This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A229962 #98 Feb 11 2025 14:14:25 %S A229962 2,1,1,9,8,5,2,8,0,0,0,0,3,8,3,2,3,8,8,7,3,9,4,4,1,0,8,5,9,0,8,5,4,7, %T A229962 4,7,2,0,6,1,3,9,5,2,7,8,8,6,3,6,2,4,6,9,6,9,8,0,0,0,3,4,3,4,6,5,5,1, %U A229962 8,8,3,5,4,6,9,2,9,3,5,6,4,5,1,8,0,2,9,5,8,6,5,8,4,3,2,1,5,2,2,2,1,6,6 %N A229962 Decimal expansion of 149896229*sqrt(2). %C A229962 Also decimal expansion of the speed b = c/sqrt(2) in SI units (meter/second), where c = 299792458 (m/s) is the speed of light in vacuum (A003678). %C A229962 A particle (or object) with speed b has the property that its relativistic momentum equals the momentum of a virtual photon whose energy equals the rest energy of the particle. Also its relativistic de Broglie wavelength equals the Compton wavelength for the particle and therefore equals the wavelength of the photon mentioned above. %C A229962 More generally it appears that the speed b is a critical speed for several relativistic magnitudes of the particle. Explanation: consider a table of relativistic magnitudes in which every formula is written as the product of a dimensionless factor and a constant with the same dimensions as the relativistic magnitude. For instance, for the relativistic momentum we write the formula p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c instead of the standard formula p = [1/(1 - v^2/c^2)^(1/2)]*m_0*v. See below: %C A229962 Table 1. %C A229962 ---------------------------------------------------- %C A229962 Relativistic %C A229962 magnitude Formula %C A229962 ---------------------------------------------------- %C A229962 Speed.........: v = [v/c]*c %C A229962 Group velocity: g = [v/c]*c %C A229962 Length........: L = [1/γ]*L_0 %C A229962 Momentum......: p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c %C A229962 Wavenumber....: k = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c/h %C A229962 Wavelength....: W = [(c^2/v^2 - 1)^(1/2)]*h/(m_0*c) %C A229962 Time interval.: t = γ*t_0 %C A229962 Mass..........: m = γ*m_0 %C A229962 Energy........: E = γ*m_0*c^2 %C A229962 Frequency.....: f = γ*m_0*c^2/h %C A229962 Phase velocity: w = [c/v]*c %C A229962 Kinetic energy: K = [γ - 1]*m_0*c^2 %C A229962 ---------------------------------------------------- %C A229962 Where: %C A229962 v is the speed of the object or particle. %C A229962 c is the speed of light in vacuum (A003678). %C A229962 h is the Planck constant (A003676). %C A229962 L_0 is the length at rest of the object or the length at rest of a virtual cube which contains the particle. %C A229962 m_0 is the mass at rest (for the electron see A081801, for the proton see A070059). %C A229962 t_0 is the time interval at rest. %C A229962 W is the relativistic de Broglie wavelength assuming that W = h/p. %C A229962 γ = [1/(1 - v^2/c^2)^(1/2)] is the Lorentz factor. %C A229962 Then table 1 can be unified as shown below: %C A229962 Table 2. Table 3. %C A229962 ------------------------------------ ----------------- %C A229962 Relativistic %C A229962 magnitude Formula Formula %C A229962 ------------------------------------ ----------------- %C A229962 Speed.........: v = sin(x) * c v = sin(x) * v’ %C A229962 Group velocity: g = sin(x) * c g = sin(x) * g’ %C A229962 Length........: L = cos(x) * L_0 L = cos(x) * L’ %C A229962 Momentum......: p = tan(x) * m_0*c p = tan(x) * p’ %C A229962 Wavenumber....: k = tan(x) * 1/W_C k = tan(x) * k’ %C A229962 Wavelength....: W = cot(x) * W_C W = cot(x) * W’ %C A229962 Time interval.: t = sec(x) * t_0 t = sec(x) * t’ %C A229962 Mass..........: m = sec(x) * m_0 m = sec(x) * m’ %C A229962 Energy........: E = sec(x) * E_0 E = sec(x) * E’ %C A229962 Frequency.....: f = sec(x) * E_0/h f = sec(x) * f’ %C A229962 Phase velocity: w = csc(x) * c w = csc(x) * w’ %C A229962 Kinetic energy: K = ese(x) * E_0 K = ese(x) * K’ %C A229962 ------------------------------------ ----------------- %C A229962 Where: %C A229962 E_0 = m_0*c^2 is the energy at rest (for the electron see A081816, for the proton see A230438). %C A229962 W_C = h/(m_0*c) is the Compton wavelength for the particle (for the electron see A230436, for the proton see A230845). %C A229962 ese(x) = sec(x) - 1. %C A229962 Table 2 is simpler than table 1 because the relativistic factors are written as trigonometric functions of the angle x assuming that sin(x) = v/c and that 0 < x < Pi/2. %C A229962 Table 3 lists the simplest formulas in which the values of the constants have been interpreted as the values of the magnitudes of a virtual photon whose energy E' = h*f' is equivalent to E_0 = m_0*c^2, the rest energy of the particle. %C A229962 A visualization of the relationship between the relativistic magnitudes, the quantum constants and the trigonometric functions is obtained using the first quadrant of the trigonometric circle according to the simplest table, see below: %C A229962 Table 4. %C A229962 ----------------------------------- %C A229962 sin(x) = v/v' = g/g' %C A229962 cos(x) = L/L' %C A229962 tan(x) = p/p' = k/k' %C A229962 cot(x) = W/W' %C A229962 sec(x) = t/t' = m/m' = E/E' = f/f' %C A229962 csc(x) = w/w' %C A229962 ese(x) = K/K' %C A229962 ----------------------------------- %C A229962 Finally we can write that b is a critical speed because: %C A229962 If v = b, for instance, we have that: %C A229962 1) v/v’ = L/L’ = sin(Pi/4) = cos(Pi/4) = 2^(1/2)/2. %C A229962 2) p/p’ = W/W’ = tan(Pi/4) = cot(Pi/4) = 1. %C A229962 3) E/E’ = w/w’ = sec(Pi/4) = csc(Pi/4) = 2^(1/2). %C A229962 Otherwise if v < b we have that: %C A229962 v/v’ < L/L’ and p/p’ < W/W’ and E/E’ < w/w’. %C A229962 Otherwise if v > b we have that: %C A229962 v/v’ > L/L’ and p/p’ > W/W’ and E/E’ > w/w’. %H A229962 G. C. Greubel, <a href="/A229962/b229962.txt">Table of n, a(n) for n = 9..10008</a> %H A229962 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polreg02.jpg">A trigonometric model of the Relativity: Fig. 1</a>, <a href="http://www.polprimos.com/imagenespub/polreg04.jpg">Fig. 2</a>, <a href="http://www.polprimos.com/imagenespub/polreg03.jpg">Fig. 3</a>, <a href="http://www.polprimos.com/imagenespub/polreg05.jpg">Fig. 4</a>, <a href="http://www.polprimos.com/imagenespub/polreg06.jpg">Fig. 5</a>, <a href="http://www.polprimos.com/imagenespub/polreg07.jpg">Fig. 6</a>, <a href="http://www.polprimos.com/imagenespub/polreg10.jpg">Fig. 7</a>, <a href="http://www.polprimos.com/imagenespub/polreg11.jpg">Fig. 8</a> %H A229962 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polreg12.jpg">A table of relativistic factors</a> %H A229962 Wikipedia, <a href="http://en.wikipedia.org/wiki/Quantum_mechanics">Quantum mechanics</a> %H A229962 Wikipedia, <a href="http://en.wikipedia.org/wiki/Special_relativity">Special relativity</a> %H A229962 Wikipedia, <a href="http://en.wikipedia.org/wiki/Trigonometry">Trigonometry</a> %H A229962 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>. %F A229962 A003678/A002193 = A003678*A010503. %e A229962 211985280.000383... m/s. %t A229962 RealDigits[149896229*Sqrt[2], 10, 100][[1]] (* _G. C. Greubel_, Jan 26 2018 *) %o A229962 (PARI) 149896229*sqrt(2) \\ _G. C. Greubel_, Jan 26 2018 %o A229962 (Magma) 149896229*Sqrt(2); // _G. C. Greubel_, Jan 26 2018 %Y A229962 Cf. A002193, A003676, A003678, A010503, A070059, A081801, A081816, A182999, A229952, A230436, A230438, A230844, A230845, A231202, A231350. %K A229962 nonn,cons %O A229962 9,1 %A A229962 _Omar E. Pol_, Nov 10 2013