What are irrational numbers?

0:47 Welcome to Maths Is Not Boring by kidopoly.com. I'm Mira! And I'm Finn! Before we jump into today's mystery, we have some special mentions. A huge happy belated ninth birthday to Finn in Leeds.

1:03 Hey, another Finn! Happy birthday! I hope your day was absolutely brilliant. And we also have a big hello to five-year-old Marvin in Massachusetts, who loves listening to all the shows, especially Maths and Science.

1:16 We are so incredibly glad you're both listening. Oh, and if you haven't already, tap follow wherever you're listening so you never miss an episode.

1:26 So, since Marvin loves science and maths so much, what exactly are we looking at today?

1:32 Well, Marvin and Finn in Leeds, and Finn right here, we are looking at an idea so wild, it literally shocked the ancient world. I want you to picture something really ordinary. Think of a perfectly normal flat square tile on a bathroom floor, or maybe a single square on a checkerboard.

1:52 Okay, I am picturing it. Just a standard square. Four equal sides, four normal corners.

1:59 Exactly. Now, imagine taking a ruler and drawing a perfectly straight line from one corner, straight across the middle, all the way to the opposite corner. That line is called the diagonal. That's easy.

2:13 I draw diagonals all the time when I'm sketching.

2:16 It looks so simple, but if you try to measure the outside edge of that square and then measure that diagonal perfectly, you tumble into a mathematical mystery that people have wondered about for 2,000 years.

2:30 Wait, a mystery? From just drawing a line on a square? How can a simple straight line be a mystery?

2:36 That is exactly what makes it so amazing. Think about how we normally measure things in everyday life. We use a ruler, right? And if the thing we are measuring doesn't perfectly match up with a whole number, what do we use instead?

2:51 Well, we use fractions. Like, if a piece of wood is smaller than an inch, we might say it's a half or a quarter, or maybe three-eighths of an inch.

3:01 Right. We just use a fraction, which is one whole number placed over another whole number. And about 2,500 years ago, around 500 BCE in ancient Greece, there was a group of thinkers called the Pythagoreans.

3:17 The Pythagoreans? That sounds like a secret society or a mysterious club.

3:23 They actually kind of were. They loved numbers more than anything else in the world. In fact, they truly believed that absolutely every single length in the entire universe could be written as a perfect, neat fraction.

3:37 So they thought the whole world was built out of neat little mathematical building blocks. Like, everything could be perfectly measured if you just found the right fraction.

3:47 Yes. They thought mathematics was this perfectly tidy set of rules. They believed everything in the universe could be pinned down exactly, with one whole number over another. But then, they looked at that simple, ordinary square and its diagonal.

4:04 Let me guess. The diagonal wasn't playing by their neat and tidy rules.

4:09 Not at all. Let's imagine the side of our square is exactly one unit long. It could be one inch or one metre, it doesn't matter. The diagonal cutting across from corner to corner is longer than one, obviously. So the Pythagoreans tried to find the exact fraction for it.

4:27 I mean, that makes sense. You just measure it and find the right fraction. If they couldn't find the fraction at first, they just needed to keep chopping their measuring ruler into smaller and smaller pieces, right? Like tenths or thousandths or even millionths.

4:42 That is exactly what they thought they could do. But here is the wild part. No matter how tiny they chopped up their ruler, no matter how small they made their fractions, the side of the square and the diagonal never, ever fit a whole number of times each.

4:59 Wait, hold on. That can't be right. There has to be some tiny fraction that measures it exactly. You just have to go small enough, maybe billionths.

5:09 That is the incredible jolt of this idea. It has been proven true. No fraction in the entire universe can ever name that diagonal exactly. It is completely impossible. If the side of the square is one, the diagonal cannot be written as one whole number over another.

5:29 What? But the line is right there! I can see it. I can draw it with a pencil on my paper. How can it not have an exact fraction to name it?

5:39 Because the number that measures that diagonal goes on forever without repeating. Its decimal starts out as 1.41421, and it literally never stops. It just keeps generating new numbers, and it never settles into a repeating pattern.

5:57 Wait. 1.41421? And it just keeps going? Like into infinity. So you can never actually write down the full complete number.

6:08 Exactly. You can get closer and closer and closer to it, but you can never, ever write the absolute exact number down on a piece of paper, because you would run out of time and space.

6:21 So the Pythagoreans just looked at a totally normal square and found a wild, never-ending number hiding right there on the floor. Yes, they did!

6:30 And because this kind of number cannot be written as a neat, tidy ratio of whole numbers, it has a special mathematical name. It is called an irrational number.

6:41 Irrational numbers! I love that name. It sounds like a number that is just refusing to listen to reason or follow the rules.

6:50 It basically just means no ratio. But honestly, it really did seem totally crazy to the Pythagoreans. A later story says that a Pythagorean named Hippasus, who lived in Greece around 500 BCE, was the person who finally figured this out.

7:07 Hippasus? Wow! How did the rest of the Pythagorean club feel about him finding an irrational number that broke all their rules?

7:16 Well, the story goes that it completely shocked them. It basically shattered their whole comforting belief that the universe was made entirely of neat, tidy fractions and perfect whole numbers.

7:28 I mean, I kind of get why they were upset. It is pretty dizzying to think that a simple diagonal line you can trace with your finger actually holds a decimal that goes on into infinity.

7:39 It really is dizzying. And you know, nobody truly knows the full factual story of Hippasus' life, or exactly how the discovery happened day by day. But the idea itself, the idea that some length in our world just cannot ever be caught by fractions, that changed mathematics forever.

7:58 It's like finding a glitch in the universe. Hippasus looked at a square, saw the wildness hiding inside it, and just broke the rules of how people thought the world worked.

8:08 That is a brilliant way to put it. And the most amazing thing is that Hipposus' diagonal isn't the only irrational number out there. Pi is another famous irrational number too, though Pi has had its very own episode on this show.

8:24 Oh yeah, I remember Pi. So there are actually lots of these wild, never-ending numbers out there in the world. They are everywhere!

8:32 In fact, Finn, here is a mind-bending fact for you. You hold this exact wild irrational number from our square in your hands all the time, without even knowing it.

8:45 I do. I don't remember holding any never-ending decimals lately. Where would I even find one?

8:52 Do you ever draw or paint or write on a regular sheet of A4 printer paper?

8:58 Yeah, of course I do. We have huge stacks of A4 paper at my house, and at school. I use it every single day.

9:06 Well, the long side of a sheet of A4 paper is exactly the short side multiplied by that very same irrational number, the square root of two.

9:16 Wait, really? The standard printer paper is based on an irrational number?

9:21 Yes. The inventors designed it that way on purpose. Because of that special mathematical ratio, if you fold a piece of A4 paper exactly in half, the new smaller rectangle you make has the exact same shape as the big one.

9:36 It's just smaller. No way. So that wild, impossible number is actually what makes the paper perfectly foldable forever. That is so cool. It's like a superpower for paper.

9:50 That is the true beauty of it. What totally shocked Hipposus 2500 years ago is now sitting quietly on your desk, helping you do your homework. Alright, Finn, after all those mind-bending facts, it is time for the quiz.

10:05 Okay, I'm ready. Let's see if I caught all the details about our wild numbers.

10:11 First question. What ancient Greek group believed absolutely every length in the universe could be named by a neat fraction? Next one. What simple, everyday shape did they look at to find the mysterious, unmeasurable diagonal line? Question three. Who does a later story say first discovered this impossible diagonal?

10:47 Here's number four. What is the real mathematical name for numbers like this that cannot ever be written as a neat fraction? Last question. What everyday type of standard paper uses this exact irrational number in its shape? Great job! Let's go through the answers and see how you did. The first answer is the Pythagoreans.

11:19 For question two, they looked at a totally normal square. Number three, the answer is Hipposus, who lived around 500 BCE in Greece. Question 4, they are called irrational numbers. And the final answer is A4 paper.

11:39 I am seriously never going to look at printer paper or bathroom floor tiles the same way again. It is so wild to think there is literally infinity hiding right there in the corners of a square.

11:54 That is exactly how mathematics works. Once you start to see the strange, wild patterns hiding in ordinary things, the whole world looks completely different.

12:05 It's like you've been handed a secret map. And it makes you wonder what other ordinary things have wild secrets hiding inside them.

12:14 Exactly! If you loved learning about the wild, irrational numbers hiding in a simple square, please leave us a five-star review.

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12:45 See you next time on Maths Is Not Boring.