Things and Unthings  An approach to negative numbers
There are several wellknown ways of explaining negative numbers  earnings and debts, east and west. These work
quite easily for addition and subtraction but are rather troublesome for multiplication and division.
Now, of course, since in this world we cannot have 3 apples, there must be some sense of unreality connected with
negative numbers. I am about to describe a universe in which you could have 3 apples. This is a little fantastic
but that, if anything should commend it to children. But once we have qot this universe imagined, there is no cheating.
In it, children could learn to add and multiply negative numbers with the same apparatus and procedures that actual
children use in Grade 1 for actual numbers.
Grade 1 arithmetic is really a series of experiments in physics. You put two apples and two apples together and
find you have four apples. Or, starting with four apples you can break them up into two apples and two apples.
I will write in the form 2 T & 2 T <> 4 T the fact of physics that two things and two things can be
combined into four things and vice versa. Children are thus led to write the arithmetical statement 2+2 = 4.
We now make our break with the accepted physics, and suppose the world contains not only things but also unthings.
If an unthing and a thing meet, they wipe each other out. An unapple wipes out an apple and an undog wipes out
a dog, and so on. It can work the other way round too. If you have a box with nothing in it, a little later you
may find that it contains a tiger and an untiger or a dollar and an undollar. (This latter fact raises special
problems in connection with forgery.)
The illustration above shows 3 things and 3 unthings. An unthing is written UT for short. 3 things and two unthings
can change, reversibly, into 2 things and 1 unthing, or simply into 1 thing. In symbols
3 T & 2 UT <> 2 T & 1 UT <> 1 T
Use of the symbol U.
If X stands for some collection of objects, UX stands for what would wipe that collection out. Thus it D stands
for dog, and C for cat, U (C & D) stands for what would wipe out a cat and a dog. Clearly, that is an uncat
and an undog, i.e.
UC & UD. so U(C & D) = UC & UD. U obeys the distributive law.
Now, what about U ( UC )? This means something that wipes out an uncat. But that is a cat; for
when a cat and an uncat meet, they combine to form emptiness.
So U (UC) = C.
UnSeven.
Th answer to, a question, "How many?". is a number. Suppose we are asked, "How many cats in this
room?" and there are in fact seven uncats in the room. I suggest that the answer should be "Unseven".
We may write this number u7. So "seven uncats" and
"unseven cats" are simply two ways of saying the same thing.
Adding and Subtracting.
Adding and subtracting can now be done in the usual way. For example, what is
3 + u5? Get 3 cats and 5 uncats. Three of the uncats will
wipe out the three cats, and in no time you be left with two uncats.
3 C & 5 UC <> 2 UC
3 + u5 = u2
.
For subtraction; what is 3  u5? You have
3 cats and you you want to give someone 5 uncats; what will you have left afterwards? Well, take your 3 cats and
stir them up a bit ; with luck they will change into 8 cats and 5 uncats. Give away the 5 uncats. You are left
with 8 cats.
Answer, eight.
Alternative method, ask, "Unfive cats and
what makes three cats?"
Multiplication.
This too is done in the usual way. It we want to, find what 3 x 2 is, we first take two things, TT; then we take
three sets of two things, TT TT TT. This gives us six things.
As 3 . (2 T) <> 6T, we say 3 x 2 = 6.
Exactly the same method will work it we want (u3)
x (u2).
We must form unthree sets of untwo things each. Untwo things is the same as two unthings (see above, in paragraph
headed "Unseven". ).
So untwo things is the set UT & UT. We want unthree such sets.
This is the same as three unsets The unset is T & T, or 2 T.
Three of these gives 6 T  Thus (u3) . (u2 T) = 6 T and we say (u3) x (u2) = 6 .
One could also carry through this argument by using the kind of diagram shown earlier, with things as mountains
and unthings as valleys into the mountains may fall. This would be probably easier as follows.
3 x (u2) and (u3) x 2 can be done in the same way.
It only remains to identify u3 with the usual
sign 3.
Since 3 things and unthree things combine to give emptiness, it is clear that
3 + (u3) = 0. Thus u3 = 0  3; u3 is the result of subtracting 3 from zero.
This is ordinarily called 3.
The use of u allowed us to distinguish between
unthree (negative 3; 3) and "subtract 3".
Unthree cats is an object. "Take away three cats" is an order.
UnThings and Electrons.
As a concluding note it may be be pointed out that unthings are perhaps not so fantastic after
all. In Dirac's theory of electrons, there were things called "electronholes" which, when they met electrons
destroyed the electrons and themselves ceased to be. Of course, this annihilation of matter produced considerable
radiation. If one ignored the radiation effects, one would have an actual physical model for things and unthings.
Copyright © W. W. Sawyer & Mark Alder 2001
This version 23th February 2001
Back
