Understanding Rational and Irrational Numbers

By the end of this video, you're easily going to be able to tell which numbers are rational and which numbers are irrational. And in about 30 seconds, here's how this video is going to get you there. We're going to start off with some quick definitions here. I'll define what a rational number is, and we'll define what an irrational number is. For those definitions, though, we're going to need to know what an integer is, and so I'm going to define that as well. After we're done with our definitions, I'm going to give you some examples. of rational numbers. I'll give you a bunch of examples here and then of course I'll give you plenty of examples of what rational numbers are not. So what irrational numbers are. And after we go through all that I'll give you some problems to try and answer in the comments to make sure that you really got this but honestly after this video you definitely will. And if you're looking for the notes for this video see if something to review after the link is going to be right in the description. So yeah let's get into this video. So let's start off with these definitions here. Now a rational number is any number that can be expressed as a ratio of two integers. And that's going to mean nothing to you if you don't know what an integer is. So we're going to go over that. The only thing I want to say before we get into one integer is, is that an irrational number that's basically just not a rational number. It's something that you cannot express as a ratio of two integers. So now that we've got that out of the way, let's get back to the idea of, well, OK, what's an integer so we can start understanding this whole ratio business? An integer is any number that you regularly see on a number line. So it's 0, 1, negative 1, negative 2, 2. These are integers and they go in both directions forever. But it's none of the numbers in between. So it's just these numbers and going in both directions. Okay, so those are what integers are. Now, when we talk about a ratio of two integers, what that means is that it's a fraction with an integer in the numerator and an integer in the denominator. So an example would be, OK, we have a fraction. We need to put an integer in the numerator, integer in the denominator, 4 over 9. What about negative 2 over 5? That still works. If we have something like 3 over 2, that works. It's still an integer in the numerator, an integer in the denominator every single time. And that is a ratio of two integers. And you can remember that that's a rational number, because it literally is the word ratio in it. So in summary, a rational number is any number that you can write like this. As a fraction with the integer in the numerator and integer in the denominator and now we can actually go through some examples where i give you both some rational numbers and some irrational numbers so again a rational number ratio of two integers four over nine is one of them but also eight is a rational number because you can write that as eight divided by one eight over one and now you have an integer in the numerator and integer in the denominator so that's a rational number Some other things that are rational numbers are terminating decimals. 0.752 is a rational number and any terminating decimal will be a rational number. Now a terminating decimal is one that terminates, it's one that stops, it doesn't go on forever. And any terminating decimal we can write that still as a ratio of two integers. we can write 0.752 as 752 over a thousand and you can check you can put that in your calculator you get 0.752 so any terminating decimal we can write as a ratio of two integers and also if the decimal doesn't terminate that doesn't mean that it's an irrational number if it repeats if it goes on forever but it repeats it's still a rational number if we have 0.3 repeating we can write that 0.3 with a little bar right here to denote that the 3 is repeating. This is actually a rational number still. So I'll write repeating decimal and 0.3. Well, 0.3 repeating is just one third. Again, it's a ratio where you have an integer in the numerator and an integer in the denominator. So any repeating decimal is a rational number. Now there's other forms of repeating decimals that you could see. You could see something like you know one two five one two five one two five as long as there's that part that's repeating like this the one two five is the thing that's repeating here it still counts as a repeating decimal so you could write this as 0.125 repeating now i should also note that with that if you saw something like maybe you're curious about this 0.2353535 where the two part isn't repeating but the three five is this is still a rational number and to prove it you can write it as 0.2 plus 0.0353535 so you have a terminating decimal here which you already know that you can write as a ratio of two integers and you have a repeating decimal which again we talked about you can write that as a ratio of two integers so this still works fine and you can count it as a repeating decimal And so now that we've talked a bunch about what rational numbers are, well, what aren't rational numbers? What are they not? Well, we've said that rational numbers are decimals that terminate and decimals that repeat. But what happens if they do neither of those? What happens if they go on forever and they don't repeat? Well, now we're starting to talk about irrational numbers. And the most famous example of an irrational number is pi. Pi is I really hope I don't mess this up for a YouTube video. 3.14159 just keeps going. And that decimal, it doesn't terminate, it keeps going on. And it also doesn't repeat. And those are the qualifications for an irrational number. Another example of an irrational number is E. It's another pretty famous number. And this is 2.71828. And it goes on, it doesn't terminate, and it doesn't repeat. So it's another example of an irrational number. Now another example of an irrational number is the square root of two. And this, if you have a square root that you can't fully get rid of, like if you can't simplify this to where the square root is gone, then it is an irrational number, it's going to be another decimal that doesn't end, and it doesn't repeat. And so there you go. You can see what the square root of 2 is. That's an irrational number. But not all square roots are irrational numbers. If you had something like the square root of 16, that you can simplify to where you get the square root to be gone. The square root of 16 is 4. And we can write that as 4 over 1. We can write it as a ratio of two integers. So that's a rational number, and that works. You could even have something like 4 minus the square root of 25 over 4. It looks pretty gross, so you might assume that it's an irrational number, but if we simplify this a little bit, we can write this, we can split apart that square root and do the square root of 25 over the square root of 4. Well, that's just 4 minus the square root of 25 is 5, and the square root of 4 is 2. So now we have a rational number minus another rational number, and that is going to be a rational number. To prove that, you could find common denominators and combine this all into one fraction. So those are rational numbers, but you got to be careful here because if we add something like 7 plus the square root of 8, we can try to simplify the square root of 8. And I have a whole video where I go through how to simplify square roots. So you can check that out if you want. But we can't simplify the square root of 8 down to where the square root is gone. Where 8 isn't a perfect square. It's not one of these special numbers 1, 4, 9, 16, 25. and so on. The square root of these numbers are nice things that square roots are going to be one, two, three, four, five, all that. But eight doesn't fall into that list. And we're not going to be able to fully get that square root to go away when we try to simplify. What we end up getting is like nine point eight two eight four. So on it just it's really gross. And that's even though that we were adding a rational number to it, it doesn't matter. As soon as we add an irrational number, it messes the whole thing up. We have an irrational number. So that's the difference between rational numbers and irrational numbers. And with all that being said, if you feel pretty comfortable with this, then here is a problem for you to try and answer in the comments. And actually, I'm giving you five separate numbers, and I want you to tell me in the comments whether each of them are rational or whether they're irrational. And so try that out. Let me know your answers in the comments. And if you have any questions on anything we talked about in this video, again, let me know in the comments and I'll try to get back to you when I can. Now again, I do have a printable version of these notes that I made and if you want to snag that by the way It's free. The link is in the description. And lastly make sure that you're subscribed to this YouTube channel Look, we're trying to get to 100k right now And I'm trying to just help that by recording as many videos as possible And I'm still trying to keep this quality the entire time. So I'm trying to balance that quality versus quantity thing But yeah, so we're trying to get to 100k It would really help if you subscribe and if you know anybody else that might need help with this topic I'd appreciate if you could share the video as well, but yeah that's gonna do for this video guys and i'll see you soon

For those definitions, though, we're going to need to know what an integer is, and so I'm going to define that as well. After we're done with our definitions, I'm going to give you some examples. of rational numbers. I'll give you a bunch of examples here and then of course I'll give you plenty of examples of what rational numbers are not.

So what irrational numbers are. And after we go through all that I'll give you some problems to try and answer in the comments to make sure that you really got this but honestly after this video you definitely will. And if you're looking for the notes for this video see if something to review after the link is going to be right in the description. So yeah let's get into this video.

So let's start off with these definitions here. Now a rational number is any number that can be expressed as a ratio of two integers. And that's going to mean nothing to you if you don't know what an integer is.

So we're going to go over that. The only thing I want to say before we get into one integer is, is that an irrational number that's basically just not a rational number. It's something that you cannot express as a ratio of two integers. So now that we've got that out of the way, let's get back to the idea of, well, OK, what's an integer so we can start understanding this whole ratio business? An integer is any number that you regularly see on a number line.

So it's 0, 1, negative 1, negative 2, 2. These are integers and they go in both directions forever. But it's none of the numbers in between. So it's just these numbers and going in both directions.

Okay, so those are what integers are. Now, when we talk about a ratio of two integers, what that means is that it's a fraction with an integer in the numerator and an integer in the denominator. So an example would be, OK, we have a fraction.

We need to put an integer in the numerator, integer in the denominator, 4 over 9. What about negative 2 over 5? That still works. If we have something like 3 over 2, that works.

It's still an integer in the numerator, an integer in the denominator every single time. And that is a ratio of two integers. And you can remember that that's a rational number, because it literally is the word ratio in it. So in summary, a rational number is any number that you can write like this. As a fraction with the integer in the numerator and integer in the denominator and now we can actually go through some examples where i give you both some rational numbers and some irrational numbers so again a rational number ratio of two integers four over nine is one of them but also eight is a rational number because you can write that as eight divided by one eight over one and now you have an integer in the numerator and integer in the denominator so that's a rational number Some other things that are rational numbers are terminating decimals.

0.752 is a rational number and any terminating decimal will be a rational number. Now a terminating decimal is one that terminates, it's one that stops, it doesn't go on forever. And any terminating decimal we can write that still as a ratio of two integers.

we can write 0.752 as 752 over a thousand and you can check you can put that in your calculator you get 0.752 so any terminating decimal we can write as a ratio of two integers and also if the decimal doesn't terminate that doesn't mean that it's an irrational number if it repeats if it goes on forever but it repeats it's still a rational number if we have 0.3 repeating we can write that 0.3 with a little bar right here to denote that the 3 is repeating. This is actually a rational number still. So I'll write repeating decimal and 0.3.

Well, 0.3 repeating is just one third. Again, it's a ratio where you have an integer in the numerator and an integer in the denominator. So any repeating decimal is a rational number.

Now there's other forms of repeating decimals that you could see. You could see something like you know one two five one two five one two five as long as there's that part that's repeating like this the one two five is the thing that's repeating here it still counts as a repeating decimal so you could write this as 0.125 repeating now i should also note that with that if you saw something like maybe you're curious about this 0.2353535 where the two part isn't repeating but the three five is this is still a rational number and to prove it you can write it as 0.2 plus 0.0353535 so you have a terminating decimal here which you already know that you can write as a ratio of two integers and you have a repeating decimal which again we talked about you can write that as a ratio of two integers so this still works fine and you can count it as a repeating decimal And so now that we've talked a bunch about what rational numbers are, well, what aren't rational numbers? What are they not?

Well, we've said that rational numbers are decimals that terminate and decimals that repeat. But what happens if they do neither of those? What happens if they go on forever and they don't repeat? Well, now we're starting to talk about irrational numbers. And the most famous example of an irrational number is pi.

Pi is I really hope I don't mess this up for a YouTube video. 3.14159 just keeps going. And that decimal, it doesn't terminate, it keeps going on. And it also doesn't repeat. And those are the qualifications for an irrational number.

Another example of an irrational number is E. It's another pretty famous number. And this is 2.71828. And it goes on, it doesn't terminate, and it doesn't repeat. So it's another example of an irrational number.

Now another example of an irrational number is the square root of two. And this, if you have a square root that you can't fully get rid of, like if you can't simplify this to where the square root is gone, then it is an irrational number, it's going to be another decimal that doesn't end, and it doesn't repeat. And so there you go. You can see what the square root of 2 is. That's an irrational number.

But not all square roots are irrational numbers. If you had something like the square root of 16, that you can simplify to where you get the square root to be gone. The square root of 16 is 4. And we can write that as 4 over 1. We can write it as a ratio of two integers. So that's a rational number, and that works.

You could even have something like 4 minus the square root of 25 over 4. It looks pretty gross, so you might assume that it's an irrational number, but if we simplify this a little bit, we can write this, we can split apart that square root and do the square root of 25 over the square root of 4. Well, that's just 4 minus the square root of 25 is 5, and the square root of 4 is 2. So now we have a rational number minus another rational number, and that is going to be a rational number. To prove that, you could find common denominators and combine this all into one fraction. So those are rational numbers, but you got to be careful here because if we add something like 7 plus the square root of 8, we can try to simplify the square root of 8. And I have a whole video where I go through how to simplify square roots. So you can check that out if you want.

But we can't simplify the square root of 8 down to where the square root is gone. Where 8 isn't a perfect square. It's not one of these special numbers 1, 4, 9, 16, 25. and so on.

The square root of these numbers are nice things that square roots are going to be one, two, three, four, five, all that. But eight doesn't fall into that list. And we're not going to be able to fully get that square root to go away when we try to simplify.

What we end up getting is like nine point eight two eight four. So on it just it's really gross. And that's even though that we were adding a rational number to it, it doesn't matter.

As soon as we add an irrational number, it messes the whole thing up. We have an irrational number. So that's the difference between rational numbers and irrational numbers. And with all that being said, if you feel pretty comfortable with this, then here is a problem for you to try and answer in the comments.

And actually, I'm giving you five separate numbers, and I want you to tell me in the comments whether each of them are rational or whether they're irrational. And so try that out. Let me know your answers in the comments. And if you have any questions on anything we talked about in this video, again, let me know in the comments and I'll try to get back to you when I can.

Now again, I do have a printable version of these notes that I made and if you want to snag that by the way It's free. The link is in the description. And lastly make sure that you're subscribed to this YouTube channel Look, we're trying to get to 100k right now And I'm trying to just help that by recording as many videos as possible And I'm still trying to keep this quality the entire time. So I'm trying to balance that quality versus quantity thing But yeah, so we're trying to get to 100k It would really help if you subscribe and if you know anybody else that might need help with this topic I'd appreciate if you could share the video as well, but yeah that's gonna do for this video guys and i'll see you soon

Transcript for:Understanding Rational and Irrational Numbers

Transcript for:
Understanding Rational and Irrational Numbers