ixe=+Masaya SasakiS&V Sensor Development Group, R&D Department, Technical Development Center. He studied mathematics and programming at the Department of Mathematics and Informatics in college before joining Rion. He is involved in the development of instrument amplifiers (preamplifiers) that connect microphones to measuring instruments.Column by Rion’s staff on their obsession with scienceInterview by Hideki Hamaguchi and article by Kyoko Hamanaka and Yuki YamakawaRion is supported by many science and math-loving staff members. In this series, our science-minded staff members write about their enthusiasm for their respective fields of interest. Part 7 is about Eul-er’s formula.20My job involves the development of preamplifiers that connect microphones to measuring instruments. Ever since I was a child, I’ve been interested in arithmetic and mathematics, and enjoyed the process of finding solutions. I majored in mathematical sciences at my university, where I first en-countered Euler’s formula, often said to be the most beautiful formula known. I’ll try to explain what Euler’s formula is.* *Euler’s formula is often called the most beautiful formula because it’s simple, easy to understand, and has applications across many different fields. For example, we can represent the waveforms of sound and vi-bration, which I work with in my job, using trigonometric functions (sine or cosine) if they are simple sine waves. Complex wave-forms can be expressed as the sum of mul-tiple simple sine waves based on the super-position principle. Without Euler’s formula, you would need to perform trigonometric calculations whenever you analyzed com-plex waveforms. Euler’s formula makes it possible to use just exponential functions. Calculations with exponentials have the ad-vantage that the products of powers can be expressed as sums or differences of the ex-ponents, which can dramatically simplify the differential and integral calculus you have to do. In other words, Euler’s formula can Euler’s formulaHere, x is an arbitrary complex number, e is Napier’s constant or Euler’s number (e = 2.718...), i the imaginary unit, cos the cosine function, and sin the sine function. Euler’s formula establishes the relationship between the exponential function and trigonometric functions in the world of complex numbers.* ** *Formulas to unravel the mystery of wavessimplify calculations dramatically.To be honest, Euler’s formula didn’t interest me much when I was a student. That’s because I was more interested in generalization while I was studying in uni-versity. Generalization is the approach of first establishing that a problem or theorem is valid in a narrow range of conditions, then showing that it remains valid across a wider set of conditions that ultimately include your target.No. 007For example, if our goal is to establish that something is valid for real numbers, we first consider natural numbers, which have a narrower range than real numbers. Natural numbers are easy to think about because they’re basically the positive integers: 1, 2, 3 ... . From there, we expand into integers, and then into rational numbers, and finally show that our approach is valid for real numbers. The proof for Euler’s formula relies on one popular method in generalization called mathematical induction. While I was in col-lege, I was more interested in the process of derivation than the conclusions we arrived at. Another reason I was less interested in the formulas ultimately obtained was that I didn’t have the opportunity at that time to see how they were applied in the real world.After joining Rion, I found myself in situations where I was putting Euler’s for-mula to actual use in my work, and that’s when I recognized the practical aspects of mathematics. I’m responsible for developing sensors that measure physical quantities like sound and vibration and convert them into signals. One method for checking sensor input/output is Fast Fourier Transform (FFT), an algorithm that converts waveform data in the time domain into spectral data in the frequency domain. Since the data is convert-ed into spectral data, we can visualize it in a simple graph format, with the vertical axis representing signal level and the horizontal axis representing frequency, allowing us to visually check the waveforms. I discovered that Euler’s formula is utilized to understand Fourier transforms. In addition to FFT, Euler’s formula also has applications to machine learning, a hot topic right now. That’s how my appreciation of the practical value of Euler’s formula has grown since I joined Rion. People often say: What use is mathematics? Well, at Rion, I was impressed when I found myself in situ-ations where mathematics proved useful in manufacturing, which convinced me of its practical value.EPILOGUE SCIENCE, SCIENCE! Echo of Mathematics: Expanding the World cosxi sinxBecause We’re Science and Math Lovers
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