m (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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==Quickest way to find 'i' without calculator==
+
==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>]]
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<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) Hassan Raza Khan is a high-school student at The City School DHA Campus, Lahore.</span>
 
<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) Hassan Raza Khan is a high-school student at The City School DHA Campus, Lahore.</span>
  
 +
== Introduction ==
  
Al-Khwarizmi (780-850), in his work "Algebra," provided solutions to various types of quadratic equations, with geometric-based proofs \cite{waerden1985history}. Under the caliph al-Ma’mun, al-Khwarizmi became a member of the "House of Wisdom," an academy of scientists in Baghdad \cite{waerden1985history}. This algebraic knowledge reached Italy through translations by Gerard of Cremona (1114-1187) and the work of Leonardo da Pisa (Fibonacci) (1170-1250).
+
Al-Khwarizmi (780-850), in his work ''Algebra,'' provided solutions to various types of quadratic equations, with geometric-based proofs. Under the caliph al-Ma’mun, al-Khwarizmi became a member of the ''House of Wisdom,'' an academy of scientists in Baghdad. This algebraic knowledge reached Italy through translations by Gerard of Cremona (1114-1187) and the work of Leonardo da Pisa (Fibonacci) (1170-1250).
  
Scipione del Ferro (d. 1526) solved the general cubic equation \(x^3 + px + q = 0\) \cite{blank1999imaginary}. His formula, passed to Antonio Maria Fiore, initiated a mathematical contest against Tartaglia, who rediscovered the formula and won \cite{blank1999imaginary}. Gerolamo Cardano, learning of this, signed an oath of secrecy and later published the formula in his "Ars Magna" (1545) \cite{blank1999imaginary}.
+
Scipione del Ferro (d. 1526) solved the general cubic equation <math>x^3 + px + q = 0</math>. His formula, passed to Antonio Maria Fiore, initiated a mathematical contest against Tartaglia, who rediscovered the formula and won. Gerolamo Cardano, learning of this, signed an oath of secrecy and later published the formula in his ''Ars Magna'' (1545).
  
\begin{equation}
+
<math>x = \sqrt[3]{\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}</math>
x = \sqrt[3]{\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} \label{eq:delFerro-Tartaglia}
+
\end{equation}
+
  
Rafael Bombelli (1526-1572) tackled cubic equations in "l’Algebra" (1572) \cite{blank1999imaginary}. He fully discussed the casus irreducibilis, demonstrating the expression \(x^3 = 15x + 4\) and Cardan's formula \cite{blank1999imaginary}.
+
Rafael Bombelli (1526-1572) tackled cubic equations in ''l’Algebra'' (1572). He fully discussed the casus irreducibilis, demonstrating the expression <math>x^3 = 15x + 4</math> and Cardan's formula.
  
\begin{equation}
+
<math>x = \sqrt[3]{\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}</math>
x = \sqrt[3]{\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} \label{eq:Bombelli-CasusIrreducibilis}
+
\end{equation}
+
  
These historical works, spanning from Al-Khwarizmi to Gauss, reflect a collective focus on unraveling complex number \(i = \sqrt{-1}\). Despite this rich history, a contemporary challenge persists—finding a swift and efficient method for computing the values of basic complex number exponents. Mathematics can alway be simpler.
+
René Descartes (1596-1650), a philosopher, applied algebra to geometry in his work "La Géométrie," laying the foundations for Cartesian geometry (Descartes, 1637). Pressed by friends, he wrote the treatise "Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences" (1637), where he associated imaginary numbers with geometric impossibility. Descartes coined the term "imaginary," emphasizing the lack of a corresponding quantity for imagined roots (Descartes, 1637).
  
 +
<math>z^2 = az - b^2</math>
  
==1 Introduction==
+
John Wallis (1616-1703) noted in his "Algebra" that negative numbers, long viewed skeptically, have a physical explanation based on a line with a zero mark. He also made progress in giving a geometric interpretation to <math>\sqrt{-1}</math> (Wallis, 1693).
 +
 
 +
Abraham de Moivre (1667-1754), seeking religious refuge in London, befriended Newton. In 1698, he mentioned Newton's knowledge of an equivalent expression to what is now known as de Moivre's theorem (Moivre, 1707).
 +
 
 +
<math>(\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)</math>
 +
 
 +
Leonhard Euler (1707-1783) introduced the notation <math>i = \sqrt{-1}</math>. He visualized complex numbers as points with rectangular coordinates and explored the roots of <math>z^n = 1</math> as vertices of a regular polygon. Euler defined the complex exponential and proved the identity <math>e^{i\theta} = \cos(\theta) + i\sin(\theta)</math> (Euler, 1748).
 +
 
 +
These historical works, spanning from Al-Khwarizmi to Gauss, reflect a collective focus on unraveling the complex number <math>i = \sqrt{-1}</math>. Despite this rich history, a contemporary challenge persists—finding a quick and efficient method for computing the values of basic complex number exponents. Mathematics can always be simpler.
 +
 
 +
==The Problem==
  
 
Say you want to find the value of  i) <math display="inline">i^{8237918273}</math>; ii) <math display="inline">i^{823718932}</math>; iii) <math display="inline">i^{1291393767}</math>; iv) <math display="inline">i^{812731274}</math>;
 
Say you want to find the value of  i) <math display="inline">i^{8237918273}</math>; ii) <math display="inline">i^{823718932}</math>; iii) <math display="inline">i^{1291393767}</math>; iv) <math display="inline">i^{812731274}</math>;
  
and there's like 20 seconds left in the exam, and this paper does not allow calculators. You start solving it something like this:
+
and there are like 20 seconds left in the exam, and this paper does not allow calculators. You start solving it something like this:
  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
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|}
 
|}
  
and before you know it, the 20 seconds are over. You couldn't even answer any of the question.
+
and before you know it, the 20 seconds are over. You couldn't even answer any of the questions.
  
The student next to you, the class topper, recognizes that we can find the value of such a large number of <math display="inline">i</math> using polar coordinates, by employing Euler's formula. He starts writing:
+
The student next to you, the class topper, recognizes that we can find the value of such a large number of <math display="inline">i</math> using polar coordinates, by using Euler's formula (Euler, 1748). He starts writing:
  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
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|}
 
|}
  
He now starts using De Moivre's Theorem <math display="inline">n = 8237918273</math>.
+
He now starts using De Moivre's Theorem (Moivre, 1707) <math display="inline">n = 8237918273</math>  
  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
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|}
 
|}
  
By now, the proctor comes and snatches his paper. He could not even solve one of the 4 parts. None of the students was able to solve any part within less than 20 seconds&#8211; except a boy named Draco, who solved all 4 of these parts within less than 20 seconds. Fun fact: he didn't even do any calculations on the paper. He only used the last two digits and performed The Algorithm in his mind.
+
By now, not only does he realize he needs a calculator to use sin and cos, but the proctor also marches down the aisle and snatches his paper. He could not even solve one of the 4 parts. None of the students was able to solve any part within less than 20 seconds. It was a shame considering how all 4 parts of this question could well be solved within less than 20 seconds using The Algorithm, and no working required.
  
==2 The Algorithm==
+
==The Algorithm==
  
 
The algorithm involves three steps:
 
The algorithm involves three steps:
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</ol>
 
</ol>
  
==3 Proving that only the last two digits are necessary==
+
==Proving that only the last two digits are necessary==
  
To demonstrate that only the last two digits are necessary for determining the value of <math display="inline">i^n</math>, where <math display="inline">n</math> is a large exponent, we can employ the periodicity of powers of <math display="inline">i</math>.
+
To demonstrate that only the last two digits are necessary for determining the value of <math display="inline">i^n</math>, where <math display="inline">n</math> is a large exponent, we can take advantage the periodicity of powers of <math display="inline">i</math>.
  
 
Consider the powers of <math display="inline">i</math> when raised to successive positive integer exponents:
 
Consider the powers of <math display="inline">i</math> when raised to successive positive integer exponents:
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|}
 
|}
  
We observe that the powers of <math display="inline">i</math> repeat in cycles of four: <math display="inline">i, -1, -i, 1</math>. This periodicity implies that the value of <math display="inline">i^n</math> depends only on the remainder when <math display="inline">n</math> is divided by 4.
+
We can see that the powers of <math display="inline">i</math> repeat in cycles of four: <math display="inline">i, -1, -i, 1</math>. This periodicity implies that the value of <math display="inline">i^n</math> depends only on the remainder when <math display="inline">n</math> is divided by 4.
  
 
Now, let's consider the exponent <math display="inline">n = 8237918273</math>. We want to find <math display="inline">i^n</math>:
 
Now, let's consider the exponent <math display="inline">n = 8237918273</math>. We want to find <math display="inline">i^n</math>:
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|}
 
|}
  
Since <math display="inline">8237918200</math> has two zeros and is divisible by 100, and since 100 is divisible by 4, we can conclude that <math display="inline">8237918200</math> is divisible by 4.
+
Since <math display="inline">8237918200</math> has two zeros and is divisible by 100, and since 100 is divisible by 4, we can deduce that <math display="inline">8237918200</math> is divisible by 4.
  
 
Therefore, <math display="inline">i^{4k+r} = i^{8237918200+73} = i^{8237918200} \cdot i^{73}.</math>
 
Therefore, <math display="inline">i^{4k+r} = i^{8237918200+73} = i^{8237918200} \cdot i^{73}.</math>
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|}
 
|}
  
So, essentially, <math display="inline">i^{8237918273}</math> is congruent to <math display="inline">i^{73}</math>. This result demonstrates a general pattern: for any large exponent <math display="inline">n</math>, <math display="inline">i^n</math> is congruent to <math display="inline">i^{r}</math>, where <math display="inline">r</math> is the remainder when <math display="inline">n</math> is divided by 4. Thus, only the last two digits of the exponent (<math display="inline">73</math> in this case) are crucial for determining the value of <math display="inline">i^n</math>.
+
So, basically, <math display="inline">i^{8237918273}</math> is congruent to <math display="inline">i^{73}</math>. This demonstrates a general pattern: for any large exponent <math display="inline">n</math>, <math display="inline">i^n</math> is congruent to <math display="inline">i^{r}</math>, where <math display="inline">r</math> is the remainder when <math display="inline">n</math> is divided by 4. Thus, only the last two digits of the exponent (<math display="inline">73</math> in this case) are necessary for determining the value of <math display="inline">i^n</math>.
  
This observation allows for a more efficient approach when faced with large exponentiation of <math display="inline">i</math> without the need to calculate the entire exponent. The periodic nature of <math display="inline">i</math> simplifies the computation, making it feasible to focus solely on the last two digits of the exponent.
+
This observation allows for a more efficient approach when dealing with large exponentiation of <math display="inline">i</math> without the need to calculate the entire exponent. The periodic nature of <math display="inline">i</math> simplifies the computation, making it feasible to focus solely on the last two digits of the exponent.
  
==4 Demonstrating the algorithm==
+
==Demonstrating the algorithm==
  
What Draco essentially did was use the above three-step algorithm like so:
+
The three-step algorithm can be used like so:
  
 
<ol>
 
<ol>
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<ol>
 
<ol>
  
</li>
 
 
<li>'''Real/Imaginary number determination'''
 
<li>'''Real/Imaginary number determination'''
  
The last two digits are 73, an odd number. So it must be an imaginary number (either <math display="inline">i</math> or <math display="inline">-i</math>). Since the number is odd, we subtract it by 1 to make it parallel to its even counterpart, resulting in 72.        </li>
+
The last two digit is 73, an odd number. So it must be an imaginary number (either <math display="inline">i</math> or <math display="inline">-i</math>). Since the number is odd, we subtract it by 1 to make it parallel to its even counterpart, resulting in 72.        </li>
 
<li>'''Division by 2'''
 
<li>'''Division by 2'''
  
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<ol>
 
<ol>
  
</li>
 
 
<li>'''Real/Imaginary number determination'''
 
<li>'''Real/Imaginary number determination'''
  
The last two digits are 32, an even number. So it must be a real number (either 1 or -1). Since the number is even, we do not subtract it, so it remains 32.        </li>
+
The last two digit is 32, an even number. So it must be a real number (either 1 or -1). Since the number is even, we do not subtract it, so it remains 32.        </li>
 
<li>'''Division by 2'''
 
<li>'''Division by 2'''
  
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<ol>
 
<ol>
  
</li>
 
 
<li>'''Real/Imaginary number determination'''
 
<li>'''Real/Imaginary number determination'''
  
The last two digits are 67, an odd number. So it must be an imaginary number (either <math display="inline">i</math> or <math display="inline">-i</math>). Since the number is odd, we subtract it by 1 to make it parallel to its even counterpart, resulting in 66.        </li>
+
The last two digit is 67, an odd number. So it must be an imaginary number (either <math display="inline">i</math> or <math display="inline">-i</math>). Since the number is odd, we subtract it by 1 to make it parallel to its even counterpart, resulting in 66.        </li>
 
<li>'''Division by 2'''
 
<li>'''Division by 2'''
  
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<ol>
 
<ol>
  
</li>
 
 
<li>'''Real/Imaginary number determination'''
 
<li>'''Real/Imaginary number determination'''
  
The last two digits are 74, an even number. So it must be a real number (either 1 or -1). Since the number is even, we do not subtract it, so it remains 74.        </li>
+
The last two digit is 74, an even number. So it must be a real number (either 1 or -1). Since the number is even, we do not subtract it, so it remains 74.        </li>
 
<li>'''Division by 2'''
 
<li>'''Division by 2'''
  
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</ol>
 
</ol>
  
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. Memorizing this table can make you the Draco in the same circumstance.
+
In Table 1, we present a matrix which shows 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 the resulting values of <math display="inline">i</math> for each combination. This matrix is basically a summary of this entire concept of 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="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
 
{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
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|-
 
|-
 
| rowspan='2' colspan='2' style="border-right: 2px solid;" |  
 
| rowspan='2' colspan='2' style="border-right: 2px solid;" |  
| colspan='2' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | Initial Number
+
| colspan='2' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" |Initial Number <math>N_0</math>
 
|-
 
|-
 
| colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | Even
 
| colspan='1' style="text-align: center;border-right: 2px solid;border-left: 2px solid;" | Even
 
| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | Odd
 
| colspan='1' style="text-align: center;border-right: 2px solid;border-right: 2px solid;" | Odd
 
|- style="border-top: 2px solid;"
 
|- style="border-top: 2px solid;"
| rowspan='2' colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 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>
 
| style="border-right: 2px solid;" | Even  
 
| 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-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-bottom: 2px solid;"
 
| style="border-right: 2px solid;" | Odd   
 
| 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-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>
  
 
|}
 
|}
  
==5 Purpose of Odd/Even Analysis==
+
==Purpose of Odd/Even Analysis==
  
The divisional remainder and its congruent complex numeber counterpart can be illustrated as below:
+
The divisional remainder and its congruent complex number counterpart can be expressed as below:
  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
 
{| class="formulaSCP" style="width: 100%; text-align: left;"  
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|}
 
|}
  
The odd/even analysis serves as a strategic approach to efficiently determine the divisional remainder of the numbers in the context of complex exponentiation. By focusing solely on the last two digits of the exponent, we can employ a systematic method to evaluate whether the number is divisible by 4.
+
The odd/even analysis serves is an approach that efficiently determine the divisional remainder of the numbers in the context of complex exponentiation. By focusing solely on the last two digits of the exponent, we can evaluate whether the number is divisible by 4.
  
 
For instance, consider an exponent with the last two digits being 32. If we divide 32 by 2, we obtain 16, an even number. This indicates that the original exponent is divisible by 2 twice, meaning it is divisible by 4. Therefore, the number is congruent to 0 modulo 4.
 
For instance, consider an exponent with the last two digits being 32. If we divide 32 by 2, we obtain 16, an even number. This indicates that the original exponent is divisible by 2 twice, meaning it is divisible by 4. Therefore, the number is congruent to 0 modulo 4.
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The same approach is applied to odd numbers, where they are subtracted by 1 in order to align them with their even counterparts. This adjustment ensures that odd numbers maintain their distinctive properties in the modulo 4 arithmetic.
 
The same approach is applied to odd numbers, where they are subtracted by 1 in order to align them with their even counterparts. This adjustment ensures that odd numbers maintain their distinctive properties in the modulo 4 arithmetic.
  
It's worth noting that one could also add 1 instead of subtracting 1 from the odd number. However, by adopting the subtraction approach, the signs become universally positive for even final numbers and negative for odd final numbers. This consistency simplifies the association of odd numbers with negative results and even numbers with positive results, aiding in a more straightforward interpretation of the outcomes.
+
One could also add 1 instead of subtracting 1 from the odd number. However, by adopting the subtraction approach, the signs no longer become universally positive for even final numbers and negative for odd final numbers. That is why subtracting helps with the association of odd numbers with negative results and even numbers with positive results, making it a more straightforward interpretation of the outcomes.
 
+
In summary, the odd/even analysis not only streamlines the computation of complex exponentiation but also provides a clear and consistent method for determining the divisional remainder based on the last two digits of the exponent.
+
  
 
==Acknowledgements==
 
==Acknowledgements==
  
I would like to express my gratitude for the support and encouragement I received from my friends, family, and teachers throughout the course of this research paper.
+
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.
  
Lastly, I owe a special debt of gratitude to Dr Javed Hussain and Dr Zarqa Bano from IBA Sukkur University for their expert guidance. As a high school student new to research papers, their assistance was invaluable. They not only helped me with the paper but also instilled in me the importance of intellectual curiosity and hard work.
+
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 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===
 
===BIBLIOGRAPHY===
 +
Khan, H. R. (2023, October). Efficient Algorithm for Identifying Repeating Patterns Modulo 4 and its Application in Complex Numbers. [https://doi.org/10.31219/osf.io/8zc7e Source]
 +
 +
Merino, O. (2006, January). A Short History of Complex Numbers. University of Rhode Island.
 +
 +
Euler, L. (1748). Introductio in analysin infinitorum. Apud Marcum-Michaelem Bousquet & Socios.
 +
 +
De Moivre, A. (1730). Miscellanea Analytica de Seriebus et Quadraturis [Analytical Miscellany on Series and Integration].
 +
 +
Zhou, X. (2017). Number Theory - Modular Arithmetic: Math for Gifted Students. CreateSpace Independent Publishing Platform.
 +
 +
Halmos, P. (1980). The Heart of Mathematics. American Mathematical Monthly, 87, 519–524.
 +
 +
Nahin, P. (1998). An Imaginary Tale: The Story of <math>\sqrt{-1}</math> New Jersey: Princeton University Press.
 +
 +
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.
 +
 +
Dunham, W. (1999). Euler, The Master of Us All, The Dolciani Mathematical Expositions, Number 22, Mathematical Association of America.
 +
 +
Argand, R. (Reprinted 1971). Essai sur une Maniere de Representer Les Quantites Imaginaires dans Les Constructions Geometriques. A. Blanchard.
 +
 +
Descartes, R. (1952). La Geometrie, translated from French and Latin by D. E. Smith and M. Latham. Open Court Publishing Company, La Salle, Illinois.
 +
 +
Crowe, M. (1967). A History of Vector Analysis. U. of Notre Dame Press, Notre Dame.
 +
 +
van der Waerden, B. L. (1985). A History of Algebra. Springer Verlag, NY.
  
Khan, H. R. (2023, October 27). Efficient Algorithm for Identifying Repeating Patterns Modulo 4 and its Application in Complex Numbers. https://doi.org/10.31219/osf.io/8zc7e
+
Wallis, J. (1685). Treatise on Algebra.
  
Euler, L. (1748). ''Introductio in analysin infinitorum''. Apud Marcum-Michaelem Bousquet & Socios.
+
Andreescu, T., & Andrica, D. (2007). Complex Numbers: An Introduction. Birkhäuser.
  
De Moivre, A. (1730). ''Miscellanea Analytica de Seriebus et Quadraturis'' [Analytical Miscellany on Series and Integration].
+
Andreescu, T., & Andrica, D. (2009). Complex Numbers from A to ...Z. Birkhäuser.
  
Zhou, X. (2017). ''Number Theory - Modular Arithmetic: Math for Gifted Students''. CreateSpace Independent Publishing Platform.
+
Richeson, D. S. (2008). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
  
Halmos, P. (1980). The heart of mathematics. ''American Mathematical Monthly, 87'', 519–524.
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Latest revision as of 10:58, 6 January 2024


Efficient Algorithm for Exponentiation of Imaginary Number

Hassan Raza Khan1

(1) Hassan Raza Khan is a high-school student at The City School DHA Campus, Lahore.

Introduction

Al-Khwarizmi (780-850), in his work Algebra, provided solutions to various types of quadratic equations, with geometric-based proofs. Under the caliph al-Ma’mun, al-Khwarizmi became a member of the House of Wisdom, an academy of scientists in Baghdad. This algebraic knowledge reached Italy through translations by Gerard of Cremona (1114-1187) and the work of Leonardo da Pisa (Fibonacci) (1170-1250).

Scipione del Ferro (d. 1526) solved the general cubic equation . His formula, passed to Antonio Maria Fiore, initiated a mathematical contest against Tartaglia, who rediscovered the formula and won. Gerolamo Cardano, learning of this, signed an oath of secrecy and later published the formula in his Ars Magna (1545).

Rafael Bombelli (1526-1572) tackled cubic equations in l’Algebra (1572). He fully discussed the casus irreducibilis, demonstrating the expression and Cardan's formula.

René Descartes (1596-1650), a philosopher, applied algebra to geometry in his work "La Géométrie," laying the foundations for Cartesian geometry (Descartes, 1637). Pressed by friends, he wrote the treatise "Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences" (1637), where he associated imaginary numbers with geometric impossibility. Descartes coined the term "imaginary," emphasizing the lack of a corresponding quantity for imagined roots (Descartes, 1637).

John Wallis (1616-1703) noted in his "Algebra" that negative numbers, long viewed skeptically, have a physical explanation based on a line with a zero mark. He also made progress in giving a geometric interpretation to (Wallis, 1693).

Abraham de Moivre (1667-1754), seeking religious refuge in London, befriended Newton. In 1698, he mentioned Newton's knowledge of an equivalent expression to what is now known as de Moivre's theorem (Moivre, 1707).

Leonhard Euler (1707-1783) introduced the notation . He visualized complex numbers as points with rectangular coordinates and explored the roots of as vertices of a regular polygon. Euler defined the complex exponential and proved the identity (Euler, 1748).

These historical works, spanning from Al-Khwarizmi to Gauss, reflect a collective focus on unraveling the complex number . Despite this rich history, a contemporary challenge persists—finding a quick and efficient method for computing the values of basic complex number exponents. Mathematics can always be simpler.

The Problem

Say you want to find the value of i) ; ii) ; iii) ; iv) ;

and there are like 20 seconds left in the exam, and this paper does not allow calculators. You start solving it something like this:

and before you know it, the 20 seconds are over. You couldn't even answer any of the questions.

The student next to you, the class topper, recognizes that we can find the value of such a large number of using polar coordinates, by using Euler's formula (Euler, 1748). He starts writing:

He now starts using De Moivre's Theorem (Moivre, 1707)

By now, not only does he realize he needs a calculator to use sin and cos, but the proctor also marches down the aisle and snatches his paper. He could not even solve one of the 4 parts. None of the students was able to solve any part within less than 20 seconds. It was a shame considering how all 4 parts of this question could well be solved within less than 20 seconds using The Algorithm, and no working required.

The Algorithm

The algorithm involves three steps:

  1. Real/Imaginary number determination If the last two digits are even, the value remains unchanged (either 1 or -1). If odd, subtract 1 (resulting in either i or -i).
  2. Division by 2 Divide the resultant number by 2.
  3. Sign determination If the remaining number after division by 2 is odd, the sign is minus; if it's even, the sign is plus.

Proving that only the last two digits are necessary

To demonstrate that only the last two digits are necessary for determining the value of , where is a large exponent, we can take advantage the periodicity of powers of .

Consider the powers of when raised to successive positive integer exponents:

We can see that the powers of repeat in cycles of four: . This periodicity implies that the value of depends only on the remainder when is divided by 4.

Now, let's consider the exponent . We want to find :

Consider . We can express this as , where is an integer and is the remainder when is divided by . In this case, .

Now, let's break down further:

Since has two zeros and is divisible by 100, and since 100 is divisible by 4, we can deduce that is divisible by 4.

Therefore,

Now, simplifies to since any power of with an exponent divisible by 4 is equal to 1. Therefore, we have:

So, basically, is congruent to . This demonstrates a general pattern: for any large exponent , is congruent to , where is the remainder when is divided by 4. Thus, only the last two digits of the exponent ( in this case) are necessary for determining the value of .

This observation allows for a more efficient approach when dealing with large exponentiation of without the need to calculate the entire exponent. The periodic nature of simplifies the computation, making it feasible to focus solely on the last two digits of the exponent.

Demonstrating the algorithm

The three-step algorithm can be used like so:

  1. ;
    1. Real/Imaginary number determination The last two digit is 73, an odd number. So it must be an imaginary number (either or ). Since the number is odd, we subtract it by 1 to make it parallel to its even counterpart, resulting in 72.
    2. Division by 2 Dividing 72 by 2 results in 36, an even number.
    3. Sign determination Since the resultant number after division by 2 is even, we can establish that the sign of this number would be positive. Since the number is both imaginary and positive, it must be .
  2. ;
    1. Real/Imaginary number determination The last two digit is 32, an even number. So it must be a real number (either 1 or -1). Since the number is even, we do not subtract it, so it remains 32.
    2. Division by 2 Dividing 32 by 2 results in 16, an even number.
    3. Sign determination Since the resultant number after division by 2 is even, we can establish that the sign of this number would be positive. Since the number is both real and positive, it must be 1.
  3. ;
    1. Real/Imaginary number determination The last two digit is 67, an odd number. So it must be an imaginary number (either or ). Since the number is odd, we subtract it by 1 to make it parallel to its even counterpart, resulting in 66.
    2. Division by 2 Dividing 66 by 2 results in 33, an odd number.
    3. Sign determination Since the resultant number after division by 2 is odd, we can establish that the sign of this number would be negative. Since the number is both imaginary and negative, it must be .
  4. ;
    1. Real/Imaginary number determination The last two digit is 74, an even number. So it must be a real number (either 1 or -1). Since the number is even, we do not subtract it, so it remains 74.
    2. Division by 2 Dividing 74 by 2 results in 37, an odd number.
    3. Sign determination Since the resultant number after division by 2 is odd, we can establish that the sign of this number would be negative. Since the number is both real and negative, it must be .

In Table 1, we present a matrix which shows the relationship between the final and initial numbers corresponding to different values of the complex unit . The table categorizes the initial and final numbers as either even or odd and the resulting values of for each combination. This matrix is basically a summary of this entire concept of periodicity of and its cyclic behavior based on the evenness or oddness of the exponents. Initial Number is the raw form of the last two digits of the exponent of , while Final Number is the value of the last two digits of the exponent of after being processed by the Algorithm ( if is even, and if is Odd)

Table. 1 Matrix of Initial and Final Numbers Corresponding to Values of
Initial Number
Even Odd
Final Number Even 1
Odd -1

Purpose of Odd/Even Analysis

The divisional remainder and its congruent complex number counterpart can be expressed as below:

The odd/even analysis serves is an approach that efficiently determine the divisional remainder of the numbers in the context of complex exponentiation. By focusing solely on the last two digits of the exponent, we can evaluate whether the number is divisible by 4.

For instance, consider an exponent with the last two digits being 32. If we divide 32 by 2, we obtain 16, an even number. This indicates that the original exponent is divisible by 2 twice, meaning it is divisible by 4. Therefore, the number is congruent to 0 modulo 4.

On the contrary, let's take another example with the last two digits being 34. If we divide 34 by 2, we get 17, an odd number. In this case, 34 can only be divided by 2 once while remaining a whole number. Consequently, it is congruent to 2 modulo 4.

The same approach is applied to odd numbers, where they are subtracted by 1 in order to align them with their even counterparts. This adjustment ensures that odd numbers maintain their distinctive properties in the modulo 4 arithmetic.

One could also add 1 instead of subtracting 1 from the odd number. However, by adopting the subtraction approach, the signs no longer become universally positive for even final numbers and negative for odd final numbers. That is why subtracting helps with the association of odd numbers with negative results and even numbers with positive results, making it a more straightforward interpretation of the outcomes.

Acknowledgements

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 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

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Halmos, P. (1980). The Heart of Mathematics. American Mathematical Monthly, 87, 519–524.

Nahin, P. (1998). An Imaginary Tale: The Story of New Jersey: Princeton University Press.

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.

Dunham, W. (1999). Euler, The Master of Us All, The Dolciani Mathematical Expositions, Number 22, Mathematical Association of America.

Argand, R. (Reprinted 1971). Essai sur une Maniere de Representer Les Quantites Imaginaires dans Les Constructions Geometriques. A. Blanchard.

Descartes, R. (1952). La Geometrie, translated from French and Latin by D. E. Smith and M. Latham. Open Court Publishing Company, La Salle, Illinois.

Crowe, M. (1967). A History of Vector Analysis. U. of Notre Dame Press, Notre Dame.

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Published on 19/12/23
Submitted on 15/11/23

Volume 5, 2023
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