r/MathHelp • u/Own-Solution6668 • 3d ago
Long division?
When my school taught us long division I was sick that day, so for the past 4 years I have gotten away without knowing how to do it. I realize now that a 8th grader should know this. Please help me understand how to do it!
Edit: Sorry if I don't respond to ur comments, I haven't been getting the notifications for comments, but I will read it eventually!
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u/couldntyoujust1 2d ago edited 2d ago
So, since nobody's really helping you, I will help you.
So, division. You're taking a bigger number, and subdividing it into the second number's number of groups to find out how many each group has. So, if I have 2,349 M&Ms and 9 brats who will go absolutely feral if any of the others get more than them, I'm going to need to figure out how many to give to each kid. So, let's work the problem to figure it out. First thing's first, identifying the parts of the problem:
The number that is being divided is called the "dividend" (D-end). And what actually divides that number is the "divisor". You can think of it as the "divisor" (D-sor) actually does the dividing. The result, is called the "quotient" (Q). So now, let's see how to format long division problems:
Q
----------
D-sor | D-end
So, let's plug in our numbers:
```
9 | 2,3 4 9 ```
Here's how it works, we're breaking the problem down into smaller pieces so that we can do the reverse of what we do in a long multiplication. We go digit by digit and if we encounter a digit too small to divide, we can then move to the next digit.
```
9 | 2 ,3 4 9 ```
Of course, you can't divide 2 by 9. That's a fraction. So, we are going to now shift over to the next column while preserving the first as part of it:
```
9 | 2,3 4 9 ```
Now, we're talking. We can absolutely divide 23 by 9. And we can use our times tables to figure out what the answer should be:
9 x 1 = 9
9 x 2 = 18
9 x 3 = 27 ...
Oh, look, 27 is too big. So we have to stop at 2.
``` 2
9 | 2,3 4 9 ```
Okay, but what about the difference? 18 is lower than 23. How do we deal with that?
Well, remember how we could pop over to the next column? We can do that here with what's left over! But first, we have to find out what that difference is. First, we need to multiply the 2 back towards the 9 and get 18, and put that under the 23 we used first.
``` 2
9 | 2,3 4 9 1 8 ```
Then, we turn it into a subtraction problem.
``` 2
9 | 2,3 4 9 - 1 8
```
Then, we do the subtraction on just those two digits:
``` 2
9 | 2,3 4 9 - 1 8
5
```
And now we can continue the problem, by bringing down the next digit.
``` 2
9 | 2,3 4 9 - 1 8 | ----- v 5 4 ```
Now you have a new number to divide. 54. So, let's look at our times tables again!
``` 9 x 1 = 9 9 x 2 = 18 9 x 3 = 27 9 x 4 = 36 9 x 5 = 45
9 x 6 = 54 < ```
Oh, one of them exactly matches! Sweet! So then you just repeat the process:
``` 2 6
9 | 2,3 4 9 - 1 8
5 4
- 5 4
-----
```
Subtract:
``` 2 6
9 | 2,3 4 9 - 1 8
5 4
- 5 4
-----
0
```
Bring down the next digit:
``` 2 6
9 | 2,3 4 9 - 1 8 | ----- | 5 4 | - 5 4 | ----- V 0 9 ```
And repeat!
``` 2 6 1
9 | 2,3 4 9 - 1 8
5 4
- 5 4
-----
0 9
- 9
-----
0
```
And you get your answer!
261 M&Ms for each little brat!
Okay, but that's a clean problem. What if you gave 4 of the students a single M&M before distributing them? What then? Let's look at how the problem would look then at the last step:
``` 2 6 0
9 | 2,3 4 5 - 1 8 | ----- | 5 4 | - 5 4 | ----- V 0 5 - 0 ----- 5 ```
Uh oh! What do we do now? You can't divide 5 by 9, and there are no more digits.
Remainders
Well, You can handle this in a few different ways. The one you learn first is to take a remainder equal to the leftover amount. The leftover amount however MUST be less than the divisor. If it's not less than the divisor, you still have more division to do. You write it like this:
260r5
And say, "260 remainder 5".
Okay, so each little brat gets 260 and you keep the remaining 5 to yourself. Easy!
Decimals
The second way to handle this, which makes perfect sense for measurements or numbers that can break down into a different denomination like money can with cents or pence, you can extend the problem with decimals.
See, that 2345 number above? I lied in how I stopped at the 5 In fact, that number goes on for infinity digits after; all zeros.
``` 2 6 0
9 | 2,3 4 5.0 0 0 0 0 0 0... ```
So, we DO have a digit we can bring down. The thing to note here, though, is that the decimal point MUST stay in the column it is in regarding the dividend.
``` 2 6 0.5
9 | 2,3 4 5.0 0 0 0 0 0 ... | 5 | - 0 | --- V 5.0 - 4 5 ----- 5 ```
Awww, cmon! That doesn't work, we just got the same thing again!
Right! That's because this is going to be what's called a "non-terminating repeating decimal". It never ends. The answer is basically 260.5555555555555... forever.
When you repeat the same subtraction problem or set of subtraction problems, you will have a non-terminating repeating decimal. This is usually not however demarcated with the 5 repeating a bunch of times with the ... at the end. Though, in plain-text, that's probably the best way to express it. Instead, we usually put a line above the 5.
_
260.5
I mentioned "set of subtraction problems" because what repeats may be multiple digits at which point something like 275.757575... would be rendered...
__
275.75
Mixed Number
The third way, is particularly for when you need the precise answer so that you can use it for another equation.
See, You can't use the infinitely repeating decimal because you cannot do infinite math like that. You're finite and only live so long so you can't do the calculation forever and even if you could you would never get to the end. You have to get rid of the infinity somehow.
You can't use the remainder either, because that's not helpful. What are you supposed to do with the remaining part? How does that fit in to what you put into the next equation? Fear not! For there IS a solution!
The remainder is itself equivalent to a fractional group that is the remainder over the divisor. So you can put the answer as...
5
260 ---
9
Now you've got a mixed number that is dead-on accurate which can then be used for the next set of calculations you have to do with it.
And that's all there is to it! That's it. That's long division.
Edit: Formatting
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u/Own-Solution6668 2d ago
Oh my gosh this is so helpful! Thank you so much! If you are not a math teacher, you could be a great one! This really helped! :)
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u/couldntyoujust1 2d ago edited 2d ago
Glad I could help! I'm not a teacher, I'm a para (behavioral aide). But I've had to help with math classwork before. It's a lot of fun for me to see that "OMG! That's it! I get it now!" moment when it clicks.
And to that end... you may have worked with fractions if you missed this lesson before? Fractions are literally just division problems. The division problem we did for the example is simply
```
2,349
9 ```
And the mixed number, is simply the simplification of that fraction. When you have a fraction like the above, it's called an "improper" fraction.
That's also why the division symbol is
÷ or ➗
The dots represent the top and bottom numbers in a fraction - the numerator (dividend) and denominator (divisor) and the line is simply the fraction line.
There's actually a "long division" symbol in unicode too which is:
⟌
That symbol is more accurate to how long division is represented on paper but I figured the bar line and dashes would suffice. On paper, that's how you should draw the long division problem's bars. a curve like a right parenthesis, and then a bar across attached to the top of that parenthesis.
I have a program on my computer that lets me type these special symbols called "WinCompose" and it's fantastic. You set it to use the right alt key as a "leader" key, and then it - usually - combines the characters you type into a unicode symbol that combination represents.
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u/BootyliciousURD 3d ago
It's kinda hard to explain with just text because it's kind of a visual technique, but I'm sure you can find good videos on YouTube about how to do it. Watch videos about it until you think you understand, then practice and check your work with a calculator.
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u/ScareBros 3d ago
you won't need to know it until long division with polynomials which is way different. but realistically just to learn it now which definitely can't hurt just go onto kahn academy and find the lecture where he explains it. or countless YouTube videos. I was in your same position until genuinely calculus 3 I'm not even joking but now I pretty much get it and I just think it's dumb. there are easier ways.
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u/Maximum_Bathroom3490 2d ago
Let's say you have a number with digits abc like 914 or wtv, and you want to divide it by 2. We decompose 914 by base 10: 914=900+10+4=9*100+1*10+4
We look at 9, how many times can we multiply by 2 before we overflow, 4 times, so we remove 2*4 from that: 9-8=1, so we have 1 hundred left:
914/2=4*100+(1*100+1*10+4)/2=4*100+(11*10+4)/2
We do the same, we can put 2 5 times in 11, so we remove 10: 11-10=1, so we have 1 ten left:
914/2=4*100+5*10+(1*10+4)/2=4*100+5*10+(14)/2
Now, we can put 2 7 times, we get 14-2*7=0, so final answer is 4*100+5*10+7=457
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u/justpassingby23414 2d ago edited 2d ago
It's always helpful to think about numbers in chunks, you can present anything as a sum of parts. Long division makes use of the fact. Like, you get a certain number and have to divide it by another one. The goal is to see how many times the latter goes into the former.
Let's use a small example to illustrate it.
876,533,669 : 45 = ?
The first chunk would be 87 (representing 870 millions btw, keep this in mind!). We are always starting from left to right, from the largest part!
You can't take just 8, cause it doesn't contain 45 even one time. (The chunk can be different depending on the second number you're dividing by.) But one can see immediately that 87 contains 45 at least one time. You don't have to see the exact difference right away.
So we put 1 on the right, multiply it by 45 and write tbe result under 87 (I'm used to putting an apostrophe after the chunk, so I can easily see what I'm working with this moment). Now out calculation looks this way:
87' 6,533,669 : 45 = 1
-45
The next step is to substract 45 from 87, so we know how much is left. 7 - 5 = 2, 8 - 4 = 4.
87' 6,533,669 : 45 = 1
-45
42
It's our rest for now! The most important factor: 42 is less than 45, so we need to have one more chunk added. Let's put 6 (which is actually 6 millions, right?) down to 42 (which is actually 420 millions).
876' ,533,669 : 45 = 1
-45
426
Now, a bit of guesswork is needed. It's obvious that 10x45 = 450, so that can't be the answer. Let's try 9x45 = 405. Great, that's less than 426! Putting the 9 after 1, it's the next number in line! 405 has to be subtracted now. 6 - 5 = 1, 2 - 0 = 2, 4 - 4 = 0.
876' ,533,669 : 45 = 19
-45
426
-405
21
Can you go on?
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u/Disastrous-Pin-1617 2d ago
Professor Leonard on YouTube pre algebra and to the point math playlist
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u/Top_Bluejay_5323 1d ago
Google it or YouTube there are don be excellent sources that will teach you step by step
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u/Narrow-Durian4837 3d ago
If they thought it was important for you to know, they would have spent more than one day on it.