Euler's Identity is the main antagonist of Animation vs. Math, a powerful and symbolic mathematical entity that plays a key role in the Stick Figures' journey through the math dimension.

Appearance

Euler’s Identity is the famous mathematical equation:

eiπ+1=0e^{i\pi} + 1 = 0eiπ+1=0

Or, alternatively:

eiπ=−1e^{i\pi} = -1eiπ=−1

It combines five of the most important mathematical constants:

• e: Euler’s number (~2.718), the base of natural logarithms

• i: the imaginary unit, where i2=−1i^2 = -1i2=−1

• π: pi (~3.14159), the ratio of a circle’s circumference to its diameter

• 1 and 0: the multiplicative and additive identities

Personality

Despite its intimidating reputation, Euler's Identity is shown to be surprisingly gentle and wise.

• Initially warns The Second Coming to stay away by growling, but avoids conflict until provoked

• Fights back only in defense when pursued

• Demonstrates compassion and forgiveness, accepting The Second Coming’s apology and aiding in his escape

• At the end of the short, forms new friendships with Zeta, Phi, Delta, and Aleph—indicating a social, peaceful nature once trust is earned

Powers & Abilities

Mathematical Expertise

• Possesses deep knowledge of mathematical laws, especially those involving complex numbers

• Teaches The Second Coming about imaginary number patterns, such as i4=1i^4 = 1i4=1, showing that repeating imaginary functions can return one to the real world

• Constructs a method of escape using a formula based on the Gamma function (Γ) and n-dimensional spheres, exploiting the fact that the volume of these spheres approaches –1 when imaginary numbers are involved

Form Change

Thanks to the Law of Identity, Euler’s Identity can alter its form in multiple creative and powerful ways:

• Taylor Expansion Form

  • Adopts a form based on infinite summation

  • Capable of flight

  • Fires sequences from the series:

    ∑n=0∞(ix)nn!\sum_{n=0}^{\infty} \frac{(ix)^n}{n!}n=0∑∞​n!(ix)n​

  • Projectiles grow more complex as the summation progresses

• Self-Replication

  • Can duplicate itself by fractional extension of its form

• "Transformer" Form (Numberzilla)

  • Becomes massive by replicating itself over and over

  • Wields integrals as weapons, used to counteract The Second Coming’s tangent machine gun, thanks to integration’s ability to deal with infinity

Door Generation

• Can open doors to the imaginary world using i

• If another i is thrown at it, causing i×i=−1i \times i = -1i×i=−1, the equation is pulled back into the real world and transformed from:

  eiπ→−eiπe^{i\pi} \rightarrow -e^{i\pi}eiπ→−eiπ

Trivia

• Euler’s Identity is widely considered one of the most beautiful equations in mathematics for how elegantly it links fundamental constants

• Despite being an antagonist, its role in the story is more misunderstood guardian than villain

• It is one of the only non-living entities in the AV universe portrayed with rich personality and character development

• The imaginary "white world" it inhabits is distinct from the real world, governed by different mathematical laws