3.5 Irrational Numbers — The Numbers That Terrified Ancient Greece | Contemporary Mathematics

• Learn about the ancient Greek Pythagoreans and the shocking discovery of irrational numbers.
• Understand the precise definitions and characteristics of rational and irrational numbers.
• Learn to identify common types of irrational numbers, like non-perfect square roots and Pi.
• Master the technique of simplifying square roots by extracting perfect squares from the radicand.
• Learn the strict rules for adding and subtracting irrational numbers with matching radicands.
• Discover the more flexible rules for multiplying and dividing irrational numbers.
• Learn to rationalize a denominator by multiplying by the irrational term to remove it.
• Master rationalizing complex denominators using the conjugate to eliminate irrational terms.
• Reflect on the Euler-Mascheroni constant and the ongoing discovery of mathematical truths.
• Up next in “Contemporary Math — Full Course (OpenStax)”3.6 Real Numbers — The Secret Behind Lightning-Fast Mental Math | Contemporary Mathematics

What do a ancient Greek philosophical cult, an infinite decimal, and a mathematical murder legend all have in common? This lesson explores irrational numbers — what they are, where they come from, and why they shook the foundations of mathematics. Drawing from Section 3.5 of OpenStax Contemporary Mathematics, we cover everything from the story of Hippasus and the Pythagoreans to the practical techniques used to work with these never-ending, never-repeating numbers. In this video: • The Pythagoreans and the legendary (and likely mythical) story of Hippasus discovering irrational numbers • How to identify irrational numbers: non-terminating, non-repeating decimals that can't be expressed as a fraction • Square roots of non-perfect squares as a key category of irrational numbers, with the Desmos calculator as a practical checking tool • Pi as the second major category: its definition as circumference divided by diameter, and why any rational multiple of pi is also irrational • Simplifying square roots using the product rule — worked example: √180 simplified to 6√5 • Adding, subtracting, multiplying, and dividing irrational numbers, including the strict rule that only matching irrational parts can be added or subtracted • Rationalizing the denominator — including the conjugate method for expressions like 5 ÷ (4 + √3) — and the Euler-Mascheroni constant as a surprising example of how long it can take to classify a number #OpenStax #IrrationalNumbers #ContemporaryMathematics #Pi #CollegeMath OpenStax Content adapted from "Contemporary Math", by OpenStax Licensed under CC BY 4.0. Content based on Web Version: Apr 23, 2026. Read the textbook online https://openstax.org/details/books/contemporary-mathematics Music first girl talking to me. by ikkun (ex. Barradeen) | https://soundcloud.com/ikkunwastaken Royalty Free Music by https://www.free-stock-music.com Creative Commons / Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) https://creativecommons.org/licenses/by-sa/3.0/deed.en_US