Comparing decimals: 9.97 and 9.798

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.