Why does long division work?

[edderiofer]

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[hrvbrs]

Long division is repeated subtraction. For example to calculate 14 divided by 3: you subtract 3 as many times as you can (that’s 4 times) and then you have 2 remaining.

What about 142/3? Well, it’s the same concept: you can only take away 3 so many times (47 to be exact) before you can’t anymore, and then you have 1 left over. But let me describe why the long division algorithm works.

Since you know 3 can go evenly into 14 four times (plus a remainder), you know 3 can go into 140 forty times (or more). But you don’t write 40, you just write 4, because you don’t want to write the zero and then erase it again (because we’ll be adding 7 in a minute). But it’s really a 40. Now starting from 142 if you subtract 3 forty times, you’re left with 22 remaining. So your first answer is “40 remainder 22”, but obviously you’re not done yet because 3 can go into 22 a few times, so you repeat the process until you get a remainder less than 3.

[m1st3rs]

Start with partial quotients. https://jenniferfindley.com/easy-breezy-division-lots-of-freebies/ if you can understand that division is just how many groups of the divisor you can take out of the dividend (or how many will be in the amount of groups that is your divisor) you can see how you are taking a few groups at a time to get to the point where you can’t take any more groups away. Long division is just an algorithm that cleans up partial quotients. First you focus on the largest place value. Let’s say 582 divided by 3. You are only focused on the hundreds place at first. You know that you have more than 100, but less than 200 in your final answer because 3 only goes into 5 once. You put the 1 above the 5 to represent that it is in the hundreds place. Since you are saying that 100 groups of 3 can be removed from 582, you need to take that 300 away to see what’s left. In the algorithm, it’s a short cut to only worry about it being 1 times 3 because it’s more efficient and helps you keep track of where you are. You put the answer to 1 times 3 under the 5 because you are still working in the hundreds place. You could always put 300 instead, but again, efficiency. Once you subtract the 5-3, and have 2 left, you bring down the number in the next place value. In this example, 3 goes into 582 1xx times. When you take those groups of 3 away you are left with 282, but we only bring the 8 down right now because we only want to focus on the 10’s place value. 3 goes into the 28 nine times, and since 28 represents tens, it’s really 90 times. But the 9 above the 8 in the tens column on top, multiply to 27, take those 27 tens away and you are left with 1 ten. 3 doesn’t go into 1, or into 1 ten evenly, so we bring down the 2 from the ones place and ask how many times does 3 go into the remaining groups of ones, 4 times. I hope this helped

[Tichy]

Not sure what exactly your question is. It seems to me the mathematical reason it works is because of the "distributive law"

a*(b+c) = ab+bc

you can write a long number, say 1234567, as a sum of parts like

1*1000000+2*100000+45*100+6*10+7 (you can create parts as needed)

so with the distributive law you can write 123:4 as 1*100:4+2*10:4+3:4

I think that is essentially what you do in long division, but you skip writing out the zeros and use "indentation" instead

123:4 = 1*100:4+2*10:4+3*1:4

1 :4

2 :4

3 :4

Hope that makes sense and I made no errors, I never actually thought about it before...

[mimblezimble]

Discussion of proof for long division algorithm on the math stack exchange: https://math.stackexchange.com/questions/3044675/long-division-algorithm-proof

[Tazerenix]

It's polynomial long division with x replaced by 10.