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Let's compare 9.97 to 9.798. So to figure out which
one of these is greater, I like to start with
the largest place values and then keep moving to
smaller and smaller ones until we actually
see a difference. So they both have nine 1's. So at least in the
ones place, they seem comparable to each other. Now let's go to
the tenths place. So this number on the left
has a 9 in the tenths place, while the number on the right
has a 7 in the tenths place. So right now, we could
view this-- let's just write the whole numbers out. So this one is 9 plus 9/10. We haven't gone to the
hundredths place yet. So far, out of the two digits,
the two places we've looked at, this one on the
right is 9 plus 7/10. So this immediately cues to
me that the one on the left is the larger number. You're like, hey, how
do I know immediately that's the larger number? I have all this other
stuff to the right. I have this 98 to the right. I have this 7 to the right. And the way to think about it
is, no matter what you have, even if you really increase
this right-hand side here as much as possible,
you're still less than 9.8. In fact, if you keep
incrementing the thousandths here, you go from
9.798 to 9.799 to 9.8. So you would have to actually
increase to get to even 9.8. And this is at 9.9. So you can really just
look at the discrepancy in the largest place
value to recognize which number is greater. This has 9/10. This has 7/10. It doesn't matter what's
going on in the hundredths and the thousandths place. And to make that clear, let's
actually add up these numbers and compare them as fractions. So let's keep on
going with this. So you have 7/100 here. And here you have 9/100. And then finally,
here you have 0/1000. And here-- let me do that
in a different color. I already used blue. And here, you have 8/1000. So plus 8/1000. So let's put everything
in terms of thousandths so that we can add
these all up and have two fractions over
thousandths, or things in terms of thousandths. So 9 is the same
thing as 9000/1000. 9/10-- well, let's see. If you multiplied it by 10,
you would get 90 over 100. Multiply by 10 again,
you get 900/1000. 7/100 multiplied
by 10 is 70/1000. And let's do that over here. Once again, 9 is 9000/1000,
and then plus 700/1000 plus 90/1000-- just multiply
the numerator and denominator by 10-- plus 8/1000. And so what is this
number on the left? This number on the left is--
how many thousandths is it? It's 9,970. So it's 9970/1000, while
this number on the right here is 9798/1000. So here, once again, you're
comparing two numbers. They have the same
number of thousandths. This has 900. This only has 700. So even though
this is almost 800, 800 is still less than 900. So no matter how you think
about it, the number on the left is greater than the
number on the right.