Appendices
A.1.1 Zeno’s Paradox
It is very easy to reject the conclusion of this paradox, but not so easy to
refute it formally. In fact, the treatment of infinities was a tough nut for
both mathematicians and scientists for many centuries, up to when modern
infinitesimal Calculus, due to both Isaac Newton (1643–1727) and Gottfried
Leibniz (1646–1716), provided a simple and very elegant tool for its general
solution. In the case of Zeno’s paradox, however, we can offer a simple arithmetical argument to explain it. Let us assume that Achilles needs, say, 1 min
to cover the 100 m. He can then run the next 10 m in 0.1 min and the
following one meter in 0.01 min and so on. Hence the total time to reach
the tortoise will be the sum of all those time intervals, i.e. 1 + 0.1 + 0.01
+ 0.001 + …. = 1.1111…., which is a finite quantity (equal to the rational
number 10/9), in spite of being the sum of an infinite number of terms.
A.1.2 Frequency/Wavelength Limits for Human
Sight and Hearing
Frequency and wavelength are reciprocal quantities. The range of wavelengths
of the electromagnetic spectrum visible to the human eye, called the visible
spectrum, runs between 0.39 and 0.7 μ (microns) approximately, where
a micron is one millionth of a metre. It is bordered to the right by the
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Switzerland AG 2021
R. Barrett and P. P. Delsanto, Don’t Be Afraid of Physics,
https://doi.org/10.1007/978-3-030-63409-4
267
A.1.1 Zeno’s Paradox
It is very easy to reject the conclusion of this paradox, but not so easy to
refute it formally. In fact, the treatment of infinities was a tough nut for
both mathematicians and scientists for many centuries, up to when modern
infinitesimal Calculus, due to both Isaac Newton (1643–1727) and Gottfried
Leibniz (1646–1716), provided a simple and very elegant tool for its general
solution. In the case of Zeno’s paradox, however, we can offer a simple arithmetical argument to explain it. Let us assume that Achilles needs, say, 1 min
to cover the 100 m. He can then run the next 10 m in 0.1 min and the
following one meter in 0.01 min and so on. Hence the total time to reach
the tortoise will be the sum of all those time intervals, i.e. 1 + 0.1 + 0.01
+ 0.001 + …. = 1.1111…., which is a finite quantity (equal to the rational
number 10/9), in spite of being the sum of an infinite number of terms.
A.1.2 Frequency/Wavelength Limits for Human
Sight and Hearing
Frequency and wavelength are reciprocal quantities. The range of wavelengths
of the electromagnetic spectrum visible to the human eye, called the visible
spectrum, runs between 0.39 and 0.7 μ (microns) approximately, where
a micron is one millionth of a metre. It is bordered to the right by the
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Switzerland AG 2021
R. Barrett and P. P. Delsanto, Don’t Be Afraid of Physics,
https://doi.org/10.1007/978-3-030-63409-4
267