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%I #29 Mar 09 2024 11:17:21
%S 2,9,7,3,0,1,7,7,8,7,5,0,6,8,0,2,6,6,6,7,9,3,7,4,9,9,2,6,4,0,1,1,8,9,
%T 7,8,8,2,3,2,4,3,0,2,3,1,1,5,9,5,4,3,5,3,2,5,4,7,5,0,5,5,6,1,7,9,8,6,
%U 6,6,6,7,0,5,6,7,2,9,2,8,9,9,6,7,6,9,5,3
%N Decimal expansion of 2^(-7/4); exact length of the A4 paper size measured in meters according to the ISO 216 standard.
%C Also exact width of the A3 paper size measured in meters.
%C According to the ISO 216 standard, the A0 paper size is defined to have an area of 1 square meter where the ratio of the length to the width is sqrt(2), so the length is 2^(1/4) m and the width is 2^(-1/4) m. For each n >= 0, the length of the size A(n+1) is equal to the width of the size A(n) and the width of the size A(n+1) is equal to half of the length of the size A(n), so the area of the size A(n+1) is half of that of A(n). Equivalently, the length of the A(n) size is 2^(-n/2 + 1/4) m and the width is 2^(-n/2 - 1/4) m. For the A4 size, the exact length and width are 2^(-7/4) m = 297.301... mm and 2^(-9/4) m = 210.224... mm (A308321), and the actual length and width are 297 mm and 210 mm.
%H Prepressure, <a href="https://www.prepressure.com/library/paper-size/din-a4">A4</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/ISO-216">ISO 216</a>.
%F Equals square root of A222066. - _R. J. Mathar_, Jan 27 2021
%e 0.29730177...
%e The exact lengths and widths (rounded to the nearest 1/10 mm) and areas of the A-series are as follows:
%e .
%e size | exact length | exact width | exact area (mm^2)
%e A0 | 2^( 1/4) m = 1189.2 mm | 2^(- 1/4) m = 840.9 mm | 1000000
%e A1 | 2^(- 1/4) m = 840.9 mm | 2^(- 3/4) m = 594.6 mm | 500000
%e A2 | 2^(- 3/4) m = 594.6 mm | 2^(- 5/4) m = 420.4 mm | 250000
%e A3 | 2^(- 5/4) m = 420.4 mm | 2^(- 7/4) m = 297.3 mm | 125000
%e A4 | 2^(- 7/4) m = 297.3 mm | 2^(- 9/4) m = 210.2 mm | 62500
%e A5 | 2^(- 9/4) m = 210.2 mm | 2^(-11/4) m = 148.7 mm | 31250
%e A6 | 2^(-11/4) m = 148.7 mm | 2^(-13/4) m = 105.1 mm | 15625
%e A7 | 2^(-13/4) m = 105.1 mm | 2^(-15/4) m = 74.3 mm | 7812.5
%e A8 | 2^(-15/4) m = 74.3 mm | 2^(-17/4) m = 52.6 mm | 3906.25
%e A9 | 2^(-17/4) m = 52.6 mm | 2^(-19/4) m = 37.2 mm | 1953.125
%e A10 | 2^(-19/4) m = 37.2 mm | 2^(-21/4) m = 26.3 mm | 976.5625
%e .
%e And the actual lengths, widths and areas (note that the actual areas are always smaller than the exact areas) are as follows:
%e .
%e size | actual length (mm) | actual width (mm) | actual area (mm^2)
%e A0 | 1189 | 841 | 999949 (99.9949%)
%e A1 | 841 | 594 | 499554 (99.9108%)
%e A2 | 594 | 420 | 249480 (99.7920%)
%e A3 | 420 | 297 | 124740 (99.7920%)
%e A4 | 297 | 210 | 62370 (99.7920%)
%e A5 | 210 | 148 | 31080 (99.4560%)
%e A6 | 148 | 105 | 15540 (99.4560%)
%e A7 | 105 | 74 | 7770 (99.4560%)
%e A8 | 74 | 52 | 3848 (98.5088%)
%e A9 | 52 | 37 | 1924 (98.5088%)
%e A10 | 37 | 26 | 962 (98.5088%)
%t RealDigits[2^(-7/4),10,88][[1]] (* _James C. McMahon_, Feb 26 2024 *)
%o (PARI) default(realprecision, 100); 2^(-7/4)
%Y Cf. A010767 (2^(1/4)), A228497 (2^(-1/4)), A308321 (2^(-9/4)).
%K nonn,cons
%O 0,1
%A _Jianing Song_, May 20 2019
%E Edited by _Jon E. Schoenfield_, Feb 25 2024
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