How to Prove Square Root 2 is Irrational
A Classic Proof by Contradiction
Today we’re throwing it waaaaay back to ancient Greece for a little story time. A story from back when numbers were the holy grail and new ideas were so paradigm shifting it hurt.
Welcome to the world irrational numbers 🌍 (cuz we’re all about to lose our minds over you).
Worshipping Numbers 🙏🏼
Ah good ole Pythagoras, the man of high school geometry class legend. Although he definitely wasn’t the first person to discover his namesake theorem, his legend lives on nonetheless. Surely, if he lived in modern times we’d probably all think he was straight-up craaazy. As you may be familiar with, Pythagoras was the leader of the Pythagoreans, a religious, philosophical society that worshipped the beauty of numbers in their wholeness.
Numbers, beautiful-amazing numbers. Descriptors of the known world, the hidden gem of mankind's intellect.
In 6th century BC, there were only two kinds of numbers that humans knew about: counting numbers and ratios of counting numbers (i.e. rational numbers). And to the Pythagoreans these numbers weren’t just cool, they were holy. They were divine. They were worthy of undergoing years of study, austerity and initiation into a special society led by their beloved Pythagoras.
The Irony of the Pythagorean Theorem
If you’ve ever had the opportunity to study right triangles and employ the Pythagorean Theorem you’re probably thinking one thing right now…
“Irrational numbers and right triangles are almost ubiquitous!”
Yep. In fact, you probably stumbled upon irrational numbers on the very first day you were introduced to the Pythagorean Theorem. You know that famous 45–45–90 degree isosceles right triangle, the one with side lengths of 1, the diagonal of a 1-by-1 square, oh yeah that hypotenuse is an irrational number → square root 2.
That’s bad news bears for the Pythagoreans. You see, their most-valued equation leads you straight into the world of irrational numbers...
Oh, the irony.
The Story Continues…
More on the Pythagoreans, that pesky square root 2 & most importantly a logical proof of how we know-as-a-matter-of-fact that the square root of 2 is irrational (because hello, we actually have to DO some math today 😝) all on this week’s episode of Math Hacks.
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❤ Brett
