Irrational Numbers

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Podcast Transcript

There is a particular type of number that is so common we have keys on 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. 

Today, they are commonplace enough to be taught in grade schools.

Learn more about irrational numbers and their place in the world of mathematics on this episode of Everything Everywhere Daily.

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 is the definition. 

I’ve tangentially mentioned irrational numbers several times before in previous episodes, but I have 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 everyone is familiar with that are also called the counting numbers: 1, 2, 3, 4, etc. 

The circle beyond the natural numbers are the integers. This includes all the natural numbers plus zero and the negative numbers. 

The circle beyond that is the rational numbers. Rational numbers include all the integers and all the possible fractions, which are one integer divided by another integer, except, of course, you can’t divide by zero.

Finally, in the circle outside of the rational numbers is the real numbers. The real numbers include all the rational numbers and, the subject of this episode, all the irrational numbers. 

An irrational number, when written in decimal expansion, is infinitely long and never repeats its digits. 

The most common irrational number, and the first one which was probably ever discovered, was the square root of 2. There are no numbers you can divide into each other to equal the square root of 2. 

In fact, the square root of every number, except for perfect squares, is irrational. 

If you remember back to my episode on Infinity, Georg Cantor proved that there were more irrational numbers 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. 

If you are incredulous and find that statement to be impossible, I recommend going back to listen to that episode. 

Irrational numbers are divided into two types. The first is called algebraic numbers. These are numbers that are the solution to any algebraic equation. The square root of two would be an algebraic number.

The second type of irrational number is 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.  7 divided by 11 is .36363636… repeating. 

In fact, any infinitely long decimal that repeats itself has to be a rational number expressed as some fraction.

So what is the big hullabaloo over irrational numbers? Why is this something that is 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 tenants of the movement here. 

The Pythagoreans believed that the world was governed by mathematics, in particular whole numbers, aka natural numbers. They regarded numbers as the fundamental elements of the universe, believing that they represented the essence and structure of reality. 

They believed that all things could be understood through mathematical principles and that numbers possessed mystical qualities.

The idea that mathematics could be used to understand the physical world is actually not a crazy idea at all and is 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 Pythagoreans but was a tenant of most ancient Greek mathematicians. 

In previous episodes, I mentioned the reluctance of the ancient Greeks to accept the concepts of zero and negative numbers. 

The Pythagoreans’ 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, particularly the square root of 2, 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 universe. 

According to legend, the discovery of the irrationality of the square root of 2 is attributed to a Pythagorean named Hippasus. It is said that he revealed this mathematical truth to the Pythagorean community, which resulted in a conflict and led to his expulsion from the group. 

In some versions of the story, Hippasus made this discovery while he was on a ship. When he told the other cult members about his discovery, he was thrown overboard.

The exact reasons for his expulsion or murder are unclear, but it has been suggested that the Pythagoreans considered the existence of irrational numbers as a threat to their philosophical framework. 

The discovery of irrational numbers challenged the Pythagoreans’ insistence on the exclusivity of rational numbers and whole-number ratios as the foundation of the universe. While their specific objections to irrational numbers are not explicitly documented, it is believed that they viewed them as disruptive to the order and harmony they sought in the numerical structure of reality.

The fact that the square root of two is irrational can be easily proven using middle school mathematics. There are many such proofs available online, and I’ll leave that as a homework assignment as it is difficult to provide audio narration for mathematical proofs.

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.

Over in India, they didn’t have the hangups that the Greeks had. 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 Aryabhata was using trigonometry and the sine function to determine astronomical distances. 

The 9th-century Persian mathematician al-Khwarizmi, who seems to make an appearance in almost every episode 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 of them.

Europeans finally accepted irrational numbers in the middle ages when it became clear that they were the solutions to algebraic equations.

The thing which really cemented the acceptance of irrational numbers was a means of expressing them. The Pythagoreans had no way of expressing such numbers as they had no means of writing decimal numbers and no mathematical symbols. They mostly used 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 Simon Stevin. 

However, 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 Al? al-Qalas?d?. He may have taken it from the third letter of the Arabic alphabet, which happens to be the first letter in the Arabic word for root.

It also has 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 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 which are noteworthy. 

The first is the previously mentioned square root of two ?2. 

If you have a square with each side having a length of one, then the diagonal of the square will be the square root of two. 

The square root of two has an approximate value of 1.4142135…

Another core irrational number, which I have previously done an episode on, is pi. 

Pi is the circumference of a circle divided by its diameter. 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. 

Another irrational number that pops up everywhere, even in things like art and nature, is phi. 

? Phi is the Greek symbol used to express the Golden Ratio. The golden ratio is an irrational number with a value approximately equal to 1.61803…. 

It can also be expressed algebraically as (1+?5)/2.

The Golden Ratio has been considered aesthetically pleasing and appears in various works of art and architecture. The golden ratio also appears in nature, such as in the proportions of plants, animals, and even the human body.

Phi and the Golden Ratio is an interesting enough subject to warrant its own episode in the future. 

Finally, perhaps the most important irrational number is known as Euler’s Number, or e. 

The value of e is approximately 2.718281828459045….

E is the number that sits behind exponential growth and radioactive decay. It has a special place in calculus, as ex is the only function whose derivative is itself. To that extent, it is the calculus equivalent of multiplying by one, or adding zero.

Despite the name, irrational numbers are, in fact, very rational and very real. Without them, we couldn’t do even the most basic mathematics that makes the world function today.

It’s just too bad about what happened to Hippasus. The Pythagoreans really overreacted to his discovery. In fact, you could say they acted very….. irrational.

The Executive Producer of Everything Everywhere Daily is Charles Daniel.

The associate producers are Thor Thomsen and Peter Bennett.

Today’s review comes from listener Henr3yuuuuu 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 – 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 believed 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! You earned my 5 stars with full honors! Hendrik from Brussels

Thanks, Hendrik! 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.  

On top of that, I also have separate ancestors who came from Holland and from Luxembourg, making me one of what I can only assume is a very small number of Benelux Americans. 

Remember, if you leave a review or send me a boostagram, you too can have it read on the show.