Irrational Numbers Explained | Math with Mr. J

Welcome to Math with Mr. J. In this video, I'm going to cover what irrational numbers are. Now, irrational numbers are numbers that cannot be written as a fraction of two integers. The decimal value, the decimal form of an irrational number does not terminate or repeat. This will make a lot more sense as we go through our examples. Now, in order to understand irrational numbers, we need to understand rational numbers. So let's go through a quick recap here of rational numbers. The link to my full video on rational numbers is in the description if you'd like a more detailed explanation and more examples. So taking a look at number one, we have 12, 9 tenths, and 0.2 repeating. These are all rational numbers. We can write them as fractions. Fractions of two integers. And the decimal form of a rational number either terminates or repeats. So for 12, how can we write this as a fraction of two integers? Well, we can write this as 12 over 1. That equals 12. About 24 over 2. That equals 12. And so on. So we can write 12 as a fraction of two integers. So it is rational. And the same thing goes for negative 12. So it doesn't matter if we have a negative. This is still rational. We can write this as negative 12 over 1. And so on. So negative 12 is also rational. Let's move on to 9 tenths. Now, this is a terminating decimal. So we can see that this is rational right away. But as far as writing this as a fraction of two integers, we can write this as 9 over 10. Or even something like 36 over 40. And so on. So we can write this as a fraction of two integers. 9 tenths is rational. And then lastly, we have 0.2 repeating. This bar over the 2 means we have 0.2222. And this continues on forever. Now, any repeating decimal is rational. It can be written as a fraction of two integers. For example, 0.2 repeating can be written as 2 over 9. Or 6 over 27. And so on. Let's move on to irrational numbers. And we will start with the square root of 2. Now, the square root of 2 is approximately. So let's put approximately. And I say approximately because this decimal does not terminate or repeat. So we can't write out the full decimal. So the square root of 2 is approximately 1.44213562. And this continues on. Again, it does not terminate or repeat. So this is irrational. There's no way to write this as a fraction of two integers. Now, you may be thinking. Well, let's just write it as the square root of 2 over 1. Yes, it is a fraction. But it has to be a fraction of two integers. The square root of 2 is not an integer. So again, we cannot write the square root of 2 as a fraction of two integers. And the decimal doesn't terminate or repeat. This is actually going to be the case for taking the square root of anything that is not a perfect square. So the square root of any non-perfect square is irrational. Let's move on to number 3 where we have 4.89897948. And then that decimal continues on. It does not terminate or repeat. So this is irrational. We cannot write this as a fraction of two integers. Now, this decimal is actually what we get when we plug in the square root of 24 into a calculator. Next, for number 4, we have pi, which is a very famous irrational number that we use when working with circles. Pi is the ratio of a circle's circumference to its diameter. So the circumference divided by the diameter. Pi is the same for every circle. Now, pi is approximately 3.1415926, and this continues on forever without repeating. Now, we cannot write pi as a fraction of two integers. Pi is irrational. Now, it's very common to use 3.14 for pi, but that's just an approximation. Let's move on to number 5 where we have the number e, also called Euler's number, named after the mathematician Leonard Euler. This is another famous irrational number that shows up throughout math with things like exponential growth, compound interest, probability, among other applications. The number e is approximately 2.71828182845, and this continues on forever without repeating. So we can't write the number e as a fraction of two integers. It's irrational. Now, let's move on to numbers 6 through 10 where we will determine if we have something that is rational or irrational. For number 6, we have negative 63. This is rational. We can write this as a fraction of two integers. For example, negative 63 over 1. So negative 63 over 1. Again, this is rational. Next, for number 7, we have 3.10723 and that decimal continues on without terminating or repeating. This is irrational. We can't write it as a fraction of two integers. This is actually the decimal that we get for the cube root of 30. So this is irrational. Next, let's move on to number 8 where we have the square root of 49. Now, 49 is a perfect square. So the square root of 49 equals 7. So the square root of 49, rational. Moving on to number 9, we have 8.71425. Now, above 71425, we have a bar. That shows us that that decimal repeats. And again, any repeating decimal is rational. So this is rational. Lastly, let's move on to number 10 where we have the square root of 50. 50 is not a perfect square. So this is not going to work out to be a rational number. The square root of 50 is approximately 7.07106 and this continues on forever without terminating or repeating. So the square root of 50 is irrational. So there you have it. There's a basic explanation of irrational numbers. I hope that helped. Thanks so much for watching. Until next time, peace.