20150911 THE MYSTERY OF NEGATIVE NUMBERS:
Kids know them, teachers don't.
From different students of different elementary schools, I heard that many in their classes knew about negative numbers, yet the teachers were telling them that it is not possible to subtract a bigger number from a smaller one.
If your student is smart enough to understand negative numbers, why would not you try to perform a forbidden subtraction? What, for example, is 203-447? I am not sure how exactly they do subtraction in reformed US schools, so I tried to make it very clear.
The first line on the illustration has position numbers. The blackened and underlined position is where we are. Click on FORTH and watch. When you reached #3, please remember that spaces in front of a positional number (as well as behind it) is, in fact, the infinite line of zeros. Without them even addition would not be possible.
Remark that FORTH is on the left from BACK. That's because we Europeans have borrowed positional numeral system from Arabs together with their writing order.After #5 the demonstration ends. Clearly, we can go like this forever. I put … in front 9 to indicate that the infinite line of 0s became the infinite line of 9s. Few extra 9s were for clarity. …9 or …0 would (and will) be enough.
The infinite line of 9s looks scary, but no scarier than a periodic fraction. I decide not to use any special mark (like vinculum on top of a periodic part). For this demo an ellipsis before 9 is good enough.
We could do subtraction in strict accordance with the rules, and we can do it for any other two numbers, putting the smaller one first, but what's the use of those forever nine numerical freaks?
Let's try to add 447 to …9756. Here is how.
Going like this forever, we will get …0203, or just 203. Remark that it was not subtraction. Just plain addition. Is …9756 secretly equal to -244?
It's no different. Try do add it to some other numbers, big and small, to make sure that we found a perfectly workable representation of negative quantity, which eliminates subtraction entirely.
Wait a moment, how about number sense? An infinite line of 9s looks like an awful lot!
No matter how it looks, the biggest whole positional number …9 (an infinitely big whole number made entirely of nines) is just one smaller that 0. Try to add 1 to …9. It becomes 0. This can only mean that …9=-1 (…9.9…, which is infinite 9 to the left and infinite 9 to the right from the decimal point, is infinitely approaching 0).
This means, in turn, that our familiar number line is actually an infinitely wide circle. The biggest number we can think of approaches 0 from the other side.
Remark though, that this observation is not about quantity or geometry. It's about positional numeral system. It would not make much sense, for example, to ask infinitely deep questions like will the wave of additions will ever catch up with the wave of subtractions. Once they traveled out of view and far enough, we can use the visible part like there is nothing to the left from it. After all, there are no infinite positional numbers in real life.
Just couple more paragraphs, OK? …90=-10, …900=-100 and so forth. The number …9756 is -1000+756=-244, precisely as expected. Or how about multiplication?
…9756+…9756+…9756=…9268
Please don't think that this chapter was a singular pointless bump on the smooth road to numeracy. If you got it, you will easily understand how machines, including computers, handle quantities. And I have more to tell.
I've never happened to teach negative numbers. Every student in my life, big or little, already knew them. The knowledge of infinity is not so common, but every kid was pretty comfortable learning it.
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