What's the Use of Measuring to 11 Significant Figures?

At the end of 2016, a topic went viral about an elementary school teacher who marked down an answer sheet that read 3.9 + 5.1 = 9.0. Now, even TV personality Osamu Hayashi and mathematicians have appeared, with proponents and opponents developing their own arguments, but in mathematics, there is no dispute that 9.0 = 9, and it was an educational debate. On the other hand, in engineering and the natural sciences, 9 and 9.0 are used distinctly. 9.0 indicates a range from 8.95 to 9.05, which is different from 9, which indicates a range from 8.5 to 9.5.

Measurement is the first step in the natural sciences. Have you ever been scolded for simply copying the answer from your calculator in a lab report? Knowing to how many significant figures a measured quantity is reliable is just as important as the measured value itself. Measurement involves devising equipment to reduce measurement uncertainty, repeating measurements and performing statistical processing, and considering uncertainties that cannot be eliminated (for example, the uncertainty of a ruler).

Frequency is currently the physical quantity that can be measured most accurately. This is because a frequency standard (a reference or ruler) with small uncertainty can be obtained. When atoms and molecules in a gas are irradiated with light or radio waves, sharp absorption spectral lines are observed, and their center frequency is constant regardless of time or place. For this reason, the spectral lines of atoms and molecules serve as excellent standards. Currently, one second is defined by the spectral line of the cesium atom in the radio wave region, and using this as a standard allows for measurements with 15 significant figures.

In our group, we observe absorption spectral lines in the infrared region caused by the vibrational motion of molecules and measure their center frequencies with 11 significant figures. Figure 1 shows the observed absorption spectrum of a methane molecule. The width of the resonance spectral line was about 300 kHz, and the center frequency was measured as (88 376 181 600.3 ± 2.1) kHz. This corresponds to a resolution (sharpness of the spectral line) of 2.9 x 10 ⁸ and a relative frequency uncertainty of 2.4 x 10 ^–11 . The ability to perform such high-precision frequency measurements outside of the standard Research Centers and Institutes of developed countries is thanks to the optical frequency comb (a device that allows the frequency standard of the radio wave region to be used in the optical region, for which its developers were awarded the 2005 Nobel Prize in Physics) and the proliferation of GPS, which provides location information to smartphones and car navigation systems.

Spectroscopy, which studies atoms and molecules using light, made a significant contribution to the establishment of quantum mechanics in the early 20th century and is still utilized throughout the natural sciences and engineering. The energy levels of atoms and molecules are largely determined by the Coulomb interaction between the nucleus and electrons. When the resolution of spectroscopic measurements in the visible light region exceeds 10 ⁴ , relativistic effects and quantum electrodynamic effects can be observed. When it exceeds 10 ⁸ , the effect of nuclear spin is observed, and when it exceeds 10 ¹¹ , the effect of the momentum carried by photons is observed. In this way, as spectroscopic measurements have become more precise, the fundamental theories of physics have been successively verified. The accelerators used in high-energy experiments already require energy on a global scale, and development at the same pace as before cannot be expected. On the other hand, laser spectroscopy can approach the fundamentals of physics with overwhelmingly lower costs than accelerators and with the ideas of researchers.

Next-generation frequency standards are being developed, primarily in Japan, and it has recently been demonstrated that measurements with 18 significant figures are possible. An improvement of three significant figures will undoubtedly reveal a new world.