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

1 Answer

Answer:

#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))))#