Binary Balance

An important source of real numbers is the physics experiment, a measurement. Therefore, before launching any theory about the reals, I think it's good to have some practical experience, with some realistic physical equipment. The most common measurement probably would be measuring a length. The concept of length, however, is very much related to pure geometry, which is branch of mathematics. We all know that non-measurable numbers have emerged in geometry, meaning that the concept of length may be too well known for our purpose already. We don't want the reader to think of anything mathematical in the first place. Our measurement should be sufficiently abstract with respect to mathematics. Yet it must be sufficiently simple. And reasonably accurate. How about weighting? An example may be how to determine the weight of a tomato. Yes, it's not so difficult to devise a simple balance (: Google up "How to Make a Homemade Weighing Scale"). I've done it myself with a few pieces of wood, a mounting strip, a paper clip, two paper cups, two (equal) pieces of rope and two little pegs. It's not so difficult either to devise a weighting scale, with calibrated weights. Read about the "Basic calibration process" in:

Calibration

Mind that our weighting experiments are meant for revealing certain properties of the real numbers, as they emerge with measurements. So we don't have to conform to all kind of measurement standards. There is no need to adopt the kilogram as a unit, for example. Our unit of weight really can be anything. Another convention that we do not need to adopt is the decimal number system; such as with decimal weights, "suitable for general laboratory, commercial, and educational use":

1kg , 500 g , 2 x 200 g , 100 g , 50 g , 2 x 20 g , 10 g , 5 g ,
2 x 2 g, 1 g , 500mg , 2 x 200mg , 100mg , 50mg , 2 x 20mg , 10mg

Decimal calibration is not what we are looking for indeed. The reason is simple: it's rather bothersome to manufacture these weights at home. We would like to suggest a much more feasible alternative: printing {\em paper}.

Paper_size

Paper, here in Europe at least, comes standard as $A4$. The weight is usually known: $75g/m^2$ is read here for common printer paper. Sixteen of these $A4$ form an $A0$ size paper, with area $ = 1 m^2$ (sic, why would that be?) and hence a weight of $75g$. Which by the way is not an unreasonable weight to start with. Eight $A4$ form an $A1$ size, Four $A4$ make an $A2$. Two $A4$ make an $A3$. But by far the greatest advantage is that it's very easy to fold one $A4$ and make two $A5$ of it, fold one $A5$ and make two $A6$ of it, fold one $A6$ and make two $A7$ of it, fold one $A7$ and make two $A8$ of it, fold one $A8$ and make two $A9$ of it (we only need one of those two). Okay, that seems to be enough for our purpose. What has been accomplished now is that we have a collection of ten "calibrated" weights, non-decimal, {\em binary} weights to be precise. A picture says more than a thousand words:

Meanwhile, we could have built our primitive balance. We take a tomato. And just start the weighting process. The tomato is put into the left cup. The paper weights are put into the right cup. In the beginning, all paper weights are on the table. We start with the heaviest ($A0$) weight and then continue with putting the lighter weights into the right cup, one by one, in this order: $A1$ , $A2$ , $A3$ , $A4$ , $A5$ , $A6$ , $A7$ , $A8$ , $A9$ . See photographs:

1 : 2 : 3 :

     A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

4 :

5 :

6 :

                       A0                              A1                                 A1 A2

7 :

8 :

9 :

                       A1 A3                           A1 A4                              A1 A4 A5

10:

11:

12:

                    A1 A4 A5 A6                     A1 A4 A5 A7                         A1 A4 A5 A8

13:

14:

9 :

                   A1 A4 A5 A9                         A1 A4 A5                           A1 A4 A5

If the left cup is closer to the ground then that means the right cup is too light; so we leave the last paper weight in that cup. If the right cup is closer to the ground then that means the cup is too heavy; so we take the last paper weight out and put it on the table again. If the balance is in equilibrium then we usually decide to stop. Photograph (9) is repeated for comparison with photograph (14): both show the balance "in equilibrium". Here is a schematic overview of weighting the tomato and some other experiments:

pi| on table |right cup  |    |Half Banana|    |Metal Strip|
ct|AAAAAAAAAA|AAAAA AAAAA|    |AAAAA AAAAA|    |AAAAA AAAAA|
nr|0123456789|01234 56789|    |01234 56789|    |01234 56789|
--|----------------------|    |-----------|    |-----------|
 3|XXXXXXXXXX|           |    |X          |    |X          |
 4| XXXXXXXXX|X          |    |XX         |    |XX         |
 5|X XXXXXXXX| X         |    |X X        |    |X X        |
 6|X  XXXXXXX| XX        |    |X  X       |    |X XX       |
 7|X X XXXXXX| X X       |    |X   X      |><  |X X X      |
 8|X XX XXXXX| X  X      |    |X   X X    |    |X X   X    |
 9|X XX  XXXX| X  X X    |><  |X   X  X   |    |X X   XX   |
10|X XX   XXX| X  X XX   |    |X   X   X  |    |X X   XXX  |><
11|X XX  X XX| X  X X X  |    |X   X    X |><  |X X   XXXX |><
12|X XX  XX X| X  X X  X |><  |X   X     X|><  |X X   XXX X|><
13|X XX  XXX | X  X X   X|><   10001.00000      10100.11100
14|X XX  XXXX| X  X X    |><   Equilibrium      Equilibrium
 9|X XX  XXXX| X  X X    |><   10001.00010      10100.11110
Binary number 01001.10000      10001.00001      10100.11101

| weight A0 |    |Zero Weight|    |Heavy Metal|
|AAAAA AAAAA|    |AAAAA AAAAA|    |AAAAA AAAAA|
|01234 56789|    |01234 56789|    |01234 56789|
|-----------|    |-----------|    |-----------|
|           |    |X          |    |X          |
| X         |    | X         |    |XX         |
| XX        |    |  X        |    |XXX        |
| XXX       |    |   X       |    |XXXX       |
| XXXX      |    |    X      |    |XXXXX      |
| XXXX X    |    |      X    |    |XXXXX X    |
| XXXX XX   |    |       X   |    |XXXXX XX   |
| XXXX XXX  |    |        X  |    |XXXXX XXX  |
| XXXX XXXX |><  |         X |><  |XXXXX XXXX |
| XXXX XXXXX|><  |          X|><  |XXXXX XXXXX|
 01111.11111      00000.00010      11111.11111
 Theoretical      00000.00001      out of scope
 10000.00000      Theoretical     no equilibrium
                  00000.00000