Lattice Method of Subtraction?

Date: 06/13/2006 at 14:43:43
From: Ashley
Subject: Lattice Method

I'm studying the Lattice Method in my math class, and I was wondering 
if it is possible to use the lattice for subtraction problems?  I know
that there is lattice addition and multiplication, but what about
subtraction?

Date: 06/14/2006 at 15:06:16
From: Doctor Peterson
Subject: Re: Lattice Method

Hi, Ashley.

You're right, lattice addition uses the format of the lattice to
"hide" the "carrying".  Here's a typical problem:

    2  3  4  4
  + 7  2  6  3
  ------------
   0/ 0/ 1/ 0/
   /9 /5 /0 /7
  ------------
 0  9  6  0  7

The number in the upper left of each cell is really the digit you
"carry", just written in a somewhat different location.

An equivalent method without the lattice would look like

    2344    or      2344
  + 7263          + 7263
  ------          ------
   09                 07
    05               10
     10             05
      07           09
  ------          ------
   09607           09607

I can't see any way to do the equivalent for subtraction, since 
subtraction works backwards, requiring you to pull a "carry" digit 
OUT of the column to the left when needed, rather than just 
overflowing INTO that column naturally.  To put it another way, in 
addition you do the "carries" AFTER the additions, but in subtraction 
you have to do the "borrowing" BEFORE subtracting, in order to set 
things up to subtract a smaller number from a larger one.

There may be a way based on the method used in computers to turn 
binary subtractions into additions so the same hardware can do both; 
but that would be very awkward for humans using decimal notation!

That's not to say that there aren't other ways to do subtraction; 
I've heard of a number of them.  But none that I have heard of, or can 
invent easily, look anything like a lattice.

If you haven't seen the Austrian Subtraction method, you might like 
it; there is some resemblance to the lattice, though because the 
"carry" is used in two different ways, it couldn't be isolated in a 
cell:

  Austrian Subtraction   
http://www.sonoma.edu/users/w/wilsonst/Courses/Math_300/Groupwork/
altsub/aust.html

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

Date: 06/14/2006 at 16:10:36
From: Ashley
Subject: Thank you (Lattice Method)

Thank you so much for your help!  I really appreciate it.  Your answer
makes sense to me, and has helped me figure this concept out.  You are
the best!  Thanks!  :-)