If you are interested in what the decimal expansions of numbers are like, then you will probably find the reciprocals of powers (more accurately, their reptends) interesting.
We will begin with the reciprocals of powers of 3. The reciprocal of 3 is just 0.333333..., and the reciprocal of 9 is just 0.111111.... 1/27, though, is where it starts getting more interesting:
1/27 = 0.037037037037037037...
37 is the only prime number that is a reptend in a reciprocal of a power of 3, because it evenly divides all further such numbers. Indeed, for obvious reasons, every reptend in a reciprocal of a power of 3 divides all further such numbers.
1/81 has a curious reciprocal:
1/81 = 0.0123456790123456790123456790...
This fraction has all digits from 0 to 9 in increasing order, except 8. 12,345,679 is equal to 37 * 333,667.
1/243 also has a neat pattern:
1/243 = 0.004115226337448559670781893004115226337448559670781893004...
In this fraction, each block of 3 digits is 111 more than the one before it. The reason each block of three is 4 more than a multiple of 111 is because 1/243 is equal to 37/8991,
1/729 also has a pattern that is easy to memorize:
1/729 = 0.0013717421124828532235939643347050754458161865569272976680384087791495198902606310013717421124828...
Each block of three in this fraction is 37 more than the one before it, starting with 137. Also, each time you advance by 9 digits, the digits are incremented by 1.
The sequence of reptends in reciprocals of powers of 3 is:
3
1
37
12,345,679
4,115,226,337,448,559,670,781,893
1,371,742,112,482,853,223,593,964,334,705,075,445,816,186,556,927,297,668,038,408,779,149,519,890,260,631
457,247,370,827,617,741,197,988,111,568,358,481,938,728,852,309,099,222,679,469,593,049,839,963,420,210,333,790,580,704,160,951,074,531,321,444,901,691,815,272,062,185,
642,432,556,012,802,926,383,173,296,753,543,667,123,913,037,494,284,407,864,654,778,235,024,148,605,395,518,975,765,889,346,135,259,716,506,630,086,877 (240 digits)
152,415,790,275,872,580,399,329,370,522,786,160,646,242,950,769,699,740,893,156,531,016,613,321,140,070,111,263,526,901,386,983,691,510,440,481,633,897,271,757,354,061,880,810,852,004,267,642,127,724,432,251,181,222,374,638,012,498,094,802,621,551,592,745,008,382,868,465,172,991,921,963,115,378,753,238,835,543,362,292,333,485,749,123,609,205,913,732,662,703,856,119,493,979,576,284,103,033,074,226,489,864,349,946,654,473,403,444,596,860,234,720,317,024,843,773,814,967,230,605,090,687,395,214,144,185,337,600,975,461,057,765,584,514,555,707,971,345,831,428,135,954,884,926,078,341,716,201,798,506,325,255,296,448,712,086,572,168,876,695,625,666,819,082,456,942,539,247,065,996,037,189,452,827,312,909,617,436,366,407,559,823,197,683,279,987,806,736,777,930,193,568,053,650,358,177,107,148,300,563,938,424,020,728,547,477,518,670,934,308,794,391,098,917,847,889,041,304,679,164,761,469,288,218,259,411,675,049,535,131,839,658,588,629,782,045,419,905,502,210,028,959 (726 digits)
50,805,263,425,290,860,133,109,790,174,262,053,548,747,650,256,566,580,297,718,843,...,489,762,739,419,803,891,683,178,377,279,886,196,209,927,348,473,301,834,070,009,653 (2,183 digits)
16,935,087,808,430,286,711,036,596,724,754,017,849,582,550,085,522,193,432,572,947,...,496,587,579,806,601,297,227,726,125,759,962,065,403,309,116,157,767,278,023,336,551 (6,557 digits, the largest number I ever typed out in full).
5,645,029,269,476,762,237,012,198,908,251,339,283,194,183,...,705,385,589,092,674,445,517 (19,678 digits)
1,881,676,423,158,920,745,670,732,969,417,113,094,398,061,...,588,461,863,030,891,481,839 (59,044 digits)
Whoa! The reptend in 1/2187 is already much greater than the googol, and the reptend in 1/177147 is too large to be calculated in the Windows calculator.
1/7 = 0.142857142857142857...
1/49 = 0.0204081632653061224489795918367346938775510204081632653061224...
The reptend in 1/49 contains all the powers of 2 up to 32, after which the pattern breaks. Also, every block of 6 digits in the reptend is 142857 more than the one before it, which is similar to what we saw in the reciprocals of powers of 3, where all the digits are incremented by 1 after n/9 digits, where n is the period.
1/343 = 0.002915451895043731778425655976676384839650145772594752186588921282798833819241982507288629737609329446064139941690962...
1/2401 = 0.0004164...
The reptends in reciprocals of powers of 7 are:
142,857
20,408,163,265,306,122,448,979,591,836,734,693,877,551
2,915,451,895,043,731,778,425,655,976,676,384,839,650,145,772,594,752,186,588,921,282,798,833,819,241,982,507,288,629,737,609,329,446,064,139,941,690,962,099,125,364,431,
486,880,466,472,303,206,997,084,548,104,956,268,221,574,344,023,323,615,160,349,854,227,405,247,813,411,078,717,201,166,180,758,017,492,711,370,262,390,670,553,935,860,
058,309,037,900,874,635,568,513,119,533,527,696,793 (292 digits)
I even invented a hierarchy based off these numbers, where f0(3, n) is equal to the reptend in the reciprocal fo 3n+2. f0(7, n) is equal to the reptend in 1/7n, and in general, f0(a, b) is the reptend in ab (3 is a special case). f1(3, 2) is the reptend in the reciprocal of 312,345,681, or a number with more than 7.1*105890385 digits. Below are selected digits of that number:
1552069995056762948617698329314415881740874443923422607361701123790799558196240252103337133830746730651886867584531516975297923903969222..................
.........200858136153223593964334705075445816186556927297668038.........2606310168...
.........137757998979012345679012345679012345679012345679012345.........0123456790123456790123456790123456790123456790123456790123456790278663789628........
.........9495792690337037037037037037037037037037037037037037037.........0370370370370370370370370370370370370370370370370370370370370370370370370370370370
525578369876046665232140203301811958544457814762712631106540482749450326189994395580704083753445043435559057128823422067900162760767292.....................
.........03569856764411111111111111111111111111111111111111111111.........111111111111111111111111111111111111111111111111111111111111111111111111111111111
2663181106167874059728809440425526992851985555034533718472812234901900669307351363214448244941857841762997978695642628086409035015080333..................
.........14680967875522222222222222222222222222222222222222222222.........222222222222222222222222222222222222222222222222222222222222222222222222222222222
3774292217278985170839920551536638103963096666145644829583923346013011780418462474325559356052968952874109089806753739197520146126191444..................
.........25792078986633333333333333333333333333333333333333333333.........333333333333333333333333333333333333333333333333333333333333333333333333333333333
4885403328390096281950031662647749215074207777256755940695034457124122891529573585436670467164080063985220200917864850308631257237302555..................
.........36903190097744444444444444444444444444444444444444444444.........444444444444444444444444444444444444444444444444444444444444444444444444444444444
5996514439501207393061142773758860326185318888367867051806145568235234002640684696547781578275191175096331312028975961419742368348413666..................
.........48014301208855555555555555555555555555555555555555555555.........555555555555555555555555555555555555555555555555555555555555555555555555555555555
7107625550612318504172253884869971437296429999478978162917256679346345113751795807658892689386302286207442423140087072530853479459524778...................
.........59125412319966666666666666666666666666666666666666666666.........666666666666666666666666666666666666666666666666666666666666666666666666666666666
8218736661723429615283364995981082548407541110590089274028367790457456224862906918770003800497413397318553534231198183641964590570635889....................
.........70236523431077777777777777777777777777777777777777777777.........777777777777777777777777777777777777777777777777777777777777777777777777777777777
9329847772834540726394476107092193659518652221701200385139478901568567335974018029881114911608524508429664645342309294753075701681747000....................
.........81347634542188888888888888888888888888888888888888888888.........888888888888888888888888888888888888888888888888888888888888888888888888888888889
0440958883945651837505587218203304770629763332812311496250590012679678447085129140992226022719635619540775756453420405864186812792858111....................
.........924587456533
(312345679 - 5890386 digits in all)
How are the middle digits of that number calculated?!?!?!? About 1/9 of the way through every reptend in a reciprocal of 3n, there are as many 1s as there are digits in the decimal expansion of 3n-2, preceded by the last digits all flipped by 1 and followed by the first digits all flipped by 1. 2/9 of the way through, it is the same, only with 2s instead of 1s, and all the nonzero digits flipped by 2 instead of 1.