Question

How do you order the following from least to greatest without a calculator `0, -sqrt2, sqrt5, 13/4`?

Answer 1

`color(blue)(ul(bar(abs(color(black)(-sqrt2, 0, sqrt5, 13/4))))`

Explanation:

No calculators are being used in this answer I'm writing!

We have four numbers to order from least to greatest:

`0, -sqrt2, sqrt5, 13/4`

Let's first see that there is only one negative number - that has to be the smallest. And then there's 0, which is certainly smaller than the remaining two positive numbers, so we have so far:

`-sqrt2, 0`

So the question is now which of the two remaining numbers, `sqrt5, 13/4`, is smaller.

Let's estimate the value of each of these.

`sqrt5` is going to be more than `sqrt4` and less than `sqrt9`, so we can write that as:

`sqrt4 < sqrt5 < sqrt9`

We know that `sqrt4=2 and sqrt9=3`, and so:

`2 < sqrt5 < 3`

Is `13/4` bigger or smaller than that? Let's do something similar with the fraction - we know that `13/4` is bigger than `12/4` and smaller than `16/4`, so we can write:

`12/4 < 13/4 < 16/4`

We know that `12/4=3 and 16/4=4`, and so:

`3 < 13/4 < 4`

With `sqrt5 < 3 and 13/4 > 3`, we can now finish ordering our numbers:

`color(blue)(ul(bar(abs(color(black)(-sqrt2, 0, sqrt5, 13/4))))`