Measurement of Mathematical Constant π and Physical Quantity Pi

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1. Introduction

The number p is a mathematical and physical constant which originated as the ratio of a circle’s circumference to its diameter. Throughout scientific history it became significant as it appeared across all fields of mathematics and physics having little to do with the geometry of circles. Some of those fields are mathematical number theory, statistics, cosmology, thermodynamics, mechanics, quantum mechanics and electromagnetism. Great efforts have been invested in finding as many decimal digits of that number as it is possible. On the contrary, in this article we focus on the measurement associated with the mathematical constant p and the physical quantity Pi. So, essentially we have to distinguish two different things, the number p and the physical quantity Pi. Many have focused exclusively on the calculation of the number p and not to measure it. To be able to measure it, we need to find a link between the number p and the physical quantity Pi.

2. Definition of the Radian

The radian is the standard unit of angular measure. By definition, the radian (symbol rad) is the plane angle unit in the International System of Units (abbreviated SI). It represents the central angle on a circumference and covers an arc whose length is equal to the radius r of the circuit (Figure 1):

. (1)

Any angle formed by two radii, measured in radians, is equal to the arc length between two spokes, divided by radius r (Figure 1). So, the full angle (360˚) is:

, (2)

where is a transcendental number, i.e., a fundamental mathematical constant with the decimal expansion 3.141592653…, tens of billions of decimal places have been calculated; just 39 places suffice to compute the circumference of the observable universe accurate to within the radius of a hydrogen atom. The number is sometimes known as Archimedes’ constant or Ludolph’s constant, after Ludolph van Ceulen (1539-1610), a German-Dutch mathematician [1] .

We should distinguish two different things; first, the mention number and a second, the mathematical and physical quantity denoted by Pi. While p is a number, Pi is a quantity defined as the ratio of the circumference of a circle () to its diameter (d = 2r). Actually, according to the definition of radian, and using Equation (2), the quantity Pi is the angle, because is defined as any angle:

. (3)

There are many formulas for calculating the number p, from the simple to the very

Figure 1. Description of radian (1 rad), straight angle, full angle, and arc length; the symbol “” in the appropriate scale means “corresponds to”.

complicated. While many researchers’ oriented to calculating the number p in the largest possible number of decimals, we will focus here our efforts at something else, namely, how to measure the mathematical and physical quantity Pi.

All the measurable quantities are physical quantities. The measurement is associated with unit (rad, in our case) and numerical value, like number p in Equation (3):

. (4)

In general, the quantities can be measured directly or indirectly. E.g. measuring the length of road, weight of body and so on, are the direct measurement. Measuring the mass of Earth, electron, and so on are indirect measurement. Measuring of the quantity Pi can be made both ways, directly and indirectly.

3. Direct Measurement of the Quantity Pi

Formula for the arc length of the curve (circle, Figure 1)

(5)

read [2] :

. (6)

The derivative of Equation (5) is equal to

, (7)

so

. (8)

When the arc length becomes equal to the radius,

, (9)

i.e.,

, (10)

then the corresponding angle, according to the definition in Equation (1), is equal to 1 rad (Figure 1). In this way chosen is the arc that belongs to the angle of one radian. With this arc, as units of measurement (, i.e., r corresponds to one), we can now measure any other arc, of course and one that belongs to a straight angle (Figure 1):

. (11)

In this way we can directly measure the length of the arc belonging to the angle

. What we measure in accordance with Equation (8) is

(12)

The numeric value in such measurement of the arc we denoted by the Greek letter.

4. Indirect Measurement of the Quantity Pi

To perform measurements the presented method uses liquid in a container of any shape which volume is V_{c} (Figure 2), relying on the document which is in the process of obtaining a patent in the UK [3] . The bottom surface of the measuring pot of any shape is

, (13)

where k is a constant dependent on the shape of the bottom surface; x_{m} is a variable, which depends on both the shapes of containers and measuring pots, and which is also the calibration parameter.

When the entire content of the liquid from the container is poured into the measuring pot, and the height of the liquid reaches the amount h, using Equation (13) (Figure 2) we get:

. (14)

The shape and dimension of the measuring pot can be chosen at our will. We can choose them such that height h of liquid in the measuring pot is:

, (15)

where this relation is a requirement for calibration of the system. Now Equation (14) reads as follows:

Figure 2. Perspective view of the container V_{c} of any shapes and the measuring pot V_{m} as well as of any shapes but with the bottom surface of which can be displayed as.

, (16)

where we get

. (17)

To perform independent measurements of the number p we should get rid of it from Expression (17). This can be achieved by choosing the container which volume V_{c} depends on the number p, what gives us to possibility to eliminate the number p from Equation (17).

5. Calibration of the Measuring System

The easiest way to measure the number p using the system shown in Figure 2, is by using Equation (4) and Equation (15):

, (18)

which means that for selected x_{m} [according to Equation (17)] and after that fixed for a given system, the number p in this system is proportional to the height of the liquid column h in the measuring pot.

A unit of measurement is a magnitude of a quantity, defined an adopted by convention or by law, that is used as a standard for measurement of the same quantity. This means that in our system we can at will choose x_{m} as a unit of measurement. This further means that x_{m} corresponds to the unit; i.e.,. Then we write Equation (18):

. (19)

Using the above described system of calibration we can made different forms of containers and measuring pots, and they can realize a few systems of measurement (Table 1).

The two combinations of system to measure the number p, resulting from Table 1, are shown in Figure 3 and Figure 4.

6. Conclusion

The direct and indirect measurements of the number p have been shown. To be able to do that, the difference between the number p and physical quantity Pi has been explained. It has been shown that physical quantity Pi is actually a stretched angle, while p is a number which characterizes this angle. The idea (in patent application process) according to which the content of the liquid from the container equated with the content of measuring pot and so allowed us to use the measuring pot scale to directly read the number p has been used. Measurement accuracy of such a system depends on the precision performance of containers and measuring pots, and the conditions in which such measurement is made (type of liquid, ambient temperature and tilt of the system). The

Table 1. Various exemplary embodiments measuring pots and containers, and their pairing.

a. Ref. [4] , b. In that case the number p must be known before the measurement.

Figure 3. Perspective view of the sphere as a container V_{c} and rectangular cuboid as a measuring pot V_{m}.

simplest implementation is the use of a cylinder as container and cuboid as a measuring pot (Figure 4).

Acknowledgements

The founding of this paper is, each in its own special way, contributed by Ms. Srebrenka Ursic, Mr. Branko Balon, Mr. Damir Vuk and especially Mr. Zlatko A. Voloder.

Figure 4. Perspective view of the cylinder as a container V_{c} and rectangular cuboid as a measuring pot V_{m}.

They all have my gratitude.

References

[1] http://mathworld.wolfram.com/Pi.html

[2] Bronstein, I.N. and Semendjajev, K.A. (1964) Matematicki prirucnik, Tehnicka knjiga, Zagreb.

[3] Perkovac, M. and Voloder, A.Z. (2016) Method and Teaching Aid for Measuring the Value of Mathematical Constant Pi Using Liquid. Patent Application Nr. GB1612615.3.