Crisis of Incommensurability

The Dangerous Discovery of \(\sqrt{2}\)

Mathematics Archive: Lesson I

The Pythagorean Worldview

The Pythagoreans believed that "All is Number."

To them, "Number" meant only integers and their ratios (rational numbers). They believed any two physical lengths were commensurable—that is, they shared a common unit of measure.

1 1 ?

The Unit Square: A Geometric Nightmare

Defining the Rational

A number \(x\) is rational if there exist integers \(p\) and \(q\) such that:

\[ x = \frac{p}{q}, \quad q \neq 0 \]

Key Assumptions

\(p\) and \(q\) are integers.
The fraction is in simplest form (no common factors).

The Implication

If a number is rational, it can be expressed as a ratio of two coprime integers.

Reductio ad Absurdum

To prove something is impossible, we:

Assume the opposite is TRUE.
Follow logical steps until we reach a CONTRADICTION.
Conclude that the original assumption must be FALSE.

"Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."

Step 1: The Assumption

Assume \(\sqrt{2}\) is rational.

Therefore, \(\sqrt{2} = \frac{p}{q}\) where \(p\) and \(q\) are integers with no common factors.

Algebraic Manipulation:

\( 2 = \frac{p^2}{q^2} \)

\( 2q^2 = p^2 \)

The Logic:

If \(p^2 = 2q^2\), then \(p^2\) must be even.

Lemma: If \(p^2\) is even, then \(p\) must be even.

Step 2: Hunting for Paradoxes

1

If \(p\) is even, let \(p = 2k\) for some integer \(k\).

2

Substitute into our equation: \(2q^2 = (2k)^2 = 4k^2\).

3

Divide by 2: \(q^2 = 2k^2\). This means \(q^2\) is even, so \(q\) is even.

🚨 Contradiction Found! 🚨

We assumed \(p\) and \(q\) have no common factors.
But we just proved both are even (they share a factor of 2).

The Fall of the Rational

Because the assumption that \(\sqrt{2}\) is rational leads to an impossible logical state, we must conclude:

\(\sqrt{2}\) is IRRATIONAL

Legend says the Pythagoreans drowned Hippasus at sea for revealing this truth. The "solid" line of numbers had gaps that no ratio could fill.

The Socratic Archive

Question 1

If irrational numbers didn't exist, could we have a unit square? What would happen to geometry?

Question 2

Does the same proof work for \(\sqrt{4}\)? Why or why not? (Hint: check the Lemma!)

Question 3

Are there more rational or irrational numbers? How could we even "count" them?

Theory and Application of Irrational Numbers

Crisis of Incommensurability

Infinite Expansions

Algorithms of Approximation

Density and Completeness

Algebraic vs Transcendental Numbers

Crisis of Incommensurability Slides

Incommensurability Proof Worksheet

Infinite Expansions Slides

Decimal Decoding Worksheet

Approximation Engine Slides

Iterative Algorithms Lab Sheet

Infinite Gaps Slides

Density Explorations Worksheet

Transcendental Quest Slides

Number Nature Worksheet

Similar Sequences

Slice Lab Fraction Division

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Ratio Lab Intervention

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Number Scouts Sequence

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Statistiques Exploratoires

Hugo Pierrard

Division in Daily Life

Saiko Otake

Radical Relationships

Calculus Foundations

Transformation Surgeons

Sigma Summit

Solution Sleuths

Period Power Remediation

GCF Visual Remediation

Angle Avengers

Integer Investigation

Big Batch Bakery

Discount Deal Detectives

Algebraic Foundations

Polynomial Powers

Fraction Mastery

Proportional Powerhouse