Okay, I need to create a Wikipedia-style document about "무리수" (irrational numbers) under the categories of Education > Mathematics > Basic Math. Let me start by understanding the requirements.

First, the structure: H1 for the title, an overview section, several sections with H2 and H3 headings, and maybe references or related documents. The content needs to be accurate, objective, detailed enough, easy to read, and use technical terms with explanations. Also, markdown formatting with proper headings, lists, tables, code blocks if needed, and at least 800 characters.

Starting with the title: "무리수" as H1. Then an introduction explaining what irrational numbers are, their significance in math. Next sections could be history, properties, examples, how they differ from rational numbers, applications, etc. Maybe include a section on famous irrational numbers like π or e. Also, mention the proof that √2 is irrational.

I should check if there's any confusion between different terms. For example, "무리수" vs "무리수" – but I think it's correct. Need to make sure all technical terms are explained. Maybe use a table comparing rational and irrational numbers. Also, include some historical context about how they were discovered.

Wait, the user mentioned "기초수학" (basic math), so maybe keep explanations simple but accurate. Avoid too advanced topics unless necessary. Ensure that each section flows logically. Use H2 for main sections like "정의 및 특성", "역사적 배경", "대표적인 무리수", etc. Under each, use H3 for sub-sections.

Need to verify facts: when were irrational numbers discovered? Pythagoras' time, around 5th century BCE. The proof of √2 being irrational is a classic example. Also, mention that they can't be expressed as fractions, have non-repeating, non-terminating decimals.

For the examples section, list π, e, φ (golden ratio), and maybe √2. Explain each briefly. In applications, talk about their use in geometry, calculus, etc.

Check for any possible errors: make sure that the explanation of irrational numbers is clear, avoid common misconceptions. Maybe include a note on how they are used in real-world scenarios.

Also, think about including a section on the difference between rational and irrational numbers, perhaps with a table. Use markdown tables for that. Ensure that code blocks aren't needed here unless there's some mathematical notation, but maybe not necessary.

Finally, add references or related documents at the end. Maybe link to articles on rational numbers, real numbers, or specific constants.

Now, structure it step by step:

1. Title: # 무리수
2. Introduction: Overview of irrational numbers.
3. Sections:
4. 정의 및 특성 (H2)
   • H3 for properties like non-repeating decimals, inability to express as fractions.
5. 역사적 배경 (H2)
   • Discovery by Pythagoreans, the story of √2.
6. 대표적인 무리수 (H2)
   • π, e, φ, etc., each with H3 sections.
7. 수학적 중요성 및 응용 (H2)
   • Applications in various fields.
8. 유리수와의 차이점 (H2)
   • Table comparing them.
9. References or related documents.

Check the length: need to make sure it's over 800 characters. Each section should have enough detail. Avoid being too brief. Use examples and explanations to meet the length requirement.

Make sure all terms are explained. For instance, when mentioning "비유리수", clarify that it's another term for irrational numbers. Also, explain what real numbers are if necessary, but since this is under basic math, maybe keep it simple.

Double-check facts: The proof of √2 being irrational involves contradiction, assuming it can be expressed as a fraction and then showing that leads to a contradiction. Mention that in the history section or properties.

Avoid markdown syntax errors. Use proper headings, lists if needed. Maybe use bullet points for examples of irrational numbers.

Okay, time to put it all together in markdown format.