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6939.60160373 = 19 × 365.242189670
Number of mean solar days in the metonic cycle of 19 tropical years or 235 synodic months. Its integer approximation (6940) can be used as the basis of a lunisolar calendar that repeats every 19 years, and with some effort you can also make the months fairly regular. For example, every year can have 12 months each of which has a specific name, 6 with 29 days and 6 with 30 days; 7 out of 19 years have a 13th month of 30 days, and 4 of those 7 years also add an extra day to one of the normal 29-day months. Other similar systems are possible but all solutions have equal complexity; actual lunisolar calendars (like the Hebrew calendar) are more complex but achieve greater accuracy.
Although it is complex, a lunisolar calendar has very strong practical motivations. The importance of the solar day is obvious; the tropical year is important to anyone living in a climate with seasons. And the synodic month tells us when it is possible to see at night by the light of the moon, and what time of day the high and low tides will take place. These things are important even in modern urban society. Under a lunisolar calendar you can agree to meet outdoors at 8PM every month on the 15th of the month and know that there will be moonlight (weather permitting), or you can agree to go fishing every 10th of the month at 5AM and know there will be a low tide (assuming that's how the tide lines up at your location).
The integer approximation of 6939.60160373. There would be this many days in 19 years, if the 19-year metonic cycle were exact.
7920 is the ratio between 11!=39916800 and 7!=5040, and the ratio 7920/5040 = 11/7 = 1.571428571428... is kind of an approximation to π/2.
One of the more memorable small powers. While I was 10 and 11 years old I memorized integer exponents of integers just for fun. I still know all of these by heart:
| 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | all the way up to | 131072 |
| 3 | 9 | 27 | 81 | 243 | 729 | 2187 | |||||
| 4 | see 2 | ||||||||||
| 5 | 25 | 125 | 625 | 3125 | 15625 | ||||||
| 6 | 36 | 216 | 1296 | 7776 | 46656 | ||||||
| 7 | 49 | 343 | 2401 | ||||||||
| 8 | see 2 | ||||||||||
| 9 | see 3 | ||||||||||
| 10 | trivial | ||||||||||
| 11 | 121 | 1331 | 14641 | ||||||||
| 12 | 144 | 1728 |
7776 is also a Kaprekar number for 5th powers: 77765=28430288029929701376, and 2843+0288+0299+2970+1376=7776.
(And better still, 7+7+7+6 = 27.)
7980 = 15×19×28, the product of three numbers that have an important relation to calendars in the Roman, Byzantine and Christian worlds. 28 years is the "solar cycle", 4×7, the number of years it takes before any date falls on the same day of the week again (that's 4 years per leap-year cycle, and 7 days per week). 19 is the length of the metonic cycle, see 19 for more. 15 years is the "indiction cycle", a period related to certain phenomena such as taxation. Since all three numbers are relatively prime to each other, the least common multiple is 7980. The "Julian day number", used in astronomy, is based on a system proposed in 1583 just after the adoption of the Gregorian calendar. Julian day 1 is January 1st 4713 BC. The year 4713 BC happened to be the most recent time that all three cycles (the 15- 18- and 28-year cycles) were aligned. It is convenient primarily because it pre-dates all recorded history, even in China, Egypt, Greece and Mesopotamia.
8000 = 203 = 113+123+133+143, the smallest cube that can be expressed as the sum of 4 consecutive cubes. See also 216.
This number has a property related to vampire numbers: if you take the first two digits and multiply by the other two, the product has the same four digits (but in a different order): 81×27=2187.
See also 8712.
(Personal: For a while during my childhood the numbers 7, 27 and 127 were my favorite 1- 2- and 3-digit numbers. I have since forgotten why the properties of 127 appealed to me, but I suspect it was partly because it ends in "27". If I had continued the series, I probably would have picked 8127, because it is divisible by 7 and by 27, and consists of the cubes 8, 1 and 27 strung together; also the "81" is 3×27.)
The smallest number that is not a palindrome, and is divisible by the number you get when you reverse its digits: 8712 = 4×2178. The next number like this is 9801=992. The numbers with this property comprise Sloane's A031877 and their factors are A008919. (Thanks to Tavi Laiu for this one). See also 1089. Another related property: 8712 × 2178 = 664.
From the 3rd century BCE to as late as the 12th century, in Greek, Coptic and Armenian texts, letters of the alphabet are sometimes used to represent numbers48:
Α=1 Β=2 Γ=3 Δ=4 Ε=5 στ=6 Ζ=7 Η=8 Θ=9
Ι=10 Κ=20 Λ=30 Μ=40 Ν=50 Ξ=60 Ο=70 Π=80 Ϙ=90
Ρ=100 Σ=200 Τ=300 Υ=400 Φ=500 Χ=600 Ψ=700 Ω=800
Π=900 ͵Α=1000 ͵Β=2000 ͵Γ=3000 ͵Δ=4000
͵Ε=5000 ͵Δ=6000 ͵Ζ=7000 ͵Η=8000 ͵Θ=9000
The necessary letters are combined to make a quantity, so for example the number 8127 would be ηρκζ. For numbers above this they used a capital letter mu (Μ) to represent 10000, whose name in Greek is myriad. To distinguish it from Μ=40, a small alpha was written above the Μ. The precise way in which larger numbers were handled differed through the centuries in ancient Greece. In the most ambitious system, a number (like ηρκζ) was written in small letters directly above the M to represent 10000 raised to that power; see 1040000. However, in much more common usage the small number placed above the M merely multiplied the value; see 108.
Myriads and powers of a myriad are also involved in Chinese and other east-Asian systems for naming large numbers; see 104096.
10080 = 60 × 24 × 7, the number of minutes in a week. 10080 is also a factor record-setter with 72 factors. Thus, there are 72 different ways to divide a week into equal parts that are whole multiples of minutes. 10080 is also 2×7!, see 40320 and 604800.
In France just after the revolution they switched to the now-standard system of weights and measures based on powers of 10 (SI, systeme internationale, or the "metric system"). They also set up (for a while) a calendar involving a 10-day week, and dividing the day into powers of 10. It didn't last long, and France switched back to the Gregorian calendar a few years later.
But there have been other times since then that people have considered the idea of measuring time in powers of 10. Clearly this "metric time" idea has some appeal, because there are dozens of web sites about it (do a search for "metric time"). Perhaps best-known is Swatch's "Internet time", based on the division of the day into 1000 parts called "beats" (with 86.4 seconds per "beat"). For subdivisions of the day it is easy, the hours and minutes we have now are just arbitrary divisions and there is no conflict with the other important environmental cycles.
When considering a metric replacement for the synodic month and tropical year, clearly there is no good practical solution. These are all very important physical cycles and have strange ratios that cannot be changed (at least not yet :-) and cannot easily be adapted to work with powers of 10.
Personally, I think the existing standard time system is just as cool as any 10-based system would be, because of the utilitarian properties of the factor record-setters and cute things like the minutes in February and the 86400000 property. However, if you insist on setting up a "metric time" system, you could do worse than to take advantage of the fact that a minute and a week are almost at an exact ratio of 10000. Such a system would leave the definition of "week" unchanged and define a special "minute" which is exactly 7/10000 of a day (which comes out to exactly 60.48 "standard" seconds). Then all weekly events would always start at the same minute-mark, which would be a 4-digit number from 0000 to 9999. You'd still have to deal with the fact that the number of weeks per year is odd (the fact that it's close to 100/2 makes this a little more bearable), the number of minutes per day is odd, and there is no suitable "month".
See also 3628800 and 86400000.
The 2nd row of Pascal's Triangle shows clearly in the 2nd power of 101. The pattern continues in higher powers, and goes further than for the powers of 11. See 14641 for more.
The name of the worker who is befriended by Freder in the 1927 movie Metropolis. Also a palindrome, and strobogrammatic.
The smaller powers of 11 (121, 1331 and 14641) give rows of Pascal's Triangle. Pascal's Triangle is a rather useful table of numbers (called binomial coefficients) that is formed by continually adding numbers in a cascade starting from a single 1, as shown here:
| row 0: | 1 | |||||||||||||||||||
| row 1: | 1 | 1 | ||||||||||||||||||
| row 2: | 1 | 2 | 1 | |||||||||||||||||
| row 3: | 1 | 3 | 3 | 1 | ||||||||||||||||
| row 4: | 1 | 4 | 6 | 4 | 1 | |||||||||||||||
| row 5: | 1 | 5 | 10 | 10 | 5 | 1 | ||||||||||||||
| row 6: | 1 | 6 | 15 | 20 | 15 | 6 | 1 | |||||||||||||
| row 7: | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | ||||||||||||
| row 8: | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | |||||||||||
| row 9: | 1 | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | ||||||||||
| etc | etc | etc |
Each is the sum of the (one or) two numbers above it.
These numbers have many applications because they count "combinations", such as "how many combinations of 3 colors can you make if you have 5 colors to choose from?" (the answer is the 3rd number in row 5 of the triangle: 10). Another question with the same answer is, "If you flip a coin 5 times, how many ways are there to end up getting heads exactly 3 times?" (the answer is also 10).
A general formula that gives the value of item N in row M is:
NCM = M!/((M-N)!×N!)
For example, 3C5 = 5!/((5-3)!×3!) = 120/(2×6) = 10. However this formula gets rather impractical to use when M is large, even when the value being computed is pretty reasonable. For example, consider 3C143=143!/(140!×3!). 143! has 248 digits, and 140!×3! has 242 digits — so you probably couldn't even get the answer on a calculator, as the calculator would overflow. However, most of the 143! in the numerator cancels out with the 140! in the denominator, leaving 3C143 = 143×142×141/3! = 477191, which you could even calculate by hand if you needed to.
Number of days in the roughly 52-year cycle of the Mayan Calendar Round. This is the least common multiple of 260=13×20 and 365=5×73. See also 5126 and 1872000.
This is 3④3 or 3^^3 by the lower (left-associative) version of the hyper4 operator. It is also a "googol" in base 3: 103=3 and 1003=9, so "10100" in base 3 is 39=19683.
See also 196883.
e10 is the highest value on the LL scale of most log-log slide rules that have such a scale. (However, a few go as high as 1010.)
As of 2003, the best known value of the Platonic Year, or precession cycle, in mean solar years (the error is ±0.0015 years — less than a day!). This is the time it takes for the orientation of Earth's poles to go one full circuit in its circular path. The movement is caused by the tidal influence of the Moon and Sun (and to far lesser extent, other planets like Venus and Jupiter) on the Earth, which has a gyroscopic precession effect because of the Earth's bulging a bit around the equator. The value is approximated as 25600, 25800, 25920 or 26000, depending on the purpose and source. The raw data value from which this is derived is the "general precession in longitude" in computer models, 5028.7955 ± 0.0003 arc-minutes per Julian Century8. A Julian Century is 36525 mean solar days, so the precession period is computed by (360 × 3600 × 100 / 5028.792) × (365.25 / 365.242189670).
An often-seen approximation to the precession cycle.
According to those ancient astronomers who knew of the "precession of the equinoxes" (now known to be movement of the Earth's axis of rotation, see 25772.1300), the ascending and descending nodes of the Sun move 1 degree through the zodiac every 72=6×12 years. This produces a cycle of 25920 years for the entire period. This isn't too far off from the modern value. An "age" of 2160 years, 1/12 of the full period, is also given much significance. See also 2592.
Approximately the number of days in an expected lifetime for most readers of this page (and 75 "Babylonian" years, see 405000).
A prime number, and one of many misspellings of the word "elite" used by those who wished to hide their conversations from automatic detection on bulletin-board systems in the 1990's. This "language" possibly evolved partly from inverted calculator-display text (see 71077345).
30240 = 25×33×5×7, and is a 4-perfect number: Its divisors add up to exactly 4 times the number itself. Curiously, it also has the same digits, and most of the same factors, as 40320. See also 120, 154345556085770649600.
The width of the English Channel at its narrowest point, in meters.
See 3.1418708596056 and 137.035.
This is 8! (8 factorial), 8×7×6×5×4×3×2×1, and is the number of minutes in 4 weeks. It would also be the number of minutes in a month, if months were exactly 4 weeks long. I'm no revolutionary — I'm not suggesting we should change the calendar — but I do strongly believe February should be designated "International Factorial Appreciation Month", to be observed in all years except leap-years. Alas, all my pleas have fallen on deaf ears (-: See also 10080, 604800, 3628800 and 86400000.
In 1963 a set of bell ringers in Loughborough, Leicestershire, England became the first in history to ring eight tower bells in all possible permutations — each bell was rung 40320 times, for a total of 8×8! = 322560 blows, with each different combination of the 8 notes occurring exactly once. The feat took almost 18 hours42.
This is a square and also a triangular number: 41616 = 2042 = 288×289/2. Such numbers are fairly rare: 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (Sloane's A001110). See 204 for more.
Often related to 86400, 432000, etc. by people studying ancient and/or mystical things. Also, it has been noted that the great pyramid in Egypt has a perimeter of 3023.16 feet at its base, and estimated height (before erosion) of about 481.40 feet. If you multiply these by 43200 you get (almost exactly) the circumference of the Earth and its polar radius. See also 432, 432000, 4320000
The number of days Jeanne Calment lived; see 122.
45360 has 100 factors (including 1 and itself); no smaller number has as many. Its prime factorization is 24×34×5×7. To get one of the 100 factors of 45360, pick one number from each column in this table:
| A | B | C | D | |
| 1 | 1 | 1 | 1 | |
| 2 | 3 | 5 | 7 | |
| 4 | 9 | |||
| 8 | 27 | |||
| 16 | 81 |
|
and multiply them together to get a factor. Since you have 5 choices in each of the first two columns and 2 choices in the other two columns, the total number of choices you can make is 5×5×2×2=100. Notice that these numbers (5,5,2,2) are one more than the exponents in the prime factorization of 45360 (24345171). You can count the number of factors of any number just by taking the exponents in its prime factorization, adding one to each, and multiplying together. For 45360, the exponents are (4,4,1,1). Any other number with a prime factorization with two 4's and two 1's for exponents will have the same number of factors, including 2×34×54×13 = 1316250 and 2×5×74×114 = 351530410 and so on. Based on this, you can see that:
For more about factors record-setters, see 12, 60, 360, 840, 1260, 720720, 3603600, 245044800, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10917.
The highest value of k for which 2k-1 + k is known to be prime, or is considered a "probable prime". The other lower values of k are 1, 3, 7, 237 and 1885. There ought to be about log(N) such k values for k < N, but since 2k grows so quickly, it is very difficult to find more. It also happens to be difficult to prove whether there is or is not an infinite number of such k's.
The largest finite value that can be represented in the 16-bit floating-point format "s10e5", also called "half" or "fp16", used in professional computer graphics. This format uses all the IEEE-754 rules with a 5-bit exponent and 10-bit mantissa. (More details here.)
The number of astronomical units in a light year. In my old Macintosh space flight program Orion, I used 216 as an approximation to this. See also 1.609344.
This is 216 = 224 = 2222, which is 2 ^^ 4 using the hyper4 operator. Also, since 2 ^^ 4 is 2 ^^ (2 ^^ 2), 65536 is 2 ^^^ 3 where ^^^ is the hyper5 operator.
Appears in Zork and several later Infocom adventure games:
On the ground is a pile of leaves.
> count leaves
There are 69,105 leaves here.
The usage of the number probably originated from the MIT hacker culture74; see 69.
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