Borrowing from Zero

Date: 06/01/2006 at 05:34:40
From: Ian
Subject: Very hard subtraction!

How do you subtract from a lot of zeros, like 9000 - 584?

Date: 06/01/2006 at 08:45:59
From: Doctor Rick
Subject: Re: Very hard subtraction!

Hi, Ian.

I suppose you know how to start: you write

    9000
   - 584
   -----

Then you try to subtract in the ones column: 0 - 4 = ?  You can't do 
that, since 4 is greater than 0, so you go to borrow from the tens 
column.

That's where you run into trouble, because the tens column doesn't 
have anything to borrow!  It's like going next door to borrow a cup 
of sugar, and the lady there is all out of sugar, too!  What do you 
do?  Well, this may not make much sense with sugar, but with numbers, 
we do this: we send the lady next door to HER neighbor to borrow 
some sugar, so she will have some to lend.

So we go on to the hundreds place to borrow a 1--but we find that this 
place doesn't have anything to lend, either!  Well, we have one last 
place to try: the thousands column.  There, finally, we find something 
to borrow!  So ... we borrow.  We reduce the 9 to 8, and change the 
1 thousand that we borrowed into 10 hundreds.

    8 10  0  0
  -    5  8  4
  ------------

We can't stop there, though: we still can't subtract in the ones 
column.  Now that the hundreds column has something to be borrowed, 
the tens column borrows one of those 10 hundreds.  That leaves 9 
hundreds, and the 1 hundred we borrowed becomes 10 tens:

    8  9 10  0
  -    5  8  4
  ------------

Now that the tens column has something to be borrowed, the ones column 
borrows one of those 10 tens.  That leaves 9 tens, and the 1 ten we 
borrowed becomes 10 ones:

    8  9  9 10
  -    5  8  4
  ------------

At last we can do the subtraction--all the way through:

    8  9  9 10
  -    5  8  4
  ------------
    8  4  1  6

The answer is 8416.

Now, that looks like a lot of work.  I don't actually write down all 
that stuff; I just go straight to this:

    8  9  9 
    9  0  0 10
  -    5  8  4
  ------------
    8  4  1  6

That's not quite how I write it: I can't cross out the digits on the 
computer, and the 1 I insert is tiny and raised ... you know how to 
write it.  What matters is that I know right away that I'm going to 
have to borrow 1 (for the ones column) from the THOUSANDS column, 
the first column to the left that has something to borrow.  Therefore 
I change 1000 into 999 + 1, leaving 8999 and one more to add into 
the ones column--making 10 in the ones column.

Does this help?

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/30/2009 at 15:45:10
From: Ted
Subject: Regarding Borrowing from Zero

I heard someone recommend a simple way to address the borrowing from 
zero routine you explained. 

The simplification of the problem could be done by taking the number 
you are subtracting from and subtracting '1' from it before executing
your computation.  You can mark your paper on the side to remind 
yourself to add the 1 back before concluding the solution.

In your example:
    9000
   - 584
   -----

Could be manipulated to show:
    8999   +1
   - 584
   -----

This is easy to subtract without needing to borrow from thousands 
place then borrow from hundreds place, then borrow from tens place.

    8999   +1
   - 584
   -----
    8415   (+1) from above = 8416 is the solution

Just a suggestion for simplifying the effort.

Date: 03/30/2009 at 16:13:27
From: Doctor Rick
Subject: Re: Regarding Borrowing from Zero

Hi, Ted.

Yes, that's true.  I might modify your trick by subtracting 1 from 
BOTH numbers:

    9000   ==>     8999
  -  584   ==>   -  583
   -----          -----
                   8416

Now I don't need to remember to add the 1 back on!

Still, the trick is pretty specific; it won't help with, say, 
90002 - 5840.  (It *could* help, with some more thought!  You'd want
to subtract 10 from each number, rather than 1.)  Students do need
to know a routine method that is guaranteed. 

At the same time, I think it's really important to learn to see 
tricks like yours, rather than just following a routine method by 
rote all the time.  In working with adults going for GEDs, I have 
seen a student insist on going through the entire multiplication 
process to multiply a number by 10!

Showing other ways of looking at a problem reinforces several ideas: 
(1) There isn't just one way to do math.  (2) Math is often about 
making work *easier*, not harder!  (3) Math is not just a bunch of 
magic formulas and rules, but a discipline -- the discipline of 
*thinking* about a problem.  Too many students never learn to think.

Thanks for your thoughts!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/