About this video
Sal sorts a list of real numbers into rational and irrational. sqrt(8)/2 is irrational because 8 isn't a perfect square, and pi never ends or repeats. 5.0, 0.325, 7.777... and 8 1/2 are all rational because each can be written as a ratio of two integers.
You'll be able to
- decide whether a number is rational or irrational.
- explain why square roots of non-perfect squares are irrational and why repeating decimals are rational.
Key words
- irrational number
- A number that can't be written as a ratio of two integers.
- rational number
- A number that can be written as one integer over another.
- perfect square
- A number that is an integer times itself, like 9 = 3 × 3.
Transcript
Which of the following real numbers are irrational? Well, irrational just means it's not rational. It means that you cannot express it as the ratio of two integers. So let's see what we have here. So we have the square root of 8 over 2. If you take the square root of a number that is not a perfect square, it is going to be irrational.
And then if you just take that irrational number and you multiply it, and you divide it by any other numbers, you're still going to get an irrational number. So square root of 8 is irrational. You divide that by 2, it is still irrational. So this is not rational.
Or in other words, I'm saying it is irrational. Now, you have pi, 3.14159-- it just keeps going on and on and on forever without ever repeating. So this is irrational, probably the most famous of all of the irrational numbers. 5.0-- well, I can represent 5.0 as 5/1.
So 5.0 is rational. It is not irrational. 0.325-- well, this is the same thing as 325/1000. So I can clearly represent it as a ratio of integers. So this is rational. Just as I could represent 5.0 as 5/1, both of these are rational. They are not irrational. Here I have 7.777777, and it just keeps going on and on and on forever.
And the way we denote that, you could just say these dots that say that the 7's keep going. Or you could say 7.7. And this line shows that the 7 part, the second 7, just keeps repeating on forever. Now, if you have a repeating decimal-- in other videos, we'll actually convert them into fractions-- but a repeating decimal can be represented as a ratio of two integers.
Just as 1/3 is equal to 0.333 on and on and on. Or I could say it like this. I could say 3 repeating. We can also do the same thing for that. I won't do it here, but this is rational. So it's not irrational. 8 and 1/2? Well, that's the same thing. 8 and 1/2 is the same thing as 17/2.
So it's clearly rational. So the only two irrational numbers are the first two right over here.
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