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!NB THIS ARTICLE IS NOT TO BE PUBLISHED AND IS NOT TO BE TAKEN SERIOUSLY. IT IS ONLY A FUN/SCHOOL PROJECT BY MYSELF AND A FRIEND.

Ådne's Laws

This article is about the mathematical, chemistry and physics related laws and their proofs, which the Norwegian mathematician Ådne Abs Åkerøy derived during his life.

Ådne Abs Åkerøy throughout his life has been responsable, yet unknown for quite a few mathematical laws and even some laws of physics and chemistry, ranging from his law on complex cube roots to (last law here). Despite his hard focus and work on these scientific subjects, he never did finish the full quadfecta, meaning he never (or at least known) did any research on the subject of biology. His influence on modern mathematics, chemistry and physics are quite small, since he never published any of his own work. It is only because of a recent excavation his papers and work, as well as his wider known history as a person has been found. These recent discoveries has given mathematics a proper founder for these important mathematical and scientific laws. Ådne Abs Åkerøy's laws were all conjured within the time span from his initial discovery, however after him doing so. Therefore, Ådne Abs Åkerøy is the one who deserves the honor for these findings.

The excavation that uncovered these papers was done in the 21st century. The excavation happened on the Norwegian island Dønna, with the purpose of searching through old houses in case there were any important and historical items about. The discoverers were Sivert Olai Sørøy Johansen and Frederik Egelund Edvardsen. The exact date of discovery was 6th of Febuary 2020. The finds icluded all of Ådne's laws and proofs to support them, as well as historical and personal items.

List of laws	Year of writing
Ådne's Law on Complex Cube Roots	1863
Ådne's Law for Solving Linear Differential Equations $n$'th Order	1864
Ådne's Law for Integrating Complex and Hypercomplex Functions	1866
Ådne's Law on Non-Integer Summation	1867
Ådne's Law for Synthesizing Methamphetamine	1869

Ådne Abs Åkerøy

A photograph of the norwegian mathematician Ådne Abs Åkerøy (1865). Found in his basement amongst all his other old possesions.

Ådne Abs Åkerøy (1832 - 1903) was a Norwegian mathematician who specialized in the fields of deep number theory and complex and hypercomplex algebra, meaning he used complex numbers ($\mathbb {C}$), quaternions (ℍ), octonions (𝕆), sedenions (𝕊) and trigintaduonions (𝕋) for algebraic mathematics. His studies on physics mainly revolved around cosmoligy and astronomy, seeing as he had built a home-made telescope. Only a handful of such papers containing his theories and proofs around mathematics and physics were recovered.

He was born on the 6th of September 1832 in the then small settlement of Sandnessjøen and later migrated to the neighbouring island Dønna, where he lived out his life. His wife, Sandra Häghaug (1833 - 1905), was of Swedish origins and had moved to the Alstahaug municipality in Nordland, Norway in the late 1850's. She and Ådne got married in 1863 (according to Alstahaug city hall). They had two children, both acquiring their father's surname (Åkerøy).

His influence on modern mathematics and physics are quite small, since he never published any of his work and theories. He also died of old age on the 19th of January 1903, meaning he lived a life of 71 years old; whilst noone ever noticed his superb work in the mathematical and scientific fields.

Quite recently however, in the 21st century, some of his old notes and papers were discovered by Sivert Olai Sørøy Johansen and Frederik Egelund Edvardsen underneath some rubble in his now decrepent house, while being on an excavation on the island; searching through old houses and ruins. The exact date of discovery was 6th of Febuary 2020. One of the more astounding finds was the proof of

Ådne's Law on Complex Cube Roots

Ådne's law on complex cube roots of real numbers were the first of Ådne's big revelations. The law states that ${\sqrt[{3}]{a}}=x$ has solutions in the complex set $L=\{n,n\omega ^{\delta }\}$, where $\delta \in \mathbb {Z}$ and $n$ is the real cube root of $a$.

Ådne's law on complex cube roots of real numbers is derived from the previously known rule that for every $a\in \mathbb {R}$$\backslash \{0\}$ has more than one cube root, three to be precise, where two of them are within the complex plane or is a complex number $\left(\mathbb {C} \right)$. These cube roots are known as the complex conjugate cube roots.

Visible proof of there always only being one real ($\mathbb {R}$) cube root of any real number $n$. This also applies to the negative real numbers ($\mathbb {R} ^{-}$).

The law works as follows. Firstly, the cube root of a number $a$ is the number $b$ so that $b^{3}=a$, where $\{a,b\}\in \mathbb {R}$ . When working within the real numbers, only one such number exists. For example, ${\sqrt[{3}]{-8}}=-2$ and only $-2$. If we shift our focus to complex numbers we get a different result. All numbers $a\in \mathbb {C}$ $\backslash \{0\}$ always have three different cube roots.

Therefore, according to Ådne's law on complex cube roots for real numbers, ${\sqrt[{3}]{a}}=x$ has solutions in the complex set $L=\{n,n\omega ^{\delta }\}$, where $\delta \in \mathbb {Z}$ and $n$ is the real cube root of $a$.

Example of Ådne's Law on Complex Cube Roots

Visible proof of the two complex conjugate cube roots and the real cube root of $8$.

The most simple example of the above statement is the cube root of $8$ which has the real solution of $2$. If we follow through with Ådne's law on complex cube roots, the equation ${\sqrt[{3}]{8}}=x$ will have its solutions in the complex set $L$ such that $L\subset \mathbb {C} :L=\{2\omega ,2\omega ^{\delta }\}$, meaning that if we raise either $2$ or $2\omega ^{\delta }$ to the third power, the result will be 8. This applies to all $\delta \in \mathbb {Z}$.

Where $\omega \triangleq {\frac {i{\sqrt {3}}-1}{2}}={\frac {i{\sqrt {3}}}{2}}+{\frac {1}{2}}=\cos 120^{\circ }+i\sin 120^{\circ }$ and $i^{2}\triangleq -1$. Keep in mind that $\omega ,{\overline {\omega }}$ and $1$ are the three cube roots of $1$, meaning that either of these raised to the third power equals $1$. This is useful when proving the law.

Algebraic Use

When working within the complex plane in algebra, Ådne's law on complex cube roots can help by giving an easier way of denoting the two complex conjugate cube roots of the number you are working with. Take for example: $z=27$, where ${\sqrt[{3}]{z}}=x$. In order to find $x$ without utilising the law you would have to firstly rewrite the equation to $x^{3}=z$ and convert the complex number $z$ to its polar form and get the equation $x^{3}=r(\cos \theta +i\sin \theta )$. This implies that if we use de Moivre's formula, you would end up with: $x^{3}={\Bigl (}r(\cos \theta +i\sin \theta ){\Bigr )}^{3}=r^{3}(\cos \theta +i\sin \theta )^{3}=r^{3}{\Bigl (}\cos(3\theta )+i\sin(3\theta ){\Bigr )}$, this gives you the following result:

$x^{3}=27\Longleftrightarrow r^{3}{\Bigl (}\cos(3\theta )+i\sin(3\theta ){\Bigr )}=27$. However, the absolute values and the arguments both have to give the result of $r^{3}=27\land 3\theta =0^{\circ }+k\cdot 360^{\circ }$and $r=3\land \theta =0^{\circ }+k\cdot 120^{\circ }$, where $k\in \mathbb {Z}$. This will then give you three values for $k$, which will give the following answers for the first cycle:

$k=0\Longrightarrow \theta =0^{\circ }\Longrightarrow x_{1}=3(\cos 0^{\circ }+i\sin 0^{\circ })=3$,

$k=1\Longrightarrow \theta =120^{\circ }\Longrightarrow x_{2}=3(\cos 120^{\circ }+i\sin 120^{\circ })=-1.5+2.6i$ and

$k=2\Longrightarrow \theta =240^{\circ }\Longrightarrow x_{3}=3(\cos 240^{\circ }+i\sin 240^{\circ })=-1.5-2.6i$. If you then insert any other value for $k$, where $k\in \mathbb {Z}$, you would just get a repeat of a cube root you have already found, this includes values for $k$ where $k\in \mathbb {Z} ^{-}$.

This then means the three cube roots of $27$ are $3$, $-1.5+2.6i$ and $-1.5-2.6i$.

Instead of doing all this you could simply use the law to immediately take $3$ (the real cube root of $27$) and multiply it with $\omega$ and $\omega ^{\delta }$. This will give you the answer: $L\subset \mathbb {C} :L=\{3,3\omega ^{\delta }\}$, where if you put in the values that define $\omega$ you will get the answers: $x_{1}=3\lor x_{2}=-1.5+2.6i\lor x_{3}=-1.5-2.6i$.

This proving Ådne's law on complex cube roots of real numbers is a much faster way of doing such equations.