A couple of "How To" math questions regarding superscripts

Factual Questions
1

reading this thread, I followed a link someone posted into a wooly tangent of science articles on Wikipedia. In one of the articles, the number 10 superscript 10 superscript 10 superscript 56 is mentioned.

Is there a better, less awkward way to say that number? What I mean is, is it really necessary to have three superscripts (supersuperscripts?) Can’t all those superscripts be combined somehow to more succinctly describe the number?

Finally, how many zeros do all those superscripts represent after the 10, what is the procedure for expanding three superscripts? sequentially, ie work out the first then apply the second script to the result and then the third script to the result of that?

2

The usual way to say it in words would be “ten to the ten to the ten to the 56”. Written as symbols, there’s no way to collapse 10[sup]10[sup]10[sup]56[/sup][/sup][/sup].

That would be 10[sup]10[sup]56[/sup][/sup] zeroes. You evaluate it from right to left (or from top to bottom, if you prefer to think of it that way).

3

Necessary? In one sense, yes. 10[sup]10[sup]10[sup]56[/sup][/sup][/sup] is a 1 followed by 10[sup]10[sup]56[/sup][/sup] zeros. The number of particles (or atoms, the net doesn’t seem to be consistent here) in the observable universe is estimated to be around 10[sup]80[/sup], so there aren’t enough particles available to be able to write your number without using exponents.

As to how to evaluate a power like that, it is, as I said, a 1 followed by 10[sup]10[sup]56[/sup][/sup] zeros. Similarly, 10[sup]10[sup]56[/sup][/sup] is a 1 followed by 10[sup]56[/sup] zeros; in turn, 10[sup]56[/sup] is a 1 followed by 56 zeros.

4

Superscripts are easy to work with. Take 2[sup]2[sup]2[sup]2[/sup][/sup][/sup]. You work from the top down.

2[sup]2[/sup] = 4. That gives you 2[sup]2[sup]4[/sup][/sup].

2[sup]4[/sup] = 16. That gives you 2[sup]16[/sup] = 65536.

2[sup]65536[/sup] = 20035299304068464649790723515602557504478254755697514192650169737.

That’s a huge number and an example of why you want a notation that can quickly and easily represent it.

Now take your number 10[sup]10[sup]10[sup]56[/sup][/sup][/sup].

We start out 10[sup]56[/sup] = 100000000000000000000000000000000000000000000000000000000.

That gives you 10[sup]10[sup]100000000000000000000000000000000000000000000000000000000[/sup][/sup].

And it ends there. 10[sup]100000000000000000000000000000000000000000000000000000000[/sup] can’t be expressed in any way. If you made zeroes the size of protons the universe wouldn’t hold that many. Not even a billion or trillion or billion trillion universes would. And that’s just an intermediate step.

There are many specialized notations for superlarge numbers, but none of them are immediately understandable to laymen. And laymen can’t begin to comprehend how LARGE large numbers can get. Superscripts are the low end of the expansion totem pole. Take them. Hold them close. Appreciate them. The rest is madness.

5

Food for thought: All of those other notations used for superlarge numbers are, when you get down to it, also on the low end of the totem pole.

And if you spent an entire 20-volume encyclopedia explaining some new notation yet, and then spent another 20 volumes using that new notation to express that number, you’d still be on the low end of the totem pole. Because even then, there would be vastly more numbers greater than your number than there are less than it.

6

LARGE numbers.

7

In the light of your statement: why bother with exponentials at all? Heck, why bother with multiplication? All numbers can be expressed as 1+1+1+…!

8

Do that for pi. :smiley:

9

There is no way a power of 2 is going to be odd. The number you’re looking for is



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3,650,140,549,586,250,615,282,588,033,022,908,385,812,478,469,315,720,323,233,601,899,469,437,647,726,721,879,376,826,431,828,382,603,564,520,699,468,630,216,048,874,528,424,363,593,558,622,333,506,235,945,002,890,558,581,611,275,341,783,750,455,936,126,130,852,640,828,051,213,873,177,490,200,249,552,738,734,585,956,405,160,830,583,053,770,732,533,971,552,620,444,705,429,573,538,361,113,677,523,169,972,740,292,941,674,204,423,248,113,875,075,631,319,078,272,188,864,053,374,694,213,842,169,928,862,940,479,635,305,150,560,788,126,366,206,497,231,257,579,019,598,873,041,195,626,227,343,728,900,516,561,111,094,111,745,277,965,482,790,471,250,581,999,077,498,063,821,559,376,885,546,498,822,938,985,408,291,325,129,076,478,386,322,494,781,016,753,491,693,489,288,104,203,015,610,283,386,143,827,378,160,946,341,335,383,578,340,765,314,321,417,150,655,877,547,820,252,454,780,657,301,342,277,470,616,744,241,968,952,613,164,274,104,695,474,621,483,756,288,299,771,804,186,785,084,546,965,619,150,908,695,874,251,184,435,837,306,590,951,460,980,451,247,409,411,373,899,927,822,492,983,367,796,011,015,387,096,129,749,705,566,301,637,307,202,750,734,759,922,943,792,393,824,427,421,186,158,236,161,317,886,392,553,095,117,188,421,298,508,307,238,259,729,144,142,251,579,403,883,011,359,083,331,651,858,234,967,221,259,621,812,507,058,113,759,495,525,022,747,274,674,369,887,131,926,670,769,299,199,084,467,161,228,738,858,457,584,622,726,573,330,753,735,572,823,951,616,964,175,198,675,012,681,745,429,323,738,294,143,824,814,377,139,861,906,716,657,572,945,807,804,820,559,511,881,687,188,075,212,971,832,636,442,155,336,787,751,274,766,940,790,117,057,509,819,575,084,563,565,217,389,544,179,875,
10

Now, take that last number and append this one to the end for the actual value:



074,523,854,455,200,133,572,033,332,379,895,074,393,905,312,918,212,255,259,833,790,909,463,630,202,185,353,848,854,825,062,897,715,616,963,860,712,382,771,725,621,313,460,549,401,770,413,581,731,931,763,370,136,332,252,819,127,547,191,443,450,920,711,848,838,366,818,174,263,342,949,611,870,091,503,049,165,339,464,763,717,766,439,120,798,347,494,627,397,822,171,502,090,670,190,302,469,762,151,278,521,956,142,070,806,461,631,373,236,517,853,976,292,092,025,500,288,962,012,970,141,379,640,038,055,734,949,269,073,535,145,961,208,674,796,547,733,692,958,773,628,635,660,143,767,964,038,430,796,864,138,563,447,801,328,261,284,589,184,898,528,048,048,844,180,821,639,423,974,014,362,903,481,665,458,114,454,366,460,032,490,618,763,039,502,356,402,044,530,748,210,241,366,895,196,644,221,339,200,757,479,128,683,805,175,150,634,662,569,391,937,740,283,512,075,666,260,829,890,491,877,287,833,852,178,522,792,045,771,846,965,855,278,790,447,562,192,663,992,008,409,302,075,673,925,363,735,628,390,829,817,577,902,153,202,106,409,617,373,283,598,494,066,652,141,198,183,810,884,515,459,772,895,164,572,131,897,797,907,491,941,013,148,368,544,639,616,904,607,030,107,596,818,933,741,217,575,988,165,127,000,761,262,789,169,510,406,315,857,637,534,787,420,070,222,051,070,891,257,612,361,658,026,806,815,858,499,852,631,465,878,086,616,800,733,264,676,830,206,391,697,203,064,894,405,628,195,406,190,685,242,003,053,463,156,621,891,327,309,069,687,353,181,641,094,514,288,036,605,995,220,248,248,886,711,554,429,104,721,929,134,248,346,438,705,368,508,648,749,099,178,812,670,565,665,387,191,049,721,820,042,371,492,740,164,460,943,459,845,392,536,706,132,210,616,533,085,662,021,188,968,234,005,752,675,486,101,476,993,688,738,209,584,552,211,571,923,479,686,888,160,853,631,615,862,880,150,395,949,418,529,489,227,074,410,828,207,169,303,387,818,084,936,204,018,255,222,271,010,985,653,444,817,207,470,756,019,245,915,599,431,072,949,578,197,878,590,578,940,052,540,122,867,517,142,511,184,356,437,184,053,563,024,181,225,473,266,093,302,710,397,968,091,064,939,272,722,683,035,410,467,632,591,355,279,683,837,705,019,855,234,621,222,858,410,557,119,921,731,717,969,804,339,317,707,750,755,627,056,047,831,779,844,447,637,560,254,637,033,369,247,114,220,815,519,973,691,371,975,163,241,302,748,712,199,863,404,548,248,524,570,118,553,342,675,264,715,978,310,731,245,663,429,805,221,455,494,156,252,724,028,915,333,354,349,341,217,862,037,007,260,315,279,870,771,872,491,234,494,477,147,909,520,734,761,385,425,485,311,552,773,301,030,342,476,835,865,496,093,722,324,007,154,518,129,732,692,081,058,424,090,557,725,645,803,681,462,234,493,189,708,138,897,143,299,831,347,617,799,679,712,453,782,310,703,739,151,473,878,692,119,187,566,700,319,321,281,896,803,322,696,594,459,286,210,607,438,827,416,919,465,162,267,632,540,665,070,881,071,030,394,178,860,564,893,769,816,734,159,025,925,194,611,823,642,945,652,669,372,203,155,504,700,213,598,846,292,758,012,527,715,422,016,629,954,863,130,324,912,311,029,627,923,723,899,766,416,803,497,141,226,527,931,907,636,326,136,814,145,516,376,656,559,839,788,489,381,733,082,668,779,901,962,886,932,296,597,379,951,931,621,187,215,455,287,394,170,243,669,885,593,888,793,316,744,533,363,119,541,518,404,088,283,815,193,421,234,122,820,030,950,313,341,050,704,760,159,987,985,472,529,190,665,222,479,319,715,440,331,794,836,837,373,220,821,885,773,341,623,856,441,380,700,541,913,530,245,943,913,502,554,531,886,454,796,252,260,251,762,928,374,330,465,102,361,057,583,514,550,739,443,339,610,216,229,675,461,415,781,127,197,001,738,611,494,279,501,411,253,280,621,254,775,810,512,972,088,465,263,158,094,806,633,687,670,147,310,733,540,717,710,876,615,935,856,814,098,212,967,730,759,197,382,973,441,445,256,688,770,855,324,570,888,958,320,993,823,432,102,718,224,114,763,732,791,357,568,615,421,252,849,657,903,335,093,152,776,925,505,845,644,010,552,192,644,505,312,073,756,287,744,998,163,646,332,835,816,140,330,175,813,967,359,427,327,690,448,920,361,880,386,754,955,751,806,890,058,532,927,201,493,923,500,525,845,146,706,982,628,548,257,883,267,398,735,220,457,228,239,290,207,144,822,219,885,587,102,896,991,935,873,074,277,815,159,757,620,764,023,951,243,860,202,032,596,596,250,212,578,349,957,710,085,626,386,118,233,813,318,509,014,686,577,064,010,676,278,617,583,772,772,895,892,746,039,403,930,337,271,873,850,536,912,957,126,715,066,896,688,493,880,885,142,943,609,962,012,966,759,079,225,082,275,313,812,849,851,526,902,931,700,263,136,328,942,095,797,577,959,327,635,531,162,066,753,488,651,317,323,872,438,748,063,513,314,512,644,889,967,589,828,812,925,480,076,425,186,586,490,241,111,127,301,357,197,181,381,602,583,178,506,932,244,007,998,656,635,371,544,088,454,866,393,181,708,395,735,780,799,059,730,839,094,881,804,060,935,959,190,907,473,960,904,410,150,516,321,749,681,412,100,765,719,177,483,767,355,751,000,733,616,922,386,537,429,079,457,803,200,042,337,452,807,566,153,042,929,014,495,780,629,634,138,383,551,783,599,764,708,851,349,004,856,973,697,965,238,695,845,994,595,592,090,709,058,956,891,451,141,412,684,505,462,117,945,026,611,750,166,928,260,250,950,770,778,211,950,432,617,383,223,562,437,601,776,799,362,796,099,368,975,191,394,965,033,358,507,155,418,436,456,852,616,674,243,688,920,371,037,495,328,425,927,131,610,537,834,980,740,739,158,633,817,967,658,425,258,036,737,206,469,351,248,652,238,481,341,663,808,061,505,704,829,059,890,696,451,936,440,018,597,120,425,723,007,316,410,009,916,987,524,260,377,362,177,763,430,621,616,744,884,930,810,929,901,009,517,974,541,564,251,204,822,086,714,586,849,255,132,444,266,777,127,863,728,211,331,536,224,301,091,824,391,243,380,214,046,242,223,349,153,559,516,890,816,288,487,989,988,273,630,445,372,432,174,280,215,755,777,967,021,666,317,047,969,728,172,483,392,841,015,642,274,507,271,779,269,399,929,740,308,072,770,395,013,581,545,142,494,049,026,536,105,825,409,373,114,653,104,943,382,484,379,718,606,937,214,444,600,826,798,002,471,229,489,405,761,853,892,203,425,608,302,697,052,876,621,377,373,594,394,224,114,707,074,072,902,725,461,307,358,541,745,691,419,446,487,624,357,682,397,065,703,184,168,467,540,733,466,346,293,673,983,620,004,041,400,714,054,277,632,480,132,742,202,685,393,698,869,787,607,009,590,048,684,650,626,771,363,070,979,821,006,557,285,101,306,601,010,780,633,743,344,773,073,478,653,881,742,681,230,743,766,066,643,312,775,356,466,578,603,715,192,922,768,440,458,273,283,243,808,212,841,218,776,132,042,460,464,900,801,054,731,426,749,260,826,922,155,637,405,486,241,717,031,027,919,996,942,645,620,955,619,816,454,547,662,045,022,411,449,404,749,349,832,206,807,191,352,767,986,747,813,458,203,859,570,413,466,177,937,228,534,940,031,631,599,544,093,684,089,572,533,438,702,986,717,829,770,373,332,806,801,764,639,502,090,023,941,931,499,115,009,105,276,821,119,510,999,063,166,150,311,585,582,835,582,607,179,410,052,528,583,611,369,961,303,442,790,173,811,787,412,061,288,182,062,023,263,849,861,515,656,451,230,047,792,967,563,618,345,768,105,043,341,769,543,067,538,041,113,928,553,792,529,241,347,339,481,050,532,025,708,728,186,307,291,158,911,335,942,014,761,872,664,291,564,036,371,927,602,306,283,840,650,425,441,742,335,464,549,987,055,318,726,887,926,424,102,147,363,698,625,463,747,159,744,354,943,443,899,730,051,742,525,110,877,357,886,390,946,812,096,673,428,152,585,919,924,857,640,488,055,071,329,814,299,359,911,463,239,919,113,959,926,752,576,359,007,446,572,810,191,805,841,807,342,227,734,721,397,723,218,231,771,716,916,400,108,826,112,549,093,361,186,780,575,722,391,018,186,168,549,108,500,885,272,274,374,212,086,524,852,372,456,248,697,662,245,384,819,298,671,129,452,945,515,497,030,585,919,307,198,497,105,414,181,636,968,976,131,126,744,027,009,648,667,545,934,567,059,936,995,464,500,558,921,628,047,976,365,686,133,316,563,907,395,703,272,034,389,175,415,267,500,915,011,198,856,872,708,848,195,531,676,931,681,272,892,143,031,376,818,016,445,477,367,518,353,497,857,924,276,463,354,162,433,601,125,960,252,109,501,612,264,110,346,083,465,648,235,597,934,274,056,868,849,224,458,745,493,776,752,120,324,703,803,035,491,157,544,831,295,275,891,939,893,680,876,327,685,438,769,557,694,881,422,844,311,998,595,700,727,521,393,176,837,831,770,339,130,423,060,958,999,137,314,684,569,010,422,095,161,967,070,506,420,256,733,873,446,115,655,276,175,992,727,151,877,660,010,238,944,760,539,789,516,945,708,802,728,736,225,121,076,224,091,810,066,700,883,474,737,605,156,285,533,943,565,843,756,271,241,244,457,651,663,064,085,939,507,947,550,920,463,932,245,202,535,463,634,444,791,755,661,725,962,187,199,279,186,575,490,857,852,950,012,840,229,035,061,514,937,310,107,009,446,151,011,613,712,423,761,426,722,541,732,055,959,202,782,129,325,725,947,146,417,224,977,321,316,381,845,326,555,279,604,270,541,871,496,236,585,252,458,648,933,254,145,062,642,337,885,651,464,670,604,298,564,781,968,461,593,663,288,954,299,780,722,542,264,790,400,616,019,751,975,007,460,545,150,060,291,806,638,271,497,016,110,987,951,336,633,771,378,434,416,194,053,121,445,291,855,180,136,575,558,667,615,019,373,029,691,932,076,120,009,255,065,081,583,275,508,499,340,768,797,252,369,987,023,567,931,026,804,136,745,718,956,641,431,852,679,054,717,169,962,990,363,015,545,645,090,044,802,789,055,701,968,328,313,630,718,997,699,153,166,679,208,958,768,572,290,600,915,472,919,636,381,673,596,673,959,975,710,326,015,571,920,237,348,580,521,128,117,458,610,065,152,598,883,843,114,511,894,880,552,129,145,775,699,146,577,530,041,384,717,124,577,965,048,175,856,395,072,895,337,539,755,822,087,777,506,072,339,445,587,895,905,719,156,736
11

Damnit.

2 to the 65,536 power

12

Yes, with tetration.

10[sup]10[sup]10[sup]10[/sup][/sup][/sup] = [sup]4[/sup]10

13

Ah, and Ok, and oh, it didn’t click that it was one of the numbers larger than the universe (maybe not entirely accurate of a descriptor) but I am familiar with that concept.

14

Landau, Dedekind* et al* will soon be around to show you how to construct any real (or even complex!) number, including pi, by mere addition (or, rather, using the relation “sequel to”, which could well be interpreted as “+1”) from the set of non-negative integers; but I suspect you know this already. It’s fascinating work, although frankly a bit tedious.

15

got lost in your link, and I think you posted while I was replying, so how does that work when you get to the exponent 56 in my number?

I follow it 10 to the tenth to the tenth, then it breaks down for me at the 56th

16

If these numbers are showing up on your tax returns … hire an accountant … you did something wrong …

17

Just for fun I tried to work out some tetrations. Sorry for any errors.

10^10^10^56 ~= [sup]1/sup
10^10^10^56 ~= [sup]2/sup
10^10^10^56 ~= [sup]3/sup
10^10^10^56 ~= [sup]4/sup
10^10^10^56 ~= [sup]5/sup
10^10^10^56 ~= [sup]6/sup

Is there a way to tetrate to a non-integer? so that, e.g.
10^10^10^56 ~= [sup]4.1/sup

Big tetrations of little numbers are easy:
[sup]1[/sup]1.4142 = 1.4142
[sup]2[/sup]1.4142 = 1.6325
[sup]4[/sup]1.4142 = 1.8409
[sup]8[/sup]1.4142 = 1.9656
[sup]16[/sup]1.4142 = 1.9656
[sup]32[/sup]1.4142 = 1.99987
[sup]32[/sup]1.415 = 2.00731
[sup]32[/sup]1.42 = 2.05737
[sup]32[/sup]1.43 = 2.18381
[sup]32[/sup]1.44 = 2.38789
[sup]32[/sup]1.45 = 3.18895

But not when the base gets much larger than √2 :

[sup]32[/sup]1.46 = 190759454.95

[sup]12[/sup]1.5 = 86.1819
[sup]13[/sup]1.5 = 1499263005586575.9
[sup]14[/sup]1.5 = e[sup]607898836642732[/sup]

18

This is all very cool. And profound.

But, using the standard keyboard, how do I enter those superscripts that just keep rising?

I want to speak the language.

Thank you.

19

If it’s just about notation, just go ahead and write ^ instead of using superscripts. Everyone will understand you just fine, and you won’t have to worry about the typographic awkwardness of deeply iterated superscripts.

20

But it ain’t purdy.