Math to the Nines

By Bobby Neal Winters

Teaching mathematics is treacherous. First off, people will walk up to you and tell you to your face that they hate mathematics.  

And they are proud.

There is constant pressure to change the way mathematics is taught in order to make it easier, but when you do, you will inevitably get pushback because you are teaching it in a way the parents weren’t taught, and they are made to feel stupid because they can’t help their children.

This is a shame because mathematics is useful, beautiful, and fun.

Let’s play a game.  If you have a pencil and paper handy, pick them up.  Let’s take two 2-digit numbers (any number of digits will work, but let’s start off simple) say 57 and 38.  

Look at the smaller of the two numbers (in this case 38).  Look at it digit by digit.  See what it takes to build up each of the digits to 9.  In this case, 6 + 3 = 9 and 1 + 8 = 9.  Take the 6 and the 1 and make a new number: 61. (61 is called the nines complement of 38 because 61 + 38 = 99.)

Now take the 57 and add it to the 61.  This is 118.  In this game, we want to remain within the world of 2-digit numbers, so that extra 1 out in the hundreds place has got to go.  To get rid of it, add it to the 8 on the other end.  This gives us 19.

Okay, what do we do with that 19?  Well, 19 + 38 is 57.  So it looks like by adding 61 to 57 and adding the extra 1 to the back, I’ve subtracted 38 from 57.  

Maybe I was just lucky.  Let’s try that again.  Let’s take 91 and 63.  Let’s turn the 63 into 36 because 3 + 6 = 9 and 6 + 3 =9. Now add 36 to 91.  This gives us 127, but adding the 1 at the front to the 7 at the back gives us 28.  Now 63 + 28 = 91.  So, yes, 91 - 63 = 28.

This is called the nines complement method of subtraction.  

Catch me with pen and paper sometime and buy me an oatmeal cookie, and I will show you why it works.  You might not understand my explanation, but at least I’ll get a cookie out of it.

There are many different kinds of people in this world. It takes all kinds of people to make the world work.  One kind will think this is kind of neat.  They will want to just play around with it and show people how to do it on napkins at cocktail parties.

Another kind will wonder what it’s good for. Some of these will get nasty and start saying things like this are what’s wrong with education in this country. Our math teachers are teaching all of these new-fangled ways to subtract when we should just teach the old fashioned way that I was taught...

I think I have that recorded on an old 45 vinyl record somewhere;  maybe on vellum; maybe pressed into clay with cuneiform.

As for the nines complement method of subtraction, it was used at least as far back as the 1600s.  Blaise Pascal, a French mathematician, invented a mechanical device to add, and it could use the nines complement method to subtract.  There were also mechanical devices in the 1800s and the early 1900s that did this.

So this isn’t new-fangled math.

And it doesn’t depend on working in base 10.  By changing what needs to be changed, it can work with any base.

This came in handy when they invented digital computers. In computers, numbers are encoded in base two. In a computer’s central processing unit, there is a cluster of transistors that is wired together in such a way that allows them to perform addition.  There are literally (and I know what that word means) thousands of transistors devoted just to adding two numbers.

To say the wiring is “complicated” is to stretch that word about as far as it can go.  I suppose they could’ve come up with some other way to wire up transistors to subtract, but somebody, undoubtedly said, “Hey, you know that the nines complement method of subtraction they use on mechanical calculators doesn’t depend on base 10. We can use it in base two as well.”  (Actually he said, “Base two too,” and giggled.)

So they used what is known as ones complement to do subtraction and knocked off work to have a beer or seven.

Anyway, I think this stuff is fun and beautiful and useful.

The End.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like'' the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube. )