Borrowing in Subtraction

Date: 12/01/98 at 23:00:14
From: Christy
Subject: Subtraction with three numbers

When I work on the problem:

   100
  - 99
  ----

I don't know why the zero in the tens place becomes a 9 when I borrow 
from it. I know that I am taking the 1 from the hundreds position to 
borrow, but why does the zero in the tens become a nine? 

Date: 12/02/98 at 12:03:30
From: Doctor Peterson
Subject: Re: Subtraction with three numbers

Hi, Christy. I like to teach "borrowing" using money. If you want, get 
out some play money ($100's, $10's, and $1's) and try acting out what 
I say. It's fun, even if you don't need it.

When we write 100, it means 1 hundred, 0 tens, and 0 ones. So lay out 
three piles (two of them will be empty!):

    $100's    $10's    $1's
    ------    -----    ----
       1        0        0

I like to have fun with this and pretend these piles represent three 
people, "Mr. Hundreds," "Mr. Tens," and "Mr. Ones." Now suppose we come 
along and want to get $99 from them (9 tens from Mr. Tens and 9 ones 
from Mr. Ones). We first go to Mr. Ones and ask for the 9 ones, but he 
doesn't have any ones today, so he says "I'll be right back," and goes 
next door to Mr. Tens to ask for a ten dollar bill that he can change 
into 10 ones. Mr. Tens doesn't have any, but he goes next door to Mr. 
Hundreds and asks for a hundred dollar bill that he can change into 10 
tens. He's in luck! He takes it, goes to the bank, and now he has 10 
tens:

    $100's    $10's    $1's
    ------    -----    ----
       0       10        0

Now he can give one of his tens to Mr. Ones, who changes it to 10 ones:

    $100's    $10's    $1's
    ------    -----    ----
       0        9       10

Now Mr. Ones can give us 9 of his 10 ones:

    $100's    $10's    $1's
    ------    -----    ----
       0        9       10
                        -9
                        --
                         1

Then we can go to Mr. Tens and ask for all of his 9 tens:

    $100's    $10's    $1's
    ------    -----    ----
       0        9        1
               -9
               --
                0

So they have nothing left but a single one.

Now let's write all this down the way we usually do:

     1 0 0
    -  9 9
    ------

We can't subtract 9 from 0, so we "go next door" to borrow a ten. Since 
there are no tens to borrow, before we can do that we have to go yet 
another place over to borrow a hundred, which turns into 10 tens:

     0 10
     / /
     1 0 0
    -  9 9
    ------

Now we can take one from the tens (leaving 9 tens), which turns into 
10 ones, from which we can subtract 9:

     0 9 10
     / / /
     1 0 0
    -  9 9
    ------
         1

Finally, we can subtract the 9 tens (and no hundreds):

     0 9 10
     / / /
     1 0 0
    -  9 9
    ------
     0 0 1

So where did that 9 come from in the tens place? It's the 10 tens we 
borrowed (which was one hundred), minus the 1 ten that was borrowed 
from it by the ones.

Try adding:

      99
    +  1
    ----

and you'll see a double carry just like the double borrow in your 
subtraction.

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