SUBTRACTION BORROWING

[Saumyojit]

suppose i am subtracting 219 from 300
3 hundred zero tens zero ones.

AS we begin subtraction from lsb

1. 2 hundred one tens 9 ones
   
   generally my target is to increase zero one of 300 to n ones where n>9 i.e ten ones so that i can subtract a smaller no from bigger no
   so the rule is to check to the second 10's place but there is zero tens then go to the hundreds places and borrow one hundred
   
   so no becomes 2 hundred ten tens zero ones .
   
   And the next work is to take borrow one ten from ten tens which produces
   2 hundred 9 tens 10 ones now my lsb is bigger from the smaller no lsb
   we do subtraction . OKAY

2. now suppose i am subtracting 218 from 300. My goal still remains the same the lsb or any digits of bigger no just have to be one greater than digits of smaller no during sub....

So why we will try to achieve ten ones instead of 9 ones which does the job . 9ones -8ones =1 one
Why everything has to be in ten.

NOw i know the position value is in powers of 10 . Every next position value is double the positional value of the previous place.
BUT still its not hitting my intution

@Dr.Peterson please explain

 

[Dr.Peterson]

The goal is not to achieve ten ones; it's only to have more ones than in the subtrahend. We work with tens (in the standard algorithm) simply because, in base ten, the ones come in groups of ten! It's like, if I want to give drinks to a couple people and they come in packs of 8, I open a pack of 8: I don't need 8, but that's how they come.

There are other ways to subtract. In mental arithmetic, I might subtract 218 from 300 by adding 2 to each number, so I need to subtract 220 from 302; and then I might add 80 to each, and subtract 300 from 382, which is 82.

Or, I might choose to subtract 1 from each number, so I need to subtract 217 from 299, which is 82.

But when you use the "borrowing" method, you use tens because that's what you can easily get from the next place to the left.

 

[Saumyojit]

that's what you can easily get from the next place to the left.
this is the anser that i need . in the case of subtracting 218 from 300. if i wanted to make lsb of 300 that is zero to 9 i have to do much more mental arithmetic to borrow groups of nines from the msb maybe .IF i did that just asking out of curiosity then 300 will become 2_9 something of this format ? right?

also THE middle digit has to be bigger than the middle digit of 218 anything above 1 .
What would be the process ?

CAN i maintain both the middle and last digit of 300 to increase to a digit just one greater than the middle and last digit of 218.
THAT MEANS I AM wanting 300 to be _29 after borrowing

 

[Dr.Peterson]

If you change a digit to one greater than the digit below it, aren't you trying to make that digit in the answer be 1? You can't do that, because that digit is not 1!

Or if you did borrow only 9 from the 3, instead of 10, then the 3 would not reduce to 2, but to 2.1! (That is, you would be reducing the 3 by 9/10 rather than by 1.) That would be silly. Your curiosity is taking you too far from rationality.

In any case, what you are doing is not what we call borrowing. Borrowing always means reducing the next digit by 1 and adding 10 to the current digit. I showed you two creative, alternate methods that make sense; but they are not called borrowing.

 

[Saumyojit]

if i am taking borrow from 3 hundreds what quantity of nine i am taking ten nines from 3 hundreds . if that so that leaves me in msb (2 ninetys 3tens) how u are gwetting 2.1

 

[Dr.Peterson]

Because it's nonsense. One would never do that.

But 3*100 + 0*10 + 0*1, if you really wanted to take 9 tens (90) rather than 10 tens (100) from the hundreds, would become 2.1*100 + 9*10 + 0*1. Check it out! There are no nines involved; we're still working in base ten.

Now, if you want to work in base 9, that's a whole different thing.

 

[Saumyojit]

whenever i am taking something from the hundred column am i not taking in quantity of hundred suppose 300 i am borrowing 1 hundred from hundred column and placing ten tens in tens column. so my no becomes 2 hundred ten tens zero ones

now if i take 9 of something from the hundreds place am i not borrowing 9 hundred (90 tens) just like i am borrowing 1 hundred above but that would be impossible because i have only 3 hundred how can i take 9 hundreds from msb! its like i am thinking about it for a moment and next second i am only clarifying myself .

another thing if i am subtracting 218 from 309 that means i am borrowing 1 hundred from hundred column and placing ten tens in tens column . NOW my question is that how the tens column able to hold 10 tens value because ten tens means it increases the next signifacnt place by 1 automatically but that is not happening after borrowing
My explanation to myself is for the sake of subtraction and defination of borrowing allows us to assume that middle digit is now ten tens.
RIGHT?

 

[Dr.Peterson]

whenever i am taking something from the hundred column am i not taking in quantity of hundred suppose 300 i am borrowing 1 hundred from hundred column and placing ten tens in tens column. so my no becomes 2 hundred ten tens zero ones

now if i take 9 of something from the hundreds place am i not borrowing 9 hundred (90 tens) just like i am borrowing 1 hundred above but that would be impossible because i have only 3 hundred how can i take 9 hundreds from msb! its like i am thinking about it for a moment and next second i am only clarifying myself .

You're the only one who is wanting to borrow 9; I certainly haven't recommended it. So you get to decide what you mean by it. But apparently what you mean by it is nonsense, so don't do it! I think what I said is the only sensible thing you could mean, and even that isn't very sensible.

another thing if i am subtracting 218 from 309 that means i am borrowing 1 hundred from hundred column and placing ten tens in tens column . NOW my question is that how the tens column able to hold 10 tens value because ten tens means it increases the next signifacnt place by 1 automatically but that is not happening after borrowing
My explanation to myself is for the sake of subtraction and defination of borrowing allows us to assume that middle digit is now ten tens.
RIGHT?

Borrowing means temporarily bending the rules that limit the amount in each place. The top number after borrowing is not a properly written number. Your final answer has to follow those rules.