Multiplying challenging decimals

let's multiply one point to one or one and 21 hundredths times 43 thousandths or 0.043 and I encourage you to pause this video and try it on your own so let's just think about a very similar problem but one where essentially we don't write the decimals let's just think about multiplying 121 times 43 which we know how to do so let's just think about this problem first as kind of a simplification and then we'll think about how to get from this product to this product so we can start with so we're going to say 3 times 1 is 3 3 times 2 is 6 3 times 1 is 3 3 times 121 is 363 and now we're going to go to the tens place so this is a 40 right over here so since we're in the tens place let's put a zero there 40 times 1 is 40 40 times 20 is 840 times a hundred is 4,000 and we've already known how to do this in the past and now we can just add all of this together and we get and we get them to a new color here 3 plus 0 is 3 6 plus 4 is 10 1 plus 3 plus 8 is 12 1 plus 4 is 5 so 121 times 43 is 5,200 and 3 now how is this useful for figuring out this product well to go from 120 to go from 1 point 2 1 to 121 we're essentially multiplying by a hundred all right we're moving the decimal two places over to the right and to go from zero point zero 4 3 2 43 what are we doing we're removing the decimal so we're multiplying by ten hundred thousand we're multiplying by 1,000 so to go from this product to this product or to this product we essentially multiply two by a hundred and we multiplied by a thousand so then to go back to this product to go back to this product we have to divide we should divide by a hundred we should divide by 100 and then divide by a thousand which is equivalent to dividing by a hundred thousand but let's do that so let's rewrite this number here so five thousand two hundred and three actually let me write it like this just so it's a little bit more aligned five thousand two hundred and three and we could imagine a decimal point right over here if we divide by a hundred C divided by ten divided by a hundred and then we want to divide by another thousand so divided by ten divided by hundred divided by a thousand so our decimal point is going to go right over there and we're done one point two one times zero point zero four three is zero point zero five two zero three so one way you could think about it is just multiply these two numbers of this as if there were no decimals there then you could count how many digits are to the right of the decimal and you see that there are one two three four five digits to the right of the decimal and so in your product you should have one two three four five digits five digits to the right of the decimal why is that the case well when you ignore the decimals when you just pretended that this was 121 times 43 you essentially multiply this times a hundred thousand by a hundred and a thousand and so to get from the product you get without the decimals to the one that you need with the decimals you have to then divide by a hundred thousand again multiply by a hundred thousand is essentially equivalent to moving the decimal place five places to the right and then dividing by a hundred thousand is equivalent to moving the decimal five digits to the left so divided by ten divided by a hundred divided by a thousand divided by 10,000 divided by a hundred thousand but either way we are done this is what this is what we get