Euler's identity

For further information about the equation and constant, see Euler's identity and Euler's number on Wikipedia.

For its clones, see Euler's identity's Clones.

For other math-related pages, see here.

"Σ" and "Sigma" redirect here. For the mech-like object created from it, see Numberzilla.

Euler's identity
Full Name	eiπ
Alias(es)	Euler's identity Negative One (−1) E to the I Pi e i pi
Species	Mathematical equation
Status	Active
Related Character(s)	Its clones † Numberzilla † (final form)
Allies	The Second Coming Zeta, Phi, Delta, and Aleph
Enemies	The Second Coming
First Appearance	Animation vs. Math (only appearance)

Euler's identity (e^iπ) is the main antagonist of Animation vs. Math.

Appearance

Euler's identity contains Euler's number, pi, and the imaginary unit, and is equal to negative one.

Euler's identity is the famous mathematical equation ${e}^{i\pi} + 1 = 0$ or ${e}^{i\pi} = -1$, where "e" is Euler's number, also known as the base of the natural logarithm, approximately equal to 2.71828182846; "i" is the imaginary number, where ${i}^{2}=-1$; and "π" is the ratio of a circle's circumference to the circle's diameter, approximately equal to 3.1415926535.

Personality

Euler's identity is on the whole a friendly character willing to help others. When The Second Coming attempted to catch Euler's identity and use it to escape the math dimension, it initially "growled" at the stick figure as a warning, then tried to run away, before resorting to fighting back as The Second Coming persisted in his chase. Yet even after The Second Coming pummeled Euler's identity and came near to destroying the math dimension, the equation still chose to accept his apology and help him escape. At the end of the video, Euler's identity made some new friends: Zeta, Phi, Delta, and Aleph.

History

Animation vs. Math

Euler's identity was first summoned when The Second Coming subtracted all the way to $-1$. Since a negative number was something new to him at the time, he knocked it in curiosity, and it turned into Euler's identity. it then ran away, multiplied itself by i, and went into a gate that leads to the imaginary world, disappearing from The Second Coming's sight.

Euler's identity was created again by The Second Coming when he calculates by multiplying the imaginary unit 3 times. Since ${i}\times {i}=-1$, which is equal to ${e}^{i\pi }$, and ${i}\times {i}\times {i}=-i$, ${ie}^{i\pi }$ also equals $-i$, summoning Euler's identity again. When The Second Coming approached it more and more, Euler felt scared and afraid, ran away and fell out of the equation, knocking an i off in the process. The Second Coming then grabbed that i, chased Euler, and threw the i towards it. While Euler attempted to escape into the imaginary gate, it got hit by that i, and because ${ie}^{i\pi }=-i$ and ${-i}\times {i}=1$, ${ie}^{i\pi }\times {i}$ creates a real number, knocking Euler out of the imaginary world. The Second Coming then pinned Euler down, and Euler transformed into its formula form ($\cos {(\pi )}+i\sin {(\pi )}$) to push him away and escape. It then threw the negative sign to fool The Second Coming by flipping him backwards, but he stood up and attempted to chase it again. Euler evaded him by converting itself to ${e}^{i\pi -\pi }$, becoming ${e}^{0}$ and traversing the complex plane's unit circle from π to 0 radians over The Second Coming. As the chase continued, The Second Coming managed to catch up to Euler by multiplying his speed and flipping Euler's direction using a neative sign. It is then that Euler, using a sword forged from his negative sign, fights against The Second Coming and his sword made from a plus sign, and they constantly reduce each other's swords through addition and subtraction. The Second Coming then creates a $2 \times 2$ bow to shoot 4's at Euler, and it uses one of them via a division slash (/) to convert to ${e}^{i{\frac {\pi }{4}}}$ and travel up the unit circle to escape The Second Coming's continuous advances.

Eventually, Euler's identity was summoned a third time after The Second Coming added $\cos {(\pi )}$ and $i\sin {(\pi )}$. It hopped out of the Second Coming's hands but was pinned by him, at which point he drew out his sword to defend itself. They resumed their sword fight, and as The Second Coming used his bow to shoot it, Euler converted into its Taylor Expansion form ($\sum_{n=0}^\infty \tfrac{(i\pi)^n}{n!}$) and returned fire with its iteration values. As it was on the offensive side, it was eventually pushed back through The Second Coming's usage of an $8\pi {r}^{2}$ cylinder, knocking Euler back into its original form. It then constructed the equation $r-\theta {e^{i\pi }}=r+\theta$ to expand the unit circle, trap The Second Coming, and corner him, but the Second Coming escaped by negating himself. Euler continued its advance in its Taylor Expansion form, but was pushed back and reverted once again by The Second Coming's use of the i•sine mallet. Angered by this, it devolved itself to its formula form $\cos {(\pi )}+i\sin {(\pi )}$, and then into ${\frac {e^{i\pi }+e^{-i\pi }}{2}}+{\frac {e^{i\pi }-e^{-i\pi }}{2i}}$ to clone itself, and repeated this process to create its own army. Together, they advanced onto The Second Coming through the usage of their abilities shown thus far and the unit circle. However, they're numbers began to dwindle, as The Second Coming used its function gun to wipe out the majority of its army.

The Euler clones began their retreat, and with four of them remaining, they returned to the expression from which they came from and began to expand and rearrange itself, forming a huge entity. In this form, it used the blast from The Second Coming's function gun to create a definite integral staff ($\int \limits _{0}^{\infty }$) to attack and overwhelm him. It then proceeded to create an army of clones from its leg through the use of the equation $2=e^{i\pi }-e^{i\pi }-e^{i\pi }-e^{i\pi }$ multiple times and head towards the Second Coming. But at that moment, He used the combination of the unit circle, his function gun, and the i•sine and cosine mallets to create an orbital laser to wipe out Euler's army, and eventually itself. As it layed defeated and powerless to stop the Second Coming, it proceeded to escape to the imaginary world using i. But The Second Coming quickly gave haste to Euler and followed it inside.

While they were in the imaginary world, they watched as the orbital laser was began destroying it. The Second Coming then escaped with Euler back into the real world by continuously adding 1 and eventually, multiplying it by i. When they were in a safe spot, Euler quickly distanced itself away from The Second Coming, only for him to apologize as he didn't want the situation to escalate this far. Seeing the damage he caused as a result, Euler put aside its quarells to accepte his apology, and eventually came understand that he wanted an exit out of the world they're in. Euler agreed to help him, and proceeded to open the imaginary gate, but put The Second Coming to a halt as it wouldn't work, since multiplying i enough times will always return them to the world they're in now ($i^{4k}=1$, where k is any integer). Euler then though of another way: he first deactivated and returned the unit circle to its original size, prompting The Second Coming to go inside. Then, using the gamma function (symbolized by Γ) and Euler's Taylor expansion form, it created the equation $e^{\pi }=\sum _{2n=0}^{\infty }{\tfrac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}}+1)}}$, representing the n-dimensional unit sphere. It proceeded to increase the dimension size to infinity, and waved its final goodbye to The Second Coming before sending him off by adding i to itself and simplifying back into ${e}^{i\pi} = -1$ (as all imaginary spheres from infinite dimensions have a total volume of −1).

Shortly after, Zeta (ζ), Phi (ϕ), Delta (δ), and Aleph (א) came to greet and walk away with Euler as its company.

Powers & Abilities

Math Expertise

When The Second Coming was "accidentally" about to destroy its world, wanted to fix it while finding an "exit" to go back home, it explained to The Second Coming about the imaginary number laws. In this case, when The Second Coming suggested using the imaginary function as many times as it can to create an "exit", the problem is that $i^{4}=1$, resulting them back to the real world again.

The best case of showing its expertise is when it thought of an "exit" within a few seconds, created and using a formula involving the gamma function (Γ), where each term becomes the volume of an n-dimensional unit sphere as $n \to \infty$. Then, it adds "i" to its power, resulting the total of −1. (All imaginary spheres from infinite dimensions have a total volume of −1)

Form change

It is capable of changing its form to different kinds, thanks to the Law of Identity.

• Taylor Expansion form: It can fly on its own and project sequences from its sigma part ($\sum_{n=0}^\infty \tfrac{(i\pi)^n}{n!}$), and the value of n below increases as it continue firing.
• Self-Replication: Can duplicate itself by extending it fractionally.
• "Transformer" form (Numberzilla): Replicates itself many times over and over to become "big". It is able to utilize the integral as a weapon as a countermeasure to The Second Coming's tangent machine gun since the integral is able to handle "infinity".

Door Generation

Euler's identity uses the i to open the door to the white imaginary world. However, it can be stopped by throwing an i at them, because $i\times {i}=-1$, which is a real number that brings it back to the real world. It also transforms Euler's identity from $e^{i\pi}$ into $-e^{i\pi}$.

Gallery

Gallery

Inverted Euler's identity. Euler's identity next to i. Pi divided by 4. Pi minus pi. e to the power of 0. The minus Euler's identity, which equals to 1. The Gamma function (Γ). Euler's identity opening the door. The "exit" (e×iτ) sign from e itself, multiplication sign, imaginary number i, and half-hidden pi. Euler's identity running to the imaginary world and leaving The Second Coming behind. Euler's identity creating its clones. Its clones. Its "Transformer" form, Numberzilla. Euler's identity and The Second Coming handshaking.

Audio

Sound	Description
	Euler's identity commanding Numberzilla to attack The Second Coming, Animation vs. Math
	Euler's identity waving goodbye to The Second Coming, Animation vs. Math

External Links

• Euler's identity on the Villains Wiki
• Euler's identity on the FC/OC VS Battles Wiki
• Euler's identity on the Villainous Benchmark Wiki