts:
 
-```
+```math
 z₁ - z₂ = (a - c) + (b - d) * i
 ```
 
 3. **Multiplication**: The product of two complex numbers is defined as:
 
-```
+```math
 z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
 4. **Division**: The division of one complex number by another is given by:
 
-```
+```math
 z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
 ```
 
 5. **Reciprocal**: The reciprocal of a non-zero complex number is given by:
 
-```
+```math
 1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
 ```
 
 6. **Exponentiation**: Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
-```
+```math
 e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))
 ```
 
-### Implementation Requirements
+## Implementation Requirements
 
 Given that you should not use built-in support for complex numbers, implement the following operations:
 

From 1d6473be7662d94e83f07215cb74f3285d81c053 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 12 Nov 2024 22:32:10 +0530
Subject: [PATCH 03/18] Updating description.md

---
 exercises/complex-numbers/description.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index 7a965c90ae..c34409c877 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -22,7 +22,7 @@ where:
 z̅ = a - b * i
 ```
 
-2. **Absolute Value**: The absolute value (or modulus) of `z` is defined as:
+1. **Absolute Value**: The absolute value (or modulus) of `z` is defined as:
 
 ```math
 |z| = sqrt(a^2 + b^2)
@@ -42,31 +42,31 @@ The square of the absolute value, `|z|²`, can be computed as the product of `z`
 z₁ + z₂ = (a + c) + (b + d) * i
 ```
 
-2. **Subtraction**: The difference of two complex numbers is obtained by subtracting their respective parts:
+1. **Subtraction**: The difference of two complex numbers is obtained by subtracting their respective parts:
 
 ```math
 z₁ - z₂ = (a - c) + (b - d) * i
 ```
 
-3. **Multiplication**: The product of two complex numbers is defined as:
+1. **Multiplication**: The product of two complex numbers is defined as:
 
 ```math
 z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
-4. **Division**: The division of one complex number by another is given by:
+1. **Division**: The division of one complex number by another is given by:
 
 ```math
 z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
 ```
 
-5. **Reciprocal**: The reciprocal of a non-zero complex number is given by:
+1. **Reciprocal**: The reciprocal of a non-zero complex number is given by:
 
 ```math
 1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
 ```
 
-6. **Exponentiation**: Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
+1. **Exponentiation**: Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
 ```math
 e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))

From d8a0cda58d355690ae9bde5622037e378214887c Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 12 Nov 2024 22:51:53 +0530
Subject: [PATCH 04/18] Updating description.md

---
 exercises/complex-numbers/description.md | 32 ++++++++++++++++++------
 1 file changed, 24 insertions(+), 8 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index c34409c877..67700d016e 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -16,13 +16,17 @@ where:
 
 ## Key Properties
 
-1. **Conjugate**: The conjugate of the complex number `z = a + b * i` is given by:
+### Conjugate
+
+The conjugate of the complex number `z = a + b * i` is given by:
 
 ```math
 z̅ = a - b * i
 ```
 
-1. **Absolute Value**: The absolute value (or modulus) of `z` is defined as:
+### Absolute Value
+
+The absolute value (or modulus) of `z` is defined as:
 
 ```math
 |z| = sqrt(a^2 + b^2)
@@ -36,37 +40,49 @@ The square of the absolute value, `|z|²`, can be computed as the product of `z`
 
 ## Operations on Complex Numbers
 
-1. **Addition**: The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
+### Addition
+
+The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
 
 ```math
 z₁ + z₂ = (a + c) + (b + d) * i
 ```
 
-1. **Subtraction**: The difference of two complex numbers is obtained by subtracting their respective parts:
+### Subtraction
+
+The difference of two complex numbers is obtained by subtracting their respective parts:
 
 ```math
 z₁ - z₂ = (a - c) + (b - d) * i
 ```
 
-1. **Multiplication**: The product of two complex numbers is defined as:
+### Multiplication 
+
+The product of two complex numbers is defined as:
 
 ```math
 z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
-1. **Division**: The division of one complex number by another is given by:
+### Division
+
+The division of one complex number by another is given by:
 
 ```math
 z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
 ```
 
-1. **Reciprocal**: The reciprocal of a non-zero complex number is given by:
+### Reciprocal
+
+The reciprocal of a non-zero complex number is given by:
 
 ```math
 1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
 ```
 
-1. **Exponentiation**: Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
+### Exponentiation
+
+Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
 ```math
 e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))

From 7ba41f37ff7f1ae3fb31a8386b5aa2864533d3b2 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 12 Nov 2024 22:54:37 +0530
Subject: [PATCH 05/18] Updating description.md

---
 exercises/complex-numbers/description.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index 67700d016e..de661e98c7 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -56,7 +56,7 @@ The difference of two complex numbers is obtained by subtracting their respectiv
 z₁ - z₂ = (a - c) + (b - d) * i
 ```
 
-### Multiplication 
+### Multiplication
 
 The product of two complex numbers is defined as:
 

From 7d46dcc2266a7adf1653e81fac26e9111d549ebb Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 12 Nov 2024 23:04:04 +0530
Subject: [PATCH 06/18] Updating description.md

---
 exercises/complex-numbers/description.md | 8 +-------
 1 file changed, 1 insertion(+), 7 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index de661e98c7..4bb0a7a195 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -1,12 +1,6 @@
 # Description
 
-A **complex number** is expressed in the form:
-
-```math
-z = a + b * i
-```
-
-where:
+A **complex number** is expressed in the form `z = a + b * i`, where:
 
 - `a` is the **real part** (a real number),
 

From 14e079de85fbbdc55c67150a7987199918db5a17 Mon Sep 17 00:00:00 2001
From: jagdish-15 <jagadishdrp@gmail.com>
Date: Wed, 13 Nov 2024 00:01:51 +0530
Subject: [PATCH 07/18] Update exercises/complex-numbers/description.md

Co-authored-by: Anastasios Chatzialexiou <16361161+tasxatzial@users.noreply.github.com>
---
 exercises/complex-numbers/description.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index 4bb0a7a195..fab409f12a 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -6,7 +6,7 @@ A **complex number** is expressed in the form `z = a + b * i`, where:
 
 - `b` is the **imaginary part** (also a real number), and
 
-- `i` is the imaginary unit satisfying `i^2 = -1`.
+- `i` is the **imaginary unit** satisfying `i^2 = -1`.
 
 ## Key Properties
 

From 738a0a15ef634d334a7d6de883150f2750b6c6ea Mon Sep 17 00:00:00 2001
From: jagdish-15 <jagadishdrp@gmail.com>
Date: Wed, 13 Nov 2024 00:26:05 +0530
Subject: [PATCH 08/18] Update exercises/complex-numbers/description.md

Co-authored-by: Anastasios Chatzialexiou <16361161+tasxatzial@users.noreply.github.com>
---
 exercises/complex-numbers/description.md | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index fab409f12a..adcfe813fc 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -86,10 +86,10 @@ e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))
 
 Given that you should not use built-in support for complex numbers, implement the following operations:
 
-- **Addition** of two complex numbers.
-- **Subtraction** of two complex numbers.
-- **Multiplication** of two complex numbers.
-- **Division** of two complex numbers.
-- Calculation of the **conjugate** of a complex number.
-- Calculation of the **absolute value** of a complex number.
-- Calculation of the **exponent** of a given complex number.
+- **addition** of two complex numbers
+- **subtraction** of two complex numbers
+- **multiplication** of two complex numbers
+- **division** of two complex numbers
+- **conjugate** of a complex number
+- **absolute value** of a complex number
+- **exponent** of a complex number

From ffb6d04a3c41e61d0e34ffe5d5db44aec31fd656 Mon Sep 17 00:00:00 2001
From: jagdish-15 <jagadishdrp@gmail.com>
Date: Wed, 13 Nov 2024 00:38:09 +0530
Subject: [PATCH 09/18] Update exercises/complex-numbers/description.md

Co-authored-by: Anastasios Chatzialexiou <16361161+tasxatzial@users.noreply.github.com>
---
 exercises/complex-numbers/description.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index adcfe813fc..e2d847ab6e 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -92,4 +92,4 @@ Given that you should not use built-in support for complex numbers, implement th
 - **division** of two complex numbers
 - **conjugate** of a complex number
 - **absolute value** of a complex number
-- **exponent** of a complex number
+- **exponentiation** of the base _e_ (the natural logarithm) to a complex number

From 5d6fc061d42ef41e66dab4dbbb6acbbb6865181d Mon Sep 17 00:00:00 2001
From: jagdish-15 <jagadishdrp@gmail.com>
Date: Wed, 13 Nov 2024 08:33:48 +0530
Subject: [PATCH 10/18] Update exercises/complex-numbers/description.md

Co-authored-by: Anastasios Chatzialexiou <16361161+tasxatzial@users.noreply.github.com>
---
 exercises/complex-numbers/description.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index e2d847ab6e..51a8c52184 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -92,4 +92,4 @@ Given that you should not use built-in support for complex numbers, implement th
 - **division** of two complex numbers
 - **conjugate** of a complex number
 - **absolute value** of a complex number
-- **exponentiation** of the base _e_ (the natural logarithm) to a complex number
+- **exponentiation** of _e_ (the base of the natural logarithm) to a complex number

From 80a5fad854b8d3419f1638ec46ffc330ca59ecb1 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Sun, 17 Nov 2024 16:43:35 +0530
Subject: [PATCH 11/18] Chnaging description of Complex-Number after
 suggestions

---
 exercises/complex-numbers/description.md | 24 ++++++++----------------
 1 file changed, 8 insertions(+), 16 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index 51a8c52184..dfa7aeefa2 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -14,7 +14,7 @@ A **complex number** is expressed in the form `z = a + b * i`, where:
 
 The conjugate of the complex number `z = a + b * i` is given by:
 
-```math
+```plaintext
 z̅ = a - b * i
 ```
 
@@ -22,13 +22,13 @@ z̅ = a - b * i
 
 The absolute value (or modulus) of `z` is defined as:
 
-```math
+```plaintext
 |z| = sqrt(a^2 + b^2)
 ```
 
 The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
 
-```math
+```plaintext
 |z|² = z * z̅ = a² + b²
 ```
 
@@ -38,7 +38,7 @@ The square of the absolute value, `|z|²`, can be computed as the product of `z`
 
 The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
 
-```math
+```plaintext
 z₁ + z₂ = (a + c) + (b + d) * i
 ```
 
@@ -46,7 +46,7 @@ z₁ + z₂ = (a + c) + (b + d) * i
 
 The difference of two complex numbers is obtained by subtracting their respective parts:
 
-```math
+```plaintext
 z₁ - z₂ = (a - c) + (b - d) * i
 ```
 
@@ -54,7 +54,7 @@ z₁ - z₂ = (a - c) + (b - d) * i
 
 The product of two complex numbers is defined as:
 
-```math
+```plaintext
 z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
@@ -62,23 +62,15 @@ z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 
 The division of one complex number by another is given by:
 
-```math
+```plaintext
 z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
 ```
 
-### Reciprocal
-
-The reciprocal of a non-zero complex number is given by:
-
-```math
-1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
-```
-
 ### Exponentiation
 
 Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
-```math
+```plaintext
 e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))
 ```
 

From f0da7180c081b0f535699a85a7ac95d13d9a6d67 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Sun, 17 Nov 2024 17:45:52 +0530
Subject: [PATCH 12/18] Chnaging description of Complex-Number after
 suggestions

---
 exercises/complex-numbers/description.md | 12 +++++++++++-
 1 file changed, 11 insertions(+), 1 deletion(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index dfa7aeefa2..d8ce6c3729 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -26,6 +26,8 @@ The absolute value (or modulus) of `z` is defined as:
 |z| = sqrt(a^2 + b^2)
 ```
 
+### Sqaure
+
 The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
 
 ```plaintext
@@ -58,12 +60,20 @@ The product of two complex numbers is defined as:
 z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
+### Reciprocal
+
+The reciprocal of a non-zero complex number is given by:
+
+```math
+1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
+```
+
 ### Division
 
 The division of one complex number by another is given by:
 
 ```plaintext
-z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
+z₁ / z₂ = z₁ * (1 / z₂) = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
 ```
 
 ### Exponentiation

From 32d3e3d335bf06a8d0fbc2bd80e02715e7f6dd97 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Sun, 17 Nov 2024 17:47:21 +0530
Subject: [PATCH 13/18] Fixing consistancy issues

---
 exercises/complex-numbers/description.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index d8ce6c3729..18b0a93305 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -64,7 +64,7 @@ z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 
 The reciprocal of a non-zero complex number is given by:
 
-```math
+```plaintext
 1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
 ```
 

From e1e9bfffbd0aada2e84beaaf20d12cbf0c386013 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Sun, 17 Nov 2024 23:25:10 +0530
Subject: [PATCH 14/18] Changing description for complex-numbers

---
 exercises/complex-numbers/description.md | 20 +++++++++-----------
 1 file changed, 9 insertions(+), 11 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index 18b0a93305..e957e5c726 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -14,7 +14,7 @@ A **complex number** is expressed in the form `z = a + b * i`, where:
 
 The conjugate of the complex number `z = a + b * i` is given by:
 
-```plaintext
+```text
 z̅ = a - b * i
 ```
 
@@ -22,15 +22,13 @@ z̅ = a - b * i
 
 The absolute value (or modulus) of `z` is defined as:
 
-```plaintext
+```text
 |z| = sqrt(a^2 + b^2)
 ```
 
-### Sqaure
-
 The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
 
-```plaintext
+```text
 |z|² = z * z̅ = a² + b²
 ```
 
@@ -40,7 +38,7 @@ The square of the absolute value, `|z|²`, can be computed as the product of `z`
 
 The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
 
-```plaintext
+```text
 z₁ + z₂ = (a + c) + (b + d) * i
 ```
 
@@ -48,7 +46,7 @@ z₁ + z₂ = (a + c) + (b + d) * i
 
 The difference of two complex numbers is obtained by subtracting their respective parts:
 
-```plaintext
+```text
 z₁ - z₂ = (a - c) + (b - d) * i
 ```
 
@@ -56,7 +54,7 @@ z₁ - z₂ = (a - c) + (b - d) * i
 
 The product of two complex numbers is defined as:
 
-```plaintext
+```text
 z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
@@ -64,7 +62,7 @@ z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 
 The reciprocal of a non-zero complex number is given by:
 
-```plaintext
+```text
 1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
 ```
 
@@ -72,7 +70,7 @@ The reciprocal of a non-zero complex number is given by:
 
 The division of one complex number by another is given by:
 
-```plaintext
+```text
 z₁ / z₂ = z₁ * (1 / z₂) = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
 ```
 
@@ -80,7 +78,7 @@ z₁ / z₂ = z₁ * (1 / z₂) = (a + b * i) / (c + d * i) = (a * c + b * d) /
 
 Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
-```plaintext
+```text
 e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))
 ```
 

From d0ca8606b56b77e17f68a4a2a2668816a5ecb2e2 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Mon, 18 Nov 2024 19:34:29 +0530
Subject: [PATCH 15/18] Updating sescription of complex-numbers exercise for
 consistancy

---
 exercises/complex-numbers/description.md | 22 ++++++++++------------
 1 file changed, 10 insertions(+), 12 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index e957e5c726..b8e440803f 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -8,14 +8,14 @@ A **complex number** is expressed in the form `z = a + b * i`, where:
 
 - `i` is the **imaginary unit** satisfying `i^2 = -1`.
 
-## Key Properties
+## Operations on Complex Numbers
 
 ### Conjugate
 
 The conjugate of the complex number `z = a + b * i` is given by:
 
 ```text
-z̅ = a - b * i
+zc = a - b * i
 ```
 
 ### Absolute Value
@@ -26,20 +26,18 @@ The absolute value (or modulus) of `z` is defined as:
 |z| = sqrt(a^2 + b^2)
 ```
 
-The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
+The square of the absolute value is computed as the product of `z` and its conjugate `zc`:
 
 ```text
-|z|² = z * z̅ = a² + b²
+|z|^2 = z * zc = a^2 + b^2
 ```
 
-## Operations on Complex Numbers
-
 ### Addition
 
-The sum of two complex numbers `z₁ = a + b * i` and `z₂ = c + d * i` is computed by adding their real and imaginary parts separately:
+The sum of two complex numbers `z1 = a + b * i` and `z2 = c + d * i` is computed by adding their real and imaginary parts separately:
 
 ```text
-z₁ + z₂ = (a + c) + (b + d) * i
+z1 + z2 = (a + c) + (b + d) * i
 ```
 
 ### Subtraction
@@ -47,7 +45,7 @@ z₁ + z₂ = (a + c) + (b + d) * i
 The difference of two complex numbers is obtained by subtracting their respective parts:
 
 ```text
-z₁ - z₂ = (a - c) + (b - d) * i
+z1 - z2 = (a - c) + (b - d) * i
 ```
 
 ### Multiplication
@@ -55,7 +53,7 @@ z₁ - z₂ = (a - c) + (b - d) * i
 The product of two complex numbers is defined as:
 
 ```text
-z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
+z1 * z2 = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 ```
 
 ### Reciprocal
@@ -63,7 +61,7 @@ z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 The reciprocal of a non-zero complex number is given by:
 
 ```text
-1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
+1 / z = 1 / (a + b * i) = a / (a^2 + b^2) - b / (a^2 + b^2) * i
 ```
 
 ### Division
@@ -71,7 +69,7 @@ The reciprocal of a non-zero complex number is given by:
 The division of one complex number by another is given by:
 
 ```text
-z₁ / z₂ = z₁ * (1 / z₂) = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
+z1 / z2 = z1 * (1 / z2) = (a + b * i) / (c + d * i) = (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i
 ```
 
 ### Exponentiation

From 2c9e279a45c60828d087d4aae01c08ea6e4f51fa Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 19 Nov 2024 14:05:40 +0530
Subject: [PATCH 16/18] Updating description of complex-numbers

---
 exercises/complex-numbers/description.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index b8e440803f..2e54752d88 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -37,7 +37,7 @@ The square of the absolute value is computed as the product of `z` and its conju
 The sum of two complex numbers `z1 = a + b * i` and `z2 = c + d * i` is computed by adding their real and imaginary parts separately:
 
 ```text
-z1 + z2 = (a + c) + (b + d) * i
+z1 + z2 = (a + b * i) + (c + d * i) = (a + c) + (b + d) * i
 ```
 
 ### Subtraction
@@ -45,7 +45,7 @@ z1 + z2 = (a + c) + (b + d) * i
 The difference of two complex numbers is obtained by subtracting their respective parts:
 
 ```text
-z1 - z2 = (a - c) + (b - d) * i
+z1 - z2 = (a + b * i) - (c + d * i) = (a - c) + (b - d) * i
 ```
 
 ### Multiplication

From ddf83b922639b7a77b2e868dd17579bc1323e88b Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 19 Nov 2024 21:25:33 +0530
Subject: [PATCH 17/18] Updating description complex-numbers

---
 exercises/complex-numbers/description.md | 19 +++++++++++++------
 1 file changed, 13 insertions(+), 6 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index 2e54752d88..b1f3ff161f 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -37,7 +37,8 @@ The square of the absolute value is computed as the product of `z` and its conju
 The sum of two complex numbers `z1 = a + b * i` and `z2 = c + d * i` is computed by adding their real and imaginary parts separately:
 
 ```text
-z1 + z2 = (a + b * i) + (c + d * i) = (a + c) + (b + d) * i
+z1 + z2 = (a + b * i) + (c + d * i) 
+        = (a + c) + (b + d) * i
 ```
 
 ### Subtraction
@@ -45,7 +46,8 @@ z1 + z2 = (a + b * i) + (c + d * i) = (a + c) + (b + d) * i
 The difference of two complex numbers is obtained by subtracting their respective parts:
 
 ```text
-z1 - z2 = (a + b * i) - (c + d * i) = (a - c) + (b - d) * i
+z1 - z2 = (a + b * i) - (c + d * i) 
+        = (a - c) + (b - d) * i
 ```
 
 ### Multiplication
@@ -53,7 +55,8 @@ z1 - z2 = (a + b * i) - (c + d * i) = (a - c) + (b - d) * i
 The product of two complex numbers is defined as:
 
 ```text
-z1 * z2 = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
+z1 * z2 = (a + b * i) * (c + d * i) 
+        = (a * c - b * d) + (b * c + a * d) * i
 ```
 
 ### Reciprocal
@@ -61,7 +64,8 @@ z1 * z2 = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
 The reciprocal of a non-zero complex number is given by:
 
 ```text
-1 / z = 1 / (a + b * i) = a / (a^2 + b^2) - b / (a^2 + b^2) * i
+1 / z = 1 / (a + b * i) 
+      = a / (a^2 + b^2) - b / (a^2 + b^2) * i
 ```
 
 ### Division
@@ -69,7 +73,9 @@ The reciprocal of a non-zero complex number is given by:
 The division of one complex number by another is given by:
 
 ```text
-z1 / z2 = z1 * (1 / z2) = (a + b * i) / (c + d * i) = (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i
+z1 / z2 = z1 * (1 / z2) 
+        = (a + b * i) / (c + d * i) 
+        = (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i
 ```
 
 ### Exponentiation
@@ -77,7 +83,8 @@ z1 / z2 = z1 * (1 / z2) = (a + b * i) / (c + d * i) = (a * c + b * d) / (c^2 + d
 Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
 ```text
-e^(a + b * i) = e^a * e^(b * i) = e^a * (cos(b) + i * sin(b))
+e^(a + b * i) = e^a * e^(b * i) 
+              = e^a * (cos(b) + i * sin(b))
 ```
 
 ## Implementation Requirements

From 06a72422bc3b4776d0ddcb7958d69125c4672ea1 Mon Sep 17 00:00:00 2001
From: Jagdish Prajapati <jagadishdrp@gmail.com>
Date: Tue, 19 Nov 2024 23:47:08 +0530
Subject: [PATCH 18/18] Updating description for complex-number to fix
 formatting errors

---
 exercises/complex-numbers/description.md | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/exercises/complex-numbers/description.md b/exercises/complex-numbers/description.md
index b1f3ff161f..63e1ccdb36 100644
--- a/exercises/complex-numbers/description.md
+++ b/exercises/complex-numbers/description.md
@@ -37,7 +37,7 @@ The square of the absolute value is computed as the product of `z` and its conju
 The sum of two complex numbers `z1 = a + b * i` and `z2 = c + d * i` is computed by adding their real and imaginary parts separately:
 
 ```text
-z1 + z2 = (a + b * i) + (c + d * i) 
+z1 + z2 = (a + b * i) + (c + d * i)
         = (a + c) + (b + d) * i
 ```
 
@@ -46,7 +46,7 @@ z1 + z2 = (a + b * i) + (c + d * i)
 The difference of two complex numbers is obtained by subtracting their respective parts:
 
 ```text
-z1 - z2 = (a + b * i) - (c + d * i) 
+z1 - z2 = (a + b * i) - (c + d * i)
         = (a - c) + (b - d) * i
 ```
 
@@ -55,7 +55,7 @@ z1 - z2 = (a + b * i) - (c + d * i)
 The product of two complex numbers is defined as:
 
 ```text
-z1 * z2 = (a + b * i) * (c + d * i) 
+z1 * z2 = (a + b * i) * (c + d * i)
         = (a * c - b * d) + (b * c + a * d) * i
 ```
 
@@ -64,7 +64,7 @@ z1 * z2 = (a + b * i) * (c + d * i)
 The reciprocal of a non-zero complex number is given by:
 
 ```text
-1 / z = 1 / (a + b * i) 
+1 / z = 1 / (a + b * i)
       = a / (a^2 + b^2) - b / (a^2 + b^2) * i
 ```
 
@@ -73,8 +73,8 @@ The reciprocal of a non-zero complex number is given by:
 The division of one complex number by another is given by:
 
 ```text
-z1 / z2 = z1 * (1 / z2) 
-        = (a + b * i) / (c + d * i) 
+z1 / z2 = z1 * (1 / z2)
+        = (a + b * i) / (c + d * i)
         = (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i
 ```
 
@@ -83,7 +83,7 @@ z1 / z2 = z1 * (1 / z2)
 Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
 
 ```text
-e^(a + b * i) = e^a * e^(b * i) 
+e^(a + b * i) = e^a * e^(b * i)
               = e^a * (cos(b) + i * sin(b))
 ```