Video: Ordering Rational and Irrational Numbers

Without using a calculator, order the numbers 3𝜋, 3√2, 𝜋², 9 and 3√3 from smallest to largest.

03:15

Video Transcript

Without using a calculator, order the numbers three 𝜋, three root two, 𝜋 squared, nine, and three root three from smallest to largest.

So to enable us to order our numbers from smallest to largest, we’re actually going to have to find out the values of some of them. Well, first of all, we have three 𝜋. And three 𝜋 is going to be equal to three multiplied by 3.14 et cetera because that’s the value of 𝜋.

Well, next, we have three root two. And what three root two means is three multiplied by root two. But if we consider the value of root two, well root two is gonna be greater than root one and less than root four. I’ve chosen these two values because they’re both square numbers. So therefore, we know that root two is going to be between one and two because it’s gonna be greater than one cause the square root of one is one. But it’s gonna be less than two because the square root of four is two.

Well, next, we have 𝜋 squared which is 3.14 multiplied by 3.14, if we round 𝜋 to two decimal places. Then, there’s nine, which is three multiplied by three and then finally three root three, which is three multiplied by root three.

And reminding ourselves of the same method as we used to find out where root two would lie, we could say that root three, well it must be bigger than root two, but it’s gonna be less than two and that’s because it’s gonna be less than root four and root four is two.

So now, we’re ready to order our numbers, remembering the question has asked us to order them from smallest to largest. Well, our smallest number is gonna be three root two and that’s because it’s three multiplied by root two. And we already know that root two is between one and two, but it’s also less than root three.

And if we look at each of the other numbers, they’re actually multiplied by a number that’s greater than two, except for the one which is three root three, but then we know already that root three is gonna be greater than root two.

Next, we have three root three. And that’s because as we explained, it’s three multiplied by a number that’s less than two and all of the others are multiplied by numbers that’s greater than two.

Next, we have nine and that’s because nine is three multiplied by three. And the other two values we have are three multiplied by 3.14 et cetera, which is gonna be bigger and also 3.14 multiplied by 3.14, again which will be greater.

Then, next, we’re gonna have three 𝜋. And that’s for the reason that three 𝜋 is greater than nine because it’s three multiplied by 3.14 that’s been rounded, but it carries on. But it’s less than 3.14 squared.

So then, we’ll have the largest number, which is 𝜋 squared.

So therefore, we can say that without using a calculator, we’ve ordered the numbers three 𝜋, three root two, 𝜋 squared, nine, and three root three from smallest to largest. And they are three root two, three root three, nine, three 𝜋, and 𝜋 squared.

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