Why $\sqrt{2}$ Is Irrational

Overview

A concise visual proof that the square root of 2 cannot be expressed as a ratio of two integers. The animation shows a unit square, the diagonal length, and the classic proof by contradiction, leaving the viewer with the key takeaway: $\sqrt{2}$ is irrational.

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Phases

#	Phase Name	Duration	Description
1	Intro	~4 s	Title fades in at the top, a unit square appears centered, and the diagonal is highlighted.
2	Assumption Setup	~6 s	A brief label "Assume $\sqrt{2}=\frac{a}{b}$ in lowest terms" appears in the bottom area with an opaque background. The square’s side length is labeled $1$ and the diagonal is labeled $\sqrt{2}$.
3	Algebraic Consequence	~8 s	The equation $2 = \frac{a^{2}}{b^{2}}$ is shown, then transformed to $a^{2}=2b^{2}$. Visual emphasis (highlight) on the fact that $a^{2}$ is even, implying $a$ is even.
4	Substitution & Contradiction	~8 s	Introduce $a=2k$ with a small animation of replacing $a$. Show resulting $b^{2}=2k^{2}$ and deduce $b$ is even. A red “X” appears over the statement "both a and b even" to signal the contradiction with the “lowest terms” assumption.
5	Conclusion / Outro	~4 s	The statement "$\sqrt{2}$ is irrational" appears in the bottom area, then fades out while the square and diagonal remain for a moment before the whole scene fades to black.

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Layout

┌─────────────────────────────────────────────┐
│                TOP AREA (Title)             │
├──────────────────────┬──────────────────────┤
│                      │                      │
│      LEFT AREA       │      RIGHT AREA      │
│  (Main visual: unit │  (Supporting labels, │
│   square & diagonal)│   short text)        │
│                      │                      │
├──────────────────────┴──────────────────────┤
│               BOTTOM AREA (Equations,      │
│               assumptions, conclusion)     │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title: "Why $\sqrt{2}$ Is Irrational"	Fades in at start of phase 1
Left	Unit square with side = 1, diagonal highlighted	Primary visual focus; animated drawing of square and diagonal
Right	Small supporting labels (e.g., "side = 1", "diagonal = $\sqrt{2}$") and brief textual cues when needed	Text appears on semi‑transparent dark background for readability
Bottom	Step‑by‑step algebraic equations, assumption statement, and final conclusion	Each equation fades in/out synchronously with the corresponding phase

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Assets & Dependencies

• Fonts: LaTeX (for all mathematical notation), a clean sans‑serif for on‑screen text (e.g., OpenSans).
• Colors: Dark background (#202020); square outline in light gray; diagonal in bright teal; text in white with 70 % opacity background; contradiction “X” in red.
• External assets: None (all shapes are generated procedurally).
• Manim version / plugins: Manim Community Edition 0.18 (or later). No additional plugins required.

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Notes

• All text boxes use an opaque dark rectangle (≈70 % opacity) to ensure contrast against the background and the square.
• Transitions between phases are simple cross‑fades (≈0.5 s) to keep the total runtime under 30 seconds.
• The proof steps are timed to give roughly 1–2 seconds per displayed equation, matching the 30‑second target.
• No narration is assumed; the visual flow should be self‑explanatory.