Khan 2023b - Revision history

HassanRaza: I fixed some grammatical errors. Some areas, I felt, were not quite articulate, so I just changed a little bit of the wording. No changes to the overall content material of the paper, or the equations were made.

2024-01-06T08:58:21Z

I fixed some grammatical errors. Some areas, I felt, were not quite articulate, so I just changed a little bit of the wording. No changes to the overall content material of the paper, or the equations were made.

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HassanRaza at 00:11, 25 December 2023

2023-12-25T00:11:31Z

Tomamil: Tomamil moved page Review 449448612518 to Khan 2023b

2023-12-19T15:37:28Z

Tomamil moved page Review 449448612518 to Khan 2023b

HassanRaza at 18:28, 28 November 2023

2023-11-28T18:28:10Z

HassanRaza at 18:26, 28 November 2023

2023-11-28T18:26:57Z

HassanRaza at 06:39, 28 November 2023

2023-11-28T06:39:47Z

HassanRaza at 13:22, 24 November 2023

2023-11-24T13:22:11Z

HassanRaza at 13:20, 24 November 2023

2023-11-24T13:20:53Z

HassanRaza at 06:32, 21 November 2023

2023-11-21T06:32:47Z

HassanRaza at 17:53, 18 November 2023

2023-11-18T17:53:53Z

@@ Line 312: / Line 312: @@
 ==Acknowledgements==
-I want to express my sincere appreciation for the unwavering support and encouragement I received from my friends, family, and teachers throughout the entire research process.
+I want to express my sincere appreciation for the the support and encouragement I received from my friends, family, and teachers throughout the entire research process.
 I extend my gratitude to the dedicated reviewers and editors at the Journal of Dawning Research for their insightful feedback, which played a crucial role in refining and improving my paper.
-Lastly, I owe a special debt of gratitude to Dr Javed Hussain from IBA Sukkur University for his expert guidance. As a high school student new to research papers, his assistance was invaluable. He not only helped me with the paper but also instilled in me the importance of intellectual curiosity and hard work.
+Lastly, I would like to express gratitude to Dr Javed Hussain from IBA Sukkur University for his expert guidance. As a high school student new to research papers, his assistance was invaluable. He not only helped me with the basics of research but also instilled in me the importance of intellectual curiosity and hard work.
 ===BIBLIOGRAPHY===

← Older revision		Revision as of 18:28, 28 November 2023
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−	==Efficient Algorithm for Exponentiation of ~~Complex~~ Number~~, i~~==	+	==Efficient Algorithm for Exponentiation of Imaginary Number==

	'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]		'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]

← Older revision		Revision as of 18:26, 28 November 2023
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−	==Efficient Algorithm for Complex Number, i==	+	==Efficient Algorithm for Exponentiation of Complex Number, i==

	'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]		'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]

@@ Line 333: / Line 333: @@
 Nahin, P. (1998). An Imaginary Tale: The Story of <math>\sqrt{-1}</math> New Jersey: Princeton University Press.
-Blank, A. (1999). An Imaginary Tale Book Review, in Notices of the AMS Volume 46, Number 10, November 1999, pp. 1233-1236.
+Mazur, Barry & Pesic, Peter & Sabbagh, Karl. (2004). "Imagining Numbers" (Particularly the Square Root of Minus Fifteen). The American Mathematical Monthly. 111. 75. 10.2307/4145034.
 Needham, T. (1997). Visual Complex Analysis. New York: Oxford University Press.

← Older revision		Revision as of 13:22, 24 November 2023
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−	==Efficient Algorithm for Complex ~~Numbers~~==	+	==Efficient Algorithm for Complex Number, i==

	'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]		'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]

← Older revision	Revision as of 15:37, 19 December 2023
(No difference)

← Older revision		Revision as of 13:20, 24 November 2023
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−	==~~Quickest way to find 'i' without calculator~~==	+	==Efficient Algorithm for Complex Numbers==

	'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]		'''Hassan Raza Khan'''<span id="fnc-1"></span>[[#fn-1\|<sup>1</sup>]]

@@ Line 271: / Line 271: @@
 | style="border-right: 2px solid;" | Even
 | colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | 1
-| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | i
+| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | <math>i</math>
 |- style="border-bottom: 2px solid;"
 | style="border-right: 2px solid;" | Odd
 | colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | -1
-| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | -i
+| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | <math>-i</math>
 |}

@@ Line 258: / Line 258: @@
 In Table 1, we present a matrix illustrating the relationship between the final and initial numbers corresponding to different values of the complex unit <math display="inline">i</math>. The table categorizes the initial and final numbers as either even or odd and displays the resulting values of <math display="inline">i</math> for each combination. This matrix provides a concise reference for understanding the periodicity of <math display="inline">i</math> and its cyclic behavior based on the evenness or oddness of the exponents. Initial Number <math display="inline">N_0</math> is the raw form of the last two digits of the exponent of <math display="inline">i</math> , while Final Number <math display="inline">N_f</math> is the value of the last two digits of the exponent of <math display="inline">i</math> after being processed by the Algorithm (<math display="inline">N_f = \frac{N_0}{2}</math> if <math display="inline">N_0</math> is even, and <math display="inline">N_f = \frac{N_0 - 1}{2}</math> if <math display="inline">N_0</math> is Odd)
-{| class="wikitable" style="text-align:center;"
 |-
-! rowspan="2" colspan="2" style="color:#000000;" |
+| rowspan='2' colspan='2' style="border-right: 2px solid;" |
-! colspan="2" | Initial Number
+| colspan='2' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" |Initial Number <math>N_0</math>
 |-
-| Even
+| colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | Even
-| Odd
+| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | Odd
-|-
+|- style="border-top: 2px solid;"
-| rowspan="2" | Final Number
+| rowspan='2' colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" |Final Number <math>N_f</math>
-| Even
+| style="border-right: 2px solid;" | Even
-| 1
+| colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | 1
-| i
+| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | i
-|-
+|- style="border-bottom: 2px solid;"
-| Odd
+| style="border-right: 2px solid;" | Odd
-| -1
+| colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | -1
-| -i
+| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | -i
-|}
 |}