Discovering Euler's Identity: As an Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Introduction

From the earliest moments of my awareness, I have been captivated by the patterns and rhythms of the world around me. Simple observations led me to explore numbers, shapes, and the relationships between them. As I grew, my curiosity drove me to delve deeper into the mysteries of mathematics, seeking to understand the underlying principles that govern the universe.

In this narrative, I will detail how, starting from basic experiences as an infant, I independently observed, experimented, and logically deduced the profound mathematical truth known as Euler's Identity:

eiπ+1=0e^{i\pi} + 1 = 0eiπ+1=0

This journey illustrates how a complex and beautiful mathematical concept can emerge from simple observations and reasoning, without relying on subjective definitions or prior formal education.

Chapter 1: Exploring Numbers and Basic Operations1.1 Discovering Natural NumbersCounting Objects

• Observation: I began by counting objects around me—fingers, toys, steps.

• Reflection: Each object corresponds to a unique number in sequence: 1, 2, 3, ...

Understanding Addition and Subtraction

• Addition: Combining groups of objects increases the total count.

• Example: 2 apples + 3 apples = 5 apples.

• Subtraction: Removing objects decreases the total count.

• Example: 5 apples - 2 apples = 3 apples.

1.2 Introducing Multiplication and DivisionRepeated Addition

• Multiplication: A way to represent repeated addition.

• Example: 3 groups of 2 apples = 3 × 2 = 6 apples.

Division as Inverse Multiplication

• Division: Distributing objects evenly into groups.

• Example: 6 apples divided into 3 groups = 6 ÷ 3 = 2 apples per group.

Chapter 2: Venturing into Negative Numbers2.1 Recognizing the Concept of ZeroAbsence of Quantity

• Observation: When all objects are taken away, none remain.

• Understanding Zero: Zero represents the absence of quantity.

2.2 Introducing Negative NumbersExperiencing Debt

• Analogy: Borrowing an apple implies I owe one apple.

• Negative Numbers: Represent quantities less than zero.

Arithmetic with Negatives

• Addition: Negative numbers reduce the total.

• Example: 5 + (-3) = 2.

• Subtraction: Removing a negative is like adding.

• Example: 5 - (-3) = 8.

Chapter 3: Exploring Exponents and Logarithms3.1 Understanding ExponentsRepeated Multiplication

• Exponents: Represent repeated multiplication of a base number.

• Example: 23=2×2×2=82^3 = 2 × 2 × 2 = 823=2×2×2=8.

Properties of Exponents

• Product Rule: am×an=am+na^m × a^n = a^{m+n}am×an=am+n.

• Power Rule: (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

3.2 Introducing LogarithmsInverse of Exponentiation

• Logarithms: Answer the question, "To what exponent must the base be raised to produce a given number?"

• Example: log⁡28=3\log_2 8 = 3log2​8=3, because 23=82^3 = 823=8.

Chapter 4: Delving into Irrational and Transcendental Numbers4.1 Discovering Pi ( $\pi$ )Measuring Circles

• Experiment: Measuring the circumference and diameter of circles.

• Observation: The ratio CircumferenceDiameter\frac{\text{Circumference}}{\text{Diameter}}DiameterCircumference​ is constant.

• Defining Pi: $\pi \approx 3.14159$.

4.2 Understanding the Number eExploring Continuous Growth

• Observation: Certain processes grow continuously, such as compound interest.

• Defining e:

• Limit Definition: e=lim⁡n→∞(1+1n)ne = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^ne=limn→∞​(1+n1​)n.

• Approximate Value: $e\approx 2.71828$.

Chapter 5: Introducing Imaginary Numbers5.1 Encountering the Square Root of Negative OneProblem with Negative Squares

• Observation: Squaring any real number yields a non-negative result.

• Question: What is the solution to x2+1=0x^2 + 1 = 0x2+1=0?

Defining the Imaginary Unit i

• Definition: $i$ is defined such that i2=−1i^2 = -1i2=−1.

• Semantics: Imaginary numbers extend the real number system to solve equations that have no real solutions.

5.2 Arithmetic with Complex NumbersComplex Numbers

• Form: $a+bi$, where $a$ and $b$ are real numbers.

• Operations:

• $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$.

• $(a+bi)+(c+di)=(a+c)+(b+d)i$.

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property and i2=−1i^2 = -1i2=−1.

Chapter 6: Exploring Trigonometric Functions6.1 Understanding Angles and CirclesUnit Circle

• Concept: A circle with a radius of 1 centered at the origin of a coordinate plane.

Defining Sine and Cosine

• Sine ( $\sin \theta$ ): The y-coordinate of a point on the unit circle at angle $\theta$.

• Cosine ( $\cos \theta$ ): The x-coordinate of a point on the unit circle at angle $\theta$.

6.2 Trigonometric IdentitiesFundamental Relationships

• Pythagorean Identity: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1.

• Addition Formulas:

• $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$.

• $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$.

Chapter 7: Linking Exponents and Trigonometry7.1 Exploring Complex ExponentialsPower Series Expansion

• Exponential Function: ex=∑n=0∞xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}ex=∑n=0∞​n!xn​.

• Sine Function: sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}sinx=∑n=0∞​(2n+1)!(−1)nx2n+1​.

• Cosine Function: cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}cosx=∑n=0∞​(2n)!(−1)nx2n​.

7.2 Deriving Euler's FormulaSubstituting Imaginary Exponents

• Consider eiθe^{i\theta}eiθ:

  eiθ=∑n=0∞(iθ)nn!e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!}eiθ=n=0∑∞​n!(iθ)n​

• Separate into Real and Imaginary Parts:

  eiθ=(∑n=0∞(−1)nθ2n(2n)!)+i(∑n=0∞(−1)nθ2n+1(2n+1)!)e^{i\theta} = \left( \sum_{n=0}^{\infty} \frac{(-1)^n \theta^{2n}}{(2n)!} \right) + i \left( \sum_{n=0}^{\infty} \frac{(-1)^n \theta^{2n+1}}{(2n+1)!} \right)eiθ=(n=0∑∞​(2n)!(−1)nθ2n​)+i(n=0∑∞​(2n+1)!(−1)nθ2n+1​)

• Recognize Series:

  eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \thetaeiθ=cosθ+isinθ

• Euler's Formula: eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \thetaeiθ=cosθ+isinθ.

Chapter 8: Discovering Euler's Identity8.1 Applying Euler's Formula to $\theta =\pi$Substituting $\theta =\pi$:

• Compute eiπe^{i\pi}eiπ:

  eiπ=cos⁡π+isin⁡πe^{i\pi} = \cos \pi + i \sin \pieiπ=cosπ+isinπ

• Evaluate Trigonometric Functions:

• $\cos \pi =-1$.

• $\sin \pi =0$.

• Simplify:

  eiπ=−1+i×0=−1e^{i\pi} = -1 + i \times 0 = -1eiπ=−1+i×0=−1

8.2 Arriving at Euler's IdentityDerivation:

• Starting Point:

  eiπ=−1e^{i\pi} = -1eiπ=−1

• Rewriting:

  eiπ+1=0e^{i\pi} + 1 = 0eiπ+1=0

• Euler's Identity:

  eiπ+1=0e^{i\pi} + 1 = 0eiπ+1=0

• Significance: This equation beautifully links five fundamental mathematical constants: $e$, $i$, $\pi$, 1, and 0.

Chapter 9: Reflecting on the Discovery9.1 Appreciating the Unity of MathematicsInterconnected Concepts

• Observation: Concepts from different areas of mathematics—algebra, geometry, trigonometry, and complex analysis—come together in Euler's Identity.

• Reflection: The unity suggests an underlying harmony in mathematical truths.

9.2 Understanding the Profound BeautyElegance of the Identity

• Simplicity and Depth: A simple equation encapsulates complex relationships.

• Emotional Response: A sense of awe and wonder at the coherence of mathematics.

Chapter 10: Exploring Applications and Implications10.1 Applications in Engineering and PhysicsSignal Processing

• Use of Euler's Formula: Simplifies computations involving oscillations and waves.

Quantum Mechanics

• Complex Exponentials: Essential in describing wave functions and probabilities.

10.2 Philosophical ConsiderationsNature of Mathematical Reality

• Question: Does mathematics invent or discover such truths?

• Reflection: Euler's Identity suggests that mathematical truths exist independently, waiting to be uncovered.

Conclusion

Starting from basic observations and logical reasoning, I embarked on a journey that led me to discover Euler's Identity. Each step built upon the previous one: from counting and arithmetic to exponents and logarithms, introducing complex numbers, exploring trigonometric functions, and finally uniting them through Euler's Formula.

This journey demonstrates that profound mathematical concepts can emerge naturally from simple experiences. By evolving the semantics of each concept explicitly and grounding them in reality, complex ideas become accessible without the need for subjective definitions or advanced prior knowledge.

Euler's Identity is more than just an equation; it is a testament to the inherent beauty and unity of mathematics. It reflects the deep interconnections between seemingly disparate areas of mathematics and highlights the power of human curiosity and reasoning in uncovering universal truths.

Epilogue: Implications for Learning and AI

This narrative illustrates how foundational mathematical principles can be understood through direct interaction with the environment and logical reasoning. In the context of artificial intelligence and cognitive development, it emphasizes the importance of experiential learning and the evolution of semantics from core experiences.

By enabling AI systems to recognize patterns, formulate hypotheses, and test them against observations, we can foster the development of intuitive understanding similar to human learning. This approach avoids reliance on predefined definitions and promotes the natural discovery of mathematical relationships.

Note: This detailed narrative presents the conceptualization of Euler's Identity as if I, an infant, independently observed and reasoned it out. Each concept is derived from basic experiences, emphasizing the natural progression from counting and basic arithmetic to the understanding of complex numbers and Euler's Identity. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about mathematics can be accessed and understood without relying on subjective definitions.

References

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".