Subtracting with the Abax (Lesson 3)

Subtracting with the Abax (Lesson 3)
Once you understand the previous lessons, and can subtract from left to right in a problem with many columns, it is then time for you to learn a little shortcut. This shortcut illustrates a basic Idea about mental math (speedmath). That Idea is the "one-ahead" principle. (I gave it this name, which I borrowed from the art of magic. There is a very powerful magical tool called "the one-ahead principle" which magicians use to accomplish some great tricks. The math method and the magic method are not the same, but they are equally powerful in their own fields.)

The lesson below illustrates how to do 227 - 89 using tens-complements and a shortcut

abax subtraction fig. 12
227

Here's the number 227 as represented on an abax. We are going to subtract 89 from it.
This time we are going to use a shortcut.
As in the previous lesson, there is nothing in the hundreds column of 89 to subtract from 227, so we start with the next highest column - the tens.

abax subtraction fig. 13
- 90

But before you do the tens, you look over one more column to the right (in this case the ones column), to see if when you subtract there you will have enough marbles to do it. That is called "looking one-ahead."
You will immediately see that you can't take 9 ones from 7 marbles. That tells you that when you eventually get to the ones column, you will have to take away a marble from the tens column, and make up for the difference with the tens-remainder of the 9 (just like we did in the last lesson).
The difference is, that in this lesson you will take that marble out of the tens column during the the step when you subtract from the tens column.
You should already know how do subtract the tens column (if not, go back to the last lesson), so when you do it, subtract one more marble than you would have normally, because you know you are going to have to to that eventually when you get to the next column's subtraction. That is why, at the head of this illustration, it says -90 instead of -80.
So, in effect, you are one step ahead of the next subtraction.

abax subtraction fig. 14
+1

You see that because we have subtracted 90 instead of 80 from 227 in the previous step, that we have already subtracted more than we needed to. The original problem is to subtract 89 from 227, and we have subtracted too much.
How much too much? Exactly the tens complement of the digit in the number of the ones column to be subtracted. That digit was 9, but we subtracted an extra ten. So we have to put back the difference, which is the 1. That is the tens-complement of 9. So just add that 1 marble to the ones column, and you are done.

abax subtraction fig. 15
=138

The final answer is 138, just like in the previous lesson.

This lesson is a little complicated to explain, and maybe you will need to read through it one or more times to "get" it. That is normal.

Like all good things, this takes a little extra effort to learn and get good at in the beginning. It will save you tons of work in the long-run, because after some practice, you will never have to worry about subtraction anymore.

If you put in just a little more effort than you need to, you will become good enough at left-to-right subtraction to do it in your head by just visualizing the abax. When you are that good, you will be able to freak people out with your "genius" ability to subtract. That is a big reward for a little more effort.

There is still another powerful shortcut for left-to-right subtraction with very large numbers. It is fairly advanced, and you will need to be very expert at subtraction before you tackle it. You can learn it on an abax, although by the time you are that good at subtraction, you will be doing most of your subtracting in your head.

The method is using the "hundreds complements" and doing the subtractions two columns at a time. You can fool around with it now if you want, of course, but don't let it distract you from learning the basics solidly, first.

By solidly, I don't mean that you can do them "well enough to get by." There is a huge difference between being able to understand how something is done and actually doing it. (It's not as big as the difference between saying you understand something and actually understanding it, but it is still huge.)

You should practice with the abax until you are at the point that you can do subtractions without it, by just visualizing how you would do them in your head. If it takes you more than a second or two to subtract a three-digit number from another three-digit number, then you haven't practiced enough.

I will post some practice pages for you to try with your own abax (if you have one) very soon. In the mean time, make them up yourself. Time yourself now and then to check out your progress. (You don't have to time yourself each time, though, because you don't want to make it a neurotic chore, do you?)

Once you understand the Idea behind subtraction with tens-complements very well, and can do it with the shortcut pretty easily, you can check yourself on this page.

This is as far as I am going to take the subtraction lesson on the abax on this site. Eventually I will make pages about the abacus. If you want to learn more about subtraction without an abax or abacus, there are pretty neat interactive lessons on left-to-right subtraction.

Math Mojo is part of Magic and Learning - a company that uses methods of magicians to teach thinking skills.