Without using a calculator, order the numbers 𝜋, root two, one, two, and root three from smallest to largest.
Now, in order to put our numbers in order without a calculator, what we need to do is find the numerical values of three of our numbers. And that’s 𝜋, root two, and root three. Well, first of all, we have 𝜋. Well, we know that 𝜋 is equal to 3.14 et cetera. So that’s the value of 𝜋. So we know that one.
Well, next, we need to consider root two. Well, to enable us to consider root two, what we can consider is the two square numbers either side of root two. So we’ve got one and four. Now, if we take the root of one and the root of four, we get the answers one and two. So therefore, we can say that root two is gonna be between one and two. So we can say that root two is greater than one, but less than two.
And then again, we can use the same principle for root three because root three must be less than two. And that’s because root four is equal to two, but it also must be greater than root two because the square root of three is gonna be a larger number than the square root of two.
So now, what we can do is use these properties or these estimates for our values to help us to order the numbers. So we want to order them. When looking at the question from smallest to largest, so therefore our smallest number is going to be one because we know that root two is greater than one.
Then, our next value is going to be root two. And that’s because we know that root two is greater than one, but less than root three. Then, our next value will be root three cause we’ve shown that root three is greater than root two, but less than two. Then, we have two because again we’ve shown that two is greater than root three. But two is less than 𝜋 because it’s less than 3.14. So therefore, our final value is 𝜋.
So we can say that without using a calculator, we’ve ordered the numbers 𝜋, root two, one, two, and root three from smallest to largest. And they are one, root two, root three, two, and 𝜋.