Question

Dear Mitch,

I'm in middle school and I usually love math and it's my best subject.  My teacher explains everything really perfect, so I know just what he means.  Then he gives us an example to try on our own and I usually get them right.  My teacher always tries to get us to understand things and not JUST get them "right", and I almost always do understand what we're learning.  But right now I don't really get something, even though he keeps going over it the way he said he said he taught it every other year.   For some reason, even though I'm getting them right,  I just don't understand it.  So he said maybe I should write in to you, and if you come up with a new way to explain it that works he'll do it that way for the whole class. 

Anyway, the question is this:

How come when you subtract a negative number from something it's the same thing as adding the positive of that number?

Sincerely,
Positively Confused 

Answer

Well, first of all, don't worry if some way that your teacher explained it doesn't work for you; if you don't get something when your teacher explains it a certain way, after a few tries you should stop frustrating each other with THAT way and try something new.  Even if he was successful with it in the past, that's the past. It's not just a cliche to say that everyone is different; it's a fact.  You're different from the students your teacher had in the past.  So what?

Anyway, the whole subject of positive and negative numbers is very abstract to many middle-schoolers, because you haven't had a lot of experience with borrowing from banks, or taking out mortgages or the kinds of things that make the idea feel real to you.  But you can still get it in a couple of minutes if you pay close attention to a true story:

Once a week at the local church service in a town less than forty minutes from here, a box is passed around for the collection of money for the members of the community who are in need.  And every week the town's people are so moved by the sermons of the inspirational preacher that they give as generously as they can.  One man in particular routinely reaches into his pocket and produces a very generous contribution.  One Sunday, when the box reaches him he stands up and clears his throat.  Then he speaks slowly:  "I do not think I'm overstating things when I say that I always give what I can and I never regret it."

People nodded. He had certainly been a consistent and generous giver.

"After all," he continued, "I have been very fortunate to see my small business prosper, growing one small step at a time for longer than I'd like to admit.  But this week I suffered an unforeseen setback.  I am hoping that I will figure out how to overcome the situation but at the moment I am in trouble."

At this point an older man sitting behind him stood up and spoke.

"Then by all means don't even think about giving.  I always sit behind you and I always admire your generosity.  Take a break -- "

The man interrupted his elder.  "Well, thank you, and that was my first thought, but I'm in worse shape than you might guess, and so, this week, for the first time, I am asking all of you to consider me one of the 'needy'.  In fact, not only was I going to refrain from giving, but I am hoping you will allow me to take."

He paused to take in the silence. People were surprised and did not know what to say.

Finally, the old man behind him stood up again, and shouted, "Absolutely.  It's the least we can do for you. Take!"

Hundreds of voices came forward to add their encouragement.

So the newly needy man took a pen and a small notebook from his pocket and wrote:  "Rather than waste everyone's time trying to get money out from the slot, and rather than ask anyone to run down to the office to get the key, I, Franklin J. Wentonce, am writing a sort of I.O.U., from the 'box' to me for fifty dollars.   I shall stop by the office this afternoon to pick it up." He read the note aloud and folded it into a square small enough to slip into the box.

Now, mathematically, as he slips the note into the box, he is not subtracting anything, as he would be doing by pulling money out of the slot.  He is ADDING something to the box, but what he is "adding" is a 'negative' value, which means its addition to the box decreases the value of the box's total contents by fifty dollars...

However, you asked about subtracting a NEGATIVE, not ADDING a NEGATIVE value.  And so, the next thing that happened that day is a very important piece of information:

JUST as the man began to slide his note into the box, a woman in front of him interrupted:

"Excuse me," she said, "but rather than have you make the trip all the way back here from your home, I have a better idea.   Since, by pure coincidence, I was just about to donate fifty dollars, why don't you let me take that note OUT of the box, and you just take this."

She handed him the crisp fifty-dollar-bill that was neatly folded in the palm of the long, white silk glove she was wearing.

The man paused, shrugged, and agreed.

And so the woman pulled out the note from the last corner remaining outside the box, and handed him the money.

QUESTION:  We said that when he "ADDED" the "NEGATIVE" quantity (or the 'take-away' note) to the box, it DECREASED the SUM of the contents of the box.  So what do you think happened NOW to the VALUE of the total SUM when she TOOK AWAY from the box THE TAKE-AWAY note (which, of course, had a decreasing -- NEGATIVE -- value to the box's contents when he put it in) and sent the box to her left to allow it to continue along its journey?
 
ANSWER:  By subtracting the negative value, she made the total in the box go up by that amount, or she ADDED it -- just as perfectly as if she had placed the fifty dollar bill in herself.

I know that's a long story, but if it helps you understand and remember the idea, it will have been worth it!

--Mitch