Ex) Article Title, Author, Keywords
New Phys.: Sae Mulli 2021; 71: 771-774
Published online September 30, 2021 https://doi.org/10.3938/NPSM.71.771
Copyright © New Physics: Sae Mulli.
Chang Uk Jung*
Department of Physics and Memory and Catalyst Research Center, Hankuk University of Foreign Studies, Yongin 17035, Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
“Significant figures” are basic elements of science. The convenient rule-of-thumb in handling significant figures is to meet the top constraint; that is, “the answer you obtain should not be more precise than the numbers you started with.” I previously demonstrated that applying this rule-of-thumb in calculating the areas of squares and the volumes of cubes could easily and widely violate the top constraint of significant figures. In most textbooks dealing with significant figures in a polar coordinate system with rotational symmetry, I recently found a “more significant” error absent in an ordinary rectangular coordinate system. Herein, I suggest a simple prescription in the notation of angles in polar coordinates.
Keywords: Significant figures, Rule-of-thumb, Error area, Polar coordinate system, Rotational symmetry
The “Wikipedia” page on “significant figures (SFs)” show the “top constraint” rule-of-thumb (ROT) in the handling of SF: “For quantities created from measured quantities by multiplication and division, the calculated result should have as many significant figures as the measured number with the least number of significant figures. For example, 1.234 × 2.0 = 2.468… ≈ 2.5.” The first chapter of most undergraduate study textbooks at universities of natural science contains the ROT in the handling of SF. This ROT is also observed in nearly all chemistry and elementary physics textbooks. [1–4] Recently, I demonstrated that the application of convenient ROT resulted in a serious violation of the top constraint in calculating the area of squares and the volume of cubes.
I have found another significant error in SF calculations given in most textbooks. When we report that a force
The special theory of relativity is based on a hypothesis that suggests that physics laws are the same among all observers who move at a constant velocity vector with each other. Similarly, the accuracy of the total force vector, equivalently, the error area, should nearly be the same and “single-valued” when using either the rectangular coordinate system or polar coordinate system.
Typical exercise problems in early chapters of nearly all elementary physics (or chemistry) university textbooks are presented in the followings. [1–3] For example, a hockey puck received two forces simultaneously. One vector
Most textbooks state that the angle of the total force is 30.0°, which has three SFs with an error angle of Δ
The physical systems in the two exercise problems, i.e.,
We may express
Furthermore, Fig. 2 shows the obscurity in handling SFs in the polar coordinate system by calculating the error area. First, we start with Exercise 1. The vector
How about the polar coordinate system? As shown in Fig. 2(b), the ROT is
Radians are more suitable for dealing with error areas in polar coordinates. Therefore, I can start from the radian angle, as shown in Exercise 1-1. By simply changing angle from degrees (30.0°) to radian (0.524), the error-area changes significantly.
If I exchange the magnitude of forces between
Therefore, handling SF during the conversion from the rectangular coordinate system to the polar coordinate system is very misleading and can easily violate the top constraint on SF if the ROT is used. When we change the position of some vector from a rectangular coordinate system to a polar coordinate system, we must carefully use the ROT on SF. One simple prescription for the examples given in this report is that the angle should be expressed in the following form:
The convenient ROT in handling SF uses the top constraint that “the answer you obtain should not be more precise than the numbers you measured with.” Each undergraduate student in the department of Physical Education after graduation will educate much more students in high school but the undergraduate students were found to have difficulty in handling SF even in conventional way.  I want to alert students that transforming a vector from the rectangular to the polar coordinate system results in unexpected errors. The simple ROT results in very unreasonable precision when a position vector is transformed from the rectangular to the polar coordinate system. I suggest a simple method for the notation of angles in polar coordinates.
The author appreciates H. J. Choi, G. C. Yi, J. Y. Park, and M. Ryu for reading and commenting on the manuscript. The author appreciates Dr. Venkata Raveendra Nallagatla for his assistance in preparing figures and discussions. This work was supported by Hankuk University of Foreign Studies Research Fund of 2021. This work was partly supported through the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT; No. 2020R1A2C2006187).