Quadratic irrationals: The mysterious numbers behind infinite patterns.

Exploring the strange world of quadratic irrationals

By Peter Teoh, Science Writer

Quadratic irrationals are irrational numbers that are roots of quadratic equations with integer coefficients. They often appear in number theory, continued fractions, and Diophantine approximation. Below are several example problems involving quadratic irrationals, ranging from basic to more advanced.

Example 1: Identify a Quadratic Irrational

Problem: XXXXXXXXXXXXXX XXXXXXXXXXXXXX XXXXXXXXXXXXXX XXXXXXXXXXXXX

Summary of Key Ideas:

• They come in conjugate pairs.
• They are central to solving Pell’s equation x² − Dy² = 1.

Side Notes

• Quadratic irrationals are dense on the real line — between any two rational numbers, you can find infinitely many of them!

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Reference Links

• Lagrange’s Theorem on Continued Fractions