Comparing decimals: 9.97 and 9.798

let's compare nine point nine seven to nine point naught to nine point seven nine eight so to compare to figure out which one of these is greater I like to start with the largest place values and then and then keep moving to smaller and smaller ones until we actually see a difference so they both have nine ones so they both have nine ones 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 nine in the tenths place well the number on the right has a seven in the tenths place so right now we could view this let's just write the whole numbers out so this one is nine plus nine tenths 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 this one on the right is nine plus seven tenths plus seven plus seven tenths so this immediately cused to me that this the one on the left is a 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 nine eight to the right I have the seven to the right and the way to think about it is no matter what you have even if you if you really increase this right hand side here as much as much as possible you're still less than nine point eight in fact if you keep incrementing the thousands here you go from nine point seven nine eight to nine point seven nine nine to nine point eight so you would have to actually increase to get to even nine point eight and this is at nine point nine so you can really just look at the discrepancy in the largest place value to recognize which number is greater this has nine tenths this has seven tenths it doesn't matter what's going on in the hundreds and the thousands 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 seven hundredths here so you have seven hundredths here and here you have nine hundredths so here you have nine hundredths and then finally here you have zero thousandths thousandths and here let me do that in a different color I use I already used blue and here you have eight thousandths so plus eight over a thousand so let's let's put everything in terms of thousandths so that we can add these all up and have fractions both or have two fractions over a thousandth or things in terms of thousands so nine is the same thing as nine thousand over a thousand nine tenths well let's see if you multiplied it by 10 you would get 90 over a hundred multiplied by ten again you get nine hundred thousandths seven hundredths multiplied by 10 is 70 thousandths and let's do that over here once again nine is nine thousand over a thousand and then plus seven hundred over a thousand plus seven hundred over a thousand plus ninety over a thousand plus 90 over a thousand just multiply the numerator and denominator by ten plus eight over a thousand plus eight over a thousand and so what is this number on the left this number on the left is how many thousandths is it it's nine thousand nine hundred and seven sorry nine thousand nine hundred seventy so it's nine thousand nine thousand nine hundred nine hundred and seventy and seventy thousands while this number on the right here is the number on the right here is nine thousand nine thousand seven hundred seven hundred ninety eight ninety eight thousands so here once again you're comparing two numbers they have the same number of thousands this has nine hundred this only has seven hundred so even though this is almost eight hundred eight hundred is still less than nine hundred so no matter how you think about it the number on the left the number on the left is greater than the number on the right