Have you ever thought that irrational numbers are called irrational because they’re insane? Well they’re not. They’re actually really cool and really amazing. To understand what an irrational number is, it helps to know what a rational number is. A rational number is any number that can be written as a ratio of two integers. Does that sound familiar? Yup, a ratio of two integers is a fraction, like 1/2 or 2/3. So any number that can be written as a fraction is a rational number. But can’t all numbers be written as a fraction? Surprisingly the answer is no. Sure, any whole number can be written as a fraction by giving it 1 as the denominator. And any decimal number that you can write, can be written as a fraction by having the proper power of 10 as the denominator. So all of these are rational numbers. And there are a lot more… infinitely more. Any number that can be accurately written as a fraction or a decimal number is a rational number. But there are numbers that can’t be written accurately as a fraction, or even as a decimal number. Those numbers are called irrational numbers because they are not rational numbers. One of the most famous irrational numbers is pi. And you might be confused because you may have heard that you can represent pi with a fraction like 22/7 or 355/113, but those are just approximations for pi. Which means they get close to the value of pi but they are not exactly equal. You can’t get the exact value of pi with a fraction no matter what numbers you use. And you’ve probably seen pi written as a decimal number like 3.14 or 3.14159 but those aren’t completely accurate values of pi either. Again, they’re approximations. If you tried to write a completely accurate decimal number for pi or any other irrational number, the decimal digits would never end and won’t repeat. The statement that the decimal digits would never end might be hard to believe, but it’s true they just keep going and going—forever. But the statement that the decimal digits won’t repeat might be hard to understand. It might even make you think something that’s not true. You might think it means that you could never have 2 of the same digit next to each other. Or you might think that a particular pattern of digits, like 123, won’t appear more than once. Or that it at least won’t be followed immediately by the same pattern. But that’s not what it means. To see that, let’s take a look at the first 500 digits of pi. There are many places where there are 2 or more of the same digit next to each other. And here’s the pattern 360 and then a little further along we have 360 again. And look at this, here we have the digits 209 followed immediately by 209! So if that’s not what’s meant by the decimal digits not repeating then what is? Well, remember that fraction that’s often used to approximate pi? 22/7? Let’s look at the first 500 digits of that number. Notice how the sequence ‘142857’ keeps repeating? Well, it keeps on going like this, repeating that same sequence forever. The digits of a rational number either end or keep repeating like this. And because of that, you can know exactly what all of those decimal digits are, even when they go on forever. Since this sequence of 6 digits repeats, you already know what the 501st digit is going to be: ’4’. But for pi, you don’t know what the 501st digit is going to be until you calculate it. So one way to understand the difference between rational and irrational numbers is that the digits of a rational number can be completely known, but an irrational number always has more digits that you don’t know… they’re kind of mysterious that way! There’s another cool thing about irrational numbers that I want to show you. In our video lesson about the Number Line we learned that you can subdivide the space between two consecutive marks on the number line. And you can subdivide the smaller space between two of those new marks. And you can keep doing that, forever! So let’s look at the value of pi on the number line. It’s right about here, between the 3 and the 4. And now let's zoom in. As we zoom in we keep dividing the spaces into smaller and smaller spaces. It might seem that if we kept zooming in, eventually the value of pi would line up exactly with one of our marks. But it won’t. Since pi is an irrational number, even if we could zoom in forever, we would never get to a point where pi exactly lines up with a mark on the number line. It might get really close, but as we keep zooming in we’ll see that it doesn’t quite line up and it never will. Pretty cool, huh? There’s one last thing I want to tell you about irrational numbers. It might seem like there are just a few special numbers that happen to be irrational, but in reality, there are infinitely many irrational numbers. In fact there are more irrational numbers than there are rational numbers! I don’t know about you, but I find that amazing!! Alright, hopefully you now understand irrational numbers a bit better. And you realize they aren’t called irrational because they’re insane, they’re called irrational because they are not rational. They can’t be represented as a ratio of two integers. So they can’t be accurately represented as a fraction. And their decimal digits go on forever and don’t end in a repeating pattern. Alright, that’s it for this video. Keep practicing and I’ll catch you in the next one. And for better access to all of our material, check out mathantics.com