Everything Everywhere Daily: History, Science, Geography & More - Irrational Numbers

Starting point is 00:00:00 There's a particular type of number that is so common that we have keys on our modern calculators to handle them. However, thousands of years ago, their discovery was so upsetting to one group that it may have led to the destruction of their religion and possibly the murder of the man who made the discovery. Nowadays, they're commonplace enough to be taught in grade school. Learn more about irrational numbers and their place in the world of mathematics on this episode of Everything Everywhere Daily.

Starting point is 00:00:38 Do you ever climb into bed ready to sleep, only to have your mind start racing the moment your head hits the pillow? Thoughts bouncing around, replaying the day or jumping ahead to tomorrow? That is exactly why Catherine Nikolai created Nothing Much Happens. Each episode is a gentle, cozy bedtime story where, well, nothing much happens. No drama, no tension, nothing you need to follow closely. Just soft narration, calming repetition, and soothing sensory details designed to help your mind slow down and your body relax. It's not about entertainment, it's about rest. And millions of listeners around the world use it every night to quiet their thoughts and finally fall asleep. If you've ever struggled to shut your brain off at night, this might be exactly what you've

Starting point is 00:01:18 been missing. You can listen to Nothing Much Happens wherever you get your podcasts. Episodes are every Monday and Thursday. Before I jump into a discussion of the history of irrational numbers, I should probably explain what an irrational number is. An irrational number is any real number that cannot be expressed as the ratio of two integers. That's it. That's it. That's the definition. I've tangentially mentioned irrational numbers several times before in previous episodes, but I've never dealt with them explicitly until now. Mathematicians have a way of classifying numbers into sets that can be visualized as a group of concentric circles. At the core are the natural numbers. These are the numbers that everyone is familiar with and are also

Starting point is 00:02:06 called the counting numbers, one, two, three, four, etc. The circle beyond the natural numbers are the integers. This includes all the natural numbers plus zero and the negative natural numbers. The circle beyond that is called the rational numbers. Rational numbers include all of the integers and all the possible fractions, which are one integer divided by another integer, except, of course, you can't abide by zero. Finally, in the circle outside of the rational numbers is the real numbers. The real numbers include all of the rational numbers and the subject of this episode, all of the irrational numbers. An irrational number, when written in decimal expansion, is infinitely long and never repeats

Starting point is 00:02:53 its digits. The most common irrational number, and the first one which was probably ever discovered, was the square root of two. There are no numbers that you can divide into each other to equal the square root of two. In fact, the square root of every number is, except for perfect squares is irrational. If you remember back to my episode on infinity, Georg Cantor proved that there were more irrational numbers

Starting point is 00:03:18 than there were rational numbers. And given that the number of rational numbers was infinite, it meant that there were some infinities larger than other infinities. And if you are incredulous and find that statement to be impossible, I recommend going back and listening to that episode. Irrational numbers are divided into two types. The first are called algebraic numbers. These are numbers that are the solution to any algebraic equation.

Starting point is 00:03:42 The square root of two would be an example of an algebraic number. The second type of irrational number are transcendental numbers. Transcendental numbers are any irrational number that cannot be produced algebraically. This would include numbers like pi as well as any infinitely long random list of decimals. Simply being an infinitely long string of decimals isn't enough to be considered irrational. Consider, 7 divided by 11 is 0.3636-36-36 repeating. In fact, any infinitely long decimal that repeats itself has to be a rational number expressed as some fraction. So what was the big hullabaloo over irrational numbers?

Starting point is 00:04:27 Why is this something that's worth doing a podcast episode about? The story of irrational numbers starts over 2,500 years ago in ancient Greece, with the cult of Pythagoras. I've previously recorded an episode on the cult of Pythagoras, but I'll briefly give an overview of the tenets of the movement here. The Pythagorean's believe that the world was governed by mathematics, in particular whole numbers, aka the natural numbers. They regarded numbers as the fundamental elements of the universe, believing that they represented the essence and structure of reality. Moreover, they believe that all things could be understood through mathematical principles and that numbers possessed mystical qualities.

Starting point is 00:05:08 The idea that mathematics could be used to understand the physical world is actually not a crazy idea, and it's largely true. The problem stemmed from the fact that they felt numbers could only be rational numbers to reflect reality. This notion wasn't confined to just the Pythagorean's, but was a tenant of most ancient Greek mathematicians. In previous episodes, I mentioned the reluctance of the ancient Greeks to accept concepts like zero and negative numbers.

Starting point is 00:05:33 The Pythagorean's philosophy was known as the doctrine of the harmony of the spheres, where harmony and order in the universe were believed to arise from numerical relationships. The discovery of irrational numbers, in particular the square root of two, posed a challenge to the Pythagorean worldview. If, in fact, all quantities and lengths could be expressed as rational numbers, then the existence of irrational numbers contradicted this belief and threatened the Pythagorean notion of a completely ordered and rational, rational universe. According to legend, the discovery of the irrationality of the square root of two

Starting point is 00:06:06 is attributed to a Pythagorean named Hepassus. It said that he revealed this mathematical truth to the Pythagorean community, which resulted in a conflict that led to his expulsion from the group. And in some versions of the story, Hapasus made the discovery while he was on a ship. When he told the other cult members about his discovery, he was thrown overboard. The exact reason for his expulsion and or murder are unclear, but it has been suggested that the Pythagorean's considered the existence of irrational numbers as a threat to their philosophical framework. The discovery of irrational numbers challenged the Pythagorean's insistence on the exclusivity of irrational numbers and whole number ratios as the foundation of the universe.

Starting point is 00:06:47 While their specific objections to irrational numbers are not explicitly documented anywhere, it's believed that they viewed them as disruptive to the order and harmony that they sought in the numerical structure of reality. The fact that the square root of two is irrational can easily be proven using middle school mathematics. There are many such proofs available online, and I will leave that as a homework assignment as it's difficult to provide audio narration for mathematical proofs.

Starting point is 00:07:13 Oddly enough, the easiest way to prove that the square root of two is irrational is by using the Pythagorean theorem, the very theorem named after Pythagoras, who didn't believe in irrational numbers. Over in India, they didn't have the hangups that the Greeks did, just as with concepts like zero and negative numbers, ancient Indian mathematicians didn't have a problem with irrational numbers. The 5th century Indian mathematician Ariabata was using trigonometry in the sign function to determine

Starting point is 00:07:41 astronomical distances, all of which used irrational numbers. The 9th century Persian mathematician Al-Corrismi, who seems to make an appearance in almost every episode I have about mathematics, regularly used irrational. numbers when solving quadratic equations. Irrational numbers came into wider acceptance faster than negative numbers did because they clearly had real-world analogs. A triangle or a circle had irrational numbers embedded inside them. Europeans finally accepted irrational numbers in the Middle Ages when it became clear that they were the solutions to algebraic equations. The thing that really cemented the acceptance of irrational numbers was a means of expressing them. The Bithagoreans had no way

Starting point is 00:08:22 of expressing such numbers as they had no means of writing decimal numbers and no mathematical symbols. They mostly use geometry. Representing irrational numbers as decimals was a huge step. This came about from the adoption of the Hindu-Arabic number system that we use today. The representation of fractions as decimals is usually credited to the 16th century Dutch mathematician Simone Staven. Roots got their own mathematical notation with the invention of the radical symbol or the root symbol. It isn't clear exactly who invented the symbol or where it came from. It might have come from the 15th century Islamic mathematician Ali al-Khalasadi.

Starting point is 00:08:59 He may have taken it from the third letter of the Arabic alphabet, which happens to be the first letter of the Arabic word for root. It's also been claimed that it comes from the letter R and the Latin word radix, which also means root. The root symbol actually consists of two parts. The front part, which looks like a checkmark, is called the radical symbol or the radix. The line over the top of the number is known as the vinculum. While there are an infinite number of irrational numbers, there are a few that are noteworthy. The first is the previously mentioned square root of two. If you have a square with each side having a length of one, then the diagonal of the square

Starting point is 00:09:36 will be the square root of two. The square root of two has an approximate value of 1.414-2135, etc. Another core irrational number, which I've previously done an episode on, is Pi. Pi is the circumference of a circle divided by its diameter. And at first, you might think that because Pi is defined as one thing divided by another thing, that would make it a rational number. The problem is, at least one of those things, the circumference or the diameter, has to itself be an irrational number.

Starting point is 00:10:09 Another irrational number that pops up almost everywhere, even in things like art and nature, is Fee. Fee is the Greek symbol used to express the golden ratio. The golden ratio is an irrational number, with an approximate value equal to 1.61803. And it also can be expressed algebraically as 1 plus a square root of five, all divided by two. The golden ratio has been considered aesthetically pleasing, and it often appears in works of art and architecture. It also appears in nature, such as in the proportion of plants, animals, and even the human body. Fee and the golden ratio is an interesting enough subject

Starting point is 00:10:48 to actually warrant its own future episode. Finally, perhaps the most important irrational number is known as Euler's number, or E. The value of E is approximately 2.7-1828-1828-1848-48-45-9045. E is the number that sits behind exponential growth and radioactive decay. It has a special place in calculus as E raised to the power of X is the only function whose derivative is itself. To that extent, it's the calculus equivalent of multiplying by 1 or adding by 0. Despite the name, irrational numbers are in fact very rational and very real. Without them, we couldn't even do the most basic mathematics that makes the world function today. It's just too bad about what happened to Hapasus.

Starting point is 00:11:36 The Pythagorean's really overreacted to his discovery. In fact, you could say that they acted irrational. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Thor Thompson and Peter Bennett. Today's review comes from listener Henry Yu over on Apple Podcasts in Belgium. They write, Too Perfect? Hi, Gary. To make it short, I love your podcast, and I listen to most of them. It inspires me every time to hear you switch between all topics.

Starting point is 00:12:07 Always well-researched, always interesting, and always eye-opening. And then comes the way you present in such a polished way. That's my only criticism. When I first heard you, I seriously believe this would be an infomercial, some kind of marketing podcast. Your voice is too perfect. But hey, that's just me and my sarcasm. Please don't stop.

Starting point is 00:12:24 You've earned my five stars with full honors, Hendrick from Brussels. Thanks, Hendrick. First, let me congratulate you for leaving the first review from Belgium on Apple Podcasts. Second, we have something in common. I actually have Belgian ancestry. My father's, mother's father's family came from Belgium. And on top of that, I also have separate ancestors who came from Holland and some from Luxembourg. Making me one of what I can only assume are a very small number of Benelux Americans. Remember, if you leave a review or send me a boostogram, you two can have it right on the show.

Everything Everywhere Daily: History, Science, Geography & More - Irrational Numbers

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