1.5.4 - Number Survey

1.5.4

Number Survey

IMPETUS & THEORY

How can one measure the "importance" of a number ? Most mathematicians and scientists consider very large numbers trivial if they serve no purpose other than being large. Some numbers are given special attention, such as the primes, while others are ignored. So what numbers do matter , and which are less important ?

A numbers "importance" could perhaps be measured by it's usage. The easiest way to measure a numbers usage is simply to search it on the web using a search engine.

Intuitively, we would expect smaller numbers to be used more than larger numbers. This is because when the pool of numbers is smaller ( say 1~10 ) there is greater likelihood under certain situations that one of these individual numbers will pop up. But on much larger ranges ( 100 ~ 1000 ) there are more individual numbers, and therefore, even within the proper context these numbers will individually be used less because the usage is being distributed more thinly. But perhaps the simplest and most convincing argument is simply this : There is only a finite number of webpages on the world wide web, and therefore only a finite number of the infinite numbers can be represented. Therefore there must be some number, n , which is not represented anywhere on the world wide web. We are so used to thinking that we can find "anything" on the web, that this statement strikes us as surprising.

Doing the actual search for numbers proves that intuition is indeed on the right track. As a general rule, the larger the actual number, the less hits it will get from a search engine. This is only the general behavior however. Certain numbers, even larger ones, are likely to get more hits than it's neighbors simply because it may have some intrinsic property that makes it stand out, or sometimes it just boils down to dumb luck. There is basically only 3 reasons a number will get a hit. It may because the number is being used in some practical or mathematical excersize. In this case the user is making use of the number for a specific purpose which has nothing to do with the number itself. A second possibility is that the number has some property which interests the user. In this case the use is completely abstract, and it is not used to serve a purpose. Instead it is the actual number itself which is of interest. Lastly a number may arrise simply by dumb luck. A person might write a long string of spam and accidently produce certain numbers. But regardless of the context, the general trend remains true, and the larger the number the less hits it gets.

Actually this topic pops up from time to time on blogs. Usually this is an opening to ask the question, what is the smallest number with no hits ?

Following the logic we have to conclude that such numbers exist, again because the web is finite. So there must be a number which is no where to be found. Based on the general principle, we can assume that as the numbers get larger and larger, there will be more and more numbers recieving no hits, until most numbers recieve no hits. One can also ask another important question, what is the largest number with at least one hit ? Again, there must be such a number, although it is extremely unlikely to find it, as it most certainly is hidden within a sea of zero hits.

This "Number Catalog" is not merely interested in these questions, but is more generally interested in the distribution of hits over the counting numbers.

HOW TO USE THIS CATALOG

This is how I conduct my survey. The numbers in this study are restricted to the counting numbers, {1,2,3, ...}. I type out the number in decimal notation without commas or puncuation marks in the google search engine. I then record the number of hits and the date the search was performed. The numbers I run are arbitrary, although I try to pick a set of entries that will properly characterize the behavior of the number of hits.

The chart below works like this ... the number on the far left is the number being searched. The second column reports how many hits it got, and the third column displays the date a search was last run on that number ( Note that the date is in international format. The year is displayed first, then the month , then the day ). Because the internet is a constantly changing entity, one also has to consider when a search was run, and it is important to reflect the most current state of the web as best as possible to get an accurate picture of what numbers are important now.

For the numbers which recieve 0 hits, I highlight them in red. This Catalog contains 258 entries, and the smallest entry with no hits is 7,351,000,094.

A SURVEY OF USAGE OF THE COUNTING NUMBERS

Counting Number Hits on Googol Date of Search

1 21,990,000,000 2008.12.23

2 18,000,000,000 2008.12.23

3 15,750,000,000 2008.12.23

4 13,590,000,000 2008.12.23

5 13,300,000,000 2008.12.23

6 11,050,000,000 2008.12.23

7 10,260,000,000 2008.12.23

8 9,870,000,000 2008.12.23

9 9,010,000,000 2008.12.23

10 14,060,000,000 2008.12.23

11 11,550,000,000 2008.12.23

12 11,700,000,000 2008.12.23

13 8,550,000,000 2008.12.23

14 8,590,000,000 2008.12.23

15 9,300,000,000 2008.12.23

16 8,390,000,000 2008.12.23

17 7,980,000,000 2008.12.23

18 8,210,000,000 2008.12.23

19 7,790,000,000 2008.12.23

20 9,660,000,000 2008.12.23

21 7,650,000,000 2008.12.23

22 7,680,000,000 2008.12.23

23 7,090,000,000 2008.12.23

24 7,170,000,000 2008.12.23

25 7,150,000,000 2008.12.23

26 6,220,000,000 2008.12.23

27 6,100,000,000 2008.12.23

28 6,160,000,000 2008.12.23

29 5,880,000,000 2008.12.23

30 8,000,000,000 2008.12.23

31 5,240,000,000 2008.12.23

32 3,150,000,000 2008.12.23

33 3,000,000,000 2008.12.23

34 2,880,000,000 2008.12.23

35 3,220,000,000 2008.12.23

36 2,920,000,000 2008.12.23

37 2,700,000,000 2008.12.23

38 2,720,000,000 2008.12.23

39 2,710,000,000 2008.12.23

40 3,950,000,000 2008.12.23

41 2,640,000,000 2008.12.23

42 2,630,000,000 2008.12.23

43 2,530,000,000 2008.12.23

44 2,620,000,000 2008.12.23

45 3,040,000,000 2008.12.23

46 2,480,000,000 2008.12.23

47 2,420,000,000 2008.12.23

48 2,620,000,000 2008.12.23

49 2,560,000,000 2008.12.23

50 4,750,000,000 2008.12.23

51 2,450,000,000 2008.12.23

52 2,430,000,000 2008.12.23

53 2,270,000,000 2008.12.23

54 2,310,000,000 2008.12.23

55 2,570,000,000 2008.12.23

56 2,280,000,000 2008.12.23

57 2,200,000,000 2008.12.23

58 2,220,000,000 2008.12.23

59 2,250,000,000 2008.12.23

60 2,240,000,000 2008.12.23

61 1,060,000,000 2008.12.23

62 1,020,000,000 2008.12.23

63 970,000,000 2008.12.23

64 1,140,000,000 2008.12.23

65 1,200,000,000 2008.12.23

66 1,040,000,000 2008.12.23

67 926,000,000 2008.12.23

68 925,000,000 2008.12.23

69 950,000,000 2008.12.23

70 1,680,000,000 2008.12.23

71 1,280,000,000 2008.12.23

72 1,720,000,000 2008.12.23

73 864,000,000 2008.12.23

74 861,000,000 2008.12.23

75 1,380,000,000 2008.12.23

76 852,000,000 2008.12.23

77 862,000,000 2008.12.23

78 843,000,000 2008.12.23

79 831,000,000 2008.12.23

80 1,920,000,000 2008.12.23

81 904,000,000 2008.12.23

82 858,000,000 2008.12.23

83 824,000,000 2008.12.23

84 841,000,000 2008.12.23

85 985,000,000 2008.12.23

86 890,000,000 2008.12.23

87 791,000,000 2008.12.23

88 878,000,000 2008.12.23

89 821,000,000 2008.12.23

90 1,780,000,000 2008.12.23

91 843,000,000 2008.12.23

92 835,000,000 2008.12.23

93 786,000,000 2008.12.23

94 779,000,000 2008.12.23

95 1,110,000,000 2008.12.23

96 828,000,000 2008.12.23

97 801,000,000 2008.12.23

98 946,000,000 2008.12.23

99 1,280,000,000 2008.12.23

100 3,900,000,000 2008.12.23

101 721,000,000 2008.12.23

102 796,000,000 2008.12.23

103 540,000,000 2008.12.23

104 551,000,000 2008.12.23

105 609,000,000 2008.12.23

106 836,000,000 2008.12.23

107 505,000,000 2008.12.23

108 526,000,000 2008.12.23

109 502,000,000 2008.12.23

110 728,000,000 2008.12.23

111 549,000,000 2008.12.23

112 509,000,000 2008.12.23

113 464,000,000 2008.12.23

114 475,000,000 2008.12.23

115 517,000,000 2008.12.23

116 457,000,000 2008.12.24

117 448,000,000 2008.12.23

118 456,000,000 2008.12.24

119 462,000,000 2008.12.23

120 936,000,000 2008.12.23

121 478,000,000 2008.12.23

123 489,000,000 2008.12.23

125 585,000,000 2008.12.23

127 412,000,000 2008.12.23

131 410,000,000 2008.12.23

133 402,000,000 2008.12.23

136 392,000,000 2008.12.23

140 579,000,000 2008.12.23

144 394,000,000 2008.12.23

150 1,010,000,000 2008.12.23

151 491,000,000 2008.12.23

160 599,000,000 2008.12.23

199 429,000,000 2008.12.23

200 1,510,000,000 2008.12.23

216 284,000,000 2008.12.23

243 306,000,000 2008.12.23

250 752,000,000 2008.12.23

255 277,000,000 2008.12.23

256 314,000,000 2008.12.23

300 1,410,000,000 2008.12.23

333 217,000,000 2008.12.23

399 185,000,000 2008.12.23

400 1,080,000,000 2008.12.23

401 194,000,000 2008.12.23

444 143,000,000 2008.12.23

499 222,000,000 2008.12.23

500 1,360,000,000 2008.12.23

501 191,000,000 2008.12.23

512 215,000,000 2008.12.23

513 119,000,000 2008.12.23

565 112,000,000 2008.12.23

599 145,000,000 2008.12.23

600 645,000,000 2008.12.23

601 135,000,000 2008.12.23

615 106,000,000 2008.12.23

616 102,000,000 2008.12.23

617 101,000,000 2008.12.23

665 90,400,000 2008.12.23

666 114,000,000 2008.12.23

667 111,000,000 2008.12.23

729 80,500,000 2008.12.23

999 189,000,000 2008.12.23

1000 1,160,000,000 2008.12.23

1001 129,000,000 2008.12.23

1002 64,400,000 2008.12.23

1003 64,500,000 2008.12.23

1066 57,600,000 2008.12.23

1331 35,700,000 2008.12.23

1699 37,300,000 2008.12.23

1998 841,000,000 2008.12.23

1999 1,070,000,000 2008.12.23

2000 2,300,000,000 2008.12.23

2001 1,640,000,000 2008.12.23

2002 1,730,000,000 2008.12.23

2003 2,200,000,000 2008.12.23

2004 2,640,000,000 2008.12.23

2005 3,670,000,000 2008.12.23

2006 5,330,000,000 2008.12.23

2007 7,350,000,000 2008.12.23

2008 15,270,000,000 2008.12.23

2009 1,980,000,000 2008.12.23

2010 488,000,000 2008.12.23

2011 290,000,000 2008.12.24

2012 261,000,000 2008.12.24

2020 123,000,000 2008.12.23

2061 26,300,000 2008.12.23

2222 33,500,000 2008.12.23

2361 20,800,000 2008.12.23

2999 37,700,000 2008.12.23

3000 586,000,000 2008.12.23

3001 37,000,000 2008.12.23

3002 19,700,000 2008.12.23

5101 15,500,000 2008.12.23

10,000 335,000,000 2008.12.23

16,384 4,000,000 2008.12.23

43,621 240,000 2008.12.23

43,622 430,000 2008.12.23

65,536 6,820,000 2008.12.23

100,000 126,000,000 2008.12.23

1,000,000 43,200,000 2008.12.23

10,000,000 1,010,000 2008.12.24

13,000,000 1,030,000 2008.12.23

13,260,000 11,700 2008.12.23

13,263,900 268 2008.12.23

13,263,952 101 2008.12.23

13,263,953 139 2008.12.23

13,965,321 445 2008.12.23

13,965,322 116 2008.12.23

13,965,323 496 2008.12.23

14,101,654 133 2008.12.23

45,691,083 142 2008.12.24

45,691,084 73 2008.12.24

45,691,085 212 2008.12.24

100,000,000 5,530,000 2008.12.23

353,354,149 3 2008.12.24

999,999,999 1,710,000 2008.12.23

1,000,000,000 1,970,000 2008.12.23

1,000,000,001 59,100 2008.12.23

1,000,000,002 64,100 2008.12.23

1,234,567,890 1,170,000 2008.12.23

7,249,000,982 4 2008.12.24

7,351,000,053 1 2008.12.24

7,351,000,093 1 2008.12.24

7,351,000,094 0 2008.12.24

7,351,000,095 0 2008.12.24

7,351,000,096 0 2008.12.24

7,351,000,097 0 2008.12.24

7,351,000,098 0 2008.12.24

7,351,000,099 0 2008.12.24

7,351,000,100 2 2008.12.24

7,351,000,101 0 2008.12.24

7,351,000,102 0 2008.12.24

7,352,344,145 0 2008.12.24

9,352,344,220 0 2008.12.24

9,352,344,221 1 2008.12.24

9,352,344,222 0 2008.12.24

9,876,543,210 99,000 2008.12.23

9,999,999,999 537,000 2008.12.24

10,000,000,000 793,000 2008.12.24

16,542,403,674 0 2008.12.23

16,542,403,675 0 2008.12.23

16,543,403,676 0 2008.12.23

100,000,000,000 425,000 2008.12.24

1,000,000,000,000 511,000 2008.12.23

10,000,000,000,000 226,000 2008.12.24

100,000,000,000,000 179,000 2008.12.24

1,000,000,000,000,000 206,000 2008.12.24

10,000,000,000,000,000 205,000 2008.12.24

100,000,000,000,000,000 107,000 2008.12.24

1,000,000,000,000,000,000 155,000 2008.12.24

10,000,000,000,000,000,000 085,300 2008.12.24

100,000,000,000,000,000,000 077,700 2008.12.24

1,000,000,000,000,000,000,000 076,200 2008.12.24

1,000,000,000,000,000,000,000,000 061,200 2008.12.24

1,000,000,000,000,000,000,000,000,000 037,200 2008.12.24

1,000,000,000,000,000,000,000,000,000,000 037,100 2008.12.24

1,000,000,000,000,000,000,000,000,000,000,000 031,000 2008.12.24

1,000,000,000,000,000,000,000,000,000,000,000,000 018,800 2008.12.24

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This survey has revealed some surprising ( or not so surprising ) facts about number usage. The most used counting number of all is of coarse 1. It has an incredible 21 billion hits ! The next highest hit count is that of 18 billion hits for the number 2.

The first 9 counting numbers literally follow a pattern of decreasing hits. Yet 10 spikes up to 14 billion hits, making it more used than the number 4 ! Yet it is still less common that 3 with over 15 billion hits.

The smallest number of hits for a counting number less than or equal to 100 is 779 million hits for the apparently unpopular number 94. 100 is of coarse much more popular than it should be given it's size. 100 recieves as many as 3.9 billion hits. Yet 101 recieves less than a billion hits.

One very unusual find was that the number 2000 is used almost twice as frequently as 1000. One would expect it to be the other way around since 1000 is a power of 10. However the hit counts really seem to peak around 2000 because we are currently in the beginning of the 21st century. Usage actually increases after 2000, and finally peaks at 2008 with a very remarkable 15 billion hits. Not only is 2008 used nearly 5 times as frequently as 2000 but it's also one of the very few numbers whose usage is higher than that of 4 ! Imagine, there are slightly more occurances of 2008 than 4 !

Interestingly, right after 2008, the hits drop drastically. 2009 only gets around 1.98 billion hits. 2010 gets less than half a billion.

Here is another interesting find. I searched to see what others had reported as the smallest counting number with no hits. This webpage ... http://iq.lycos.co.uk/qa/show/24929/What-is-a-dull-number-A-number-is-dull-if-no-one-uses-it.--What%E2%80%99s-the-smallest-number-with-no-hits-o/ dating from around december of 2006 reported that the smallest known number with no hits was 13,965,321. I ran a search on this number to see if this was true, but instead I got 445 hits ! In otherwords this gap has now been filled. Numbers surrounding this one also seem to get hits around the hundreds. This can not be a coincidence. This may very well be a result of the fact that the internet has been growing exponentially over the years since it's inception. 2 years ago the web may have been much smaller, which means the smallest number with no hits, would have been smaller than it is today.

Here is another example ... at another blog page [1] someone mentions that the number produced by evaluating 2x2x1153x9907 recieves no hits. When multiplying this together you get 45,691,084 ( An 8 digit number ). The date of this comment was July 11 , 2007. Only about a year and a half ago. I also ran a search on this number and found that it now has 73 hits ! It's neighboring numbers also hit in the hundreds. It may be that the era of "8-digit no hitters" is already over !

There is an article ... http://www.cnn.com/2006/TECH/internet/11/01/100millionwebsites/

that discusses the growth of the web. Here it is reported that only around 18,000 websites existed back in august of 1995. By october of 2006 however, it is reported that the internet reached 100 million websites ! That's 4 orders of magnitude in only 11 years. Very roughly that means that the web may grow 10 fold every 2 years. If this is correct, then it is not surprising that a number in the 13 millions that once had no hits is now getting in the hundreds of hits.

The smallest number I found with no hits was 7,351,000,094. That is 2 orders of magnitude above the 13 million figure. However, my search was far from thorough. Most likely the smallest counting number with no hits may be one magnitude less than this. I am guessing this because less than 2 years have gone by, which means the web shouldn't have grown more than ten fold, and consequently, the smallest no hitter should therefore be in the hundred millions. Unfortunately I have not succeeded in finding a 9-digit no hitter. One problem is that there are massive phone number listings in this range, which seem to cover a large number of the possibilities. Yet, there are roughly speaking, a billion 9 digit numbers. I assume that there must be less phone numbers than this, which means there must be 9 digit numbers which do not correspond to any phone number. Yet, it is kind of like finding a needle in a hay stack. The missing numbers would be random. If anyone finds a 9-digit no hitter, inform me. I'll check it myself, and if it's a no-hitter I'll add it to the list.

All of this is very interesting, but what does it show us ? It shows us an usual inverse relationship between numbers and usage. Think about this for a moment. Most numbers are large ones. Take a sample, say from 1 to a trillion. The vast majority of these numbers are larger than a million. In other words, small numbers, the ones we feel comfortable with, are actually rare creatures. Yet despite the large numbers being more common, individually, a very large number has much less chance of being represented than a small one.

Also there is another interesting point that this brings out. As numbers get larger, the number of practical applications diminishes. Thus most large numbers are trivial. The hit count measures the usefulness of the number and confirms this basic notion. Both mathematicians and scientists believe that large numbers for their own sake are meaningless.

At the same time however, there are certain very large numbers that will maintain large attention. For example an undecillion in decimal form, still manages to recieve 18,800 hits. Although these kinds of examples become increasingly rare, and even the hit counts for larger powers of 10 would begin to diminish, it still is impressive to see that these numbers still get hits at all. The powers of a 1000 remain relatively strong ( despite the fact that they inhabit an endless sea of zero hitters ) mainly because there are many sites which list out them. But as the number of zeroes climb, the number of eccentrics willing to write them all out in decimal diminishes.

For now this concludes our survey of the counting numbers. You can now continue to the summary if you like ...

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Source Material :

[1] http://mathfactor.uark.edu/2007/06/06/qa-numerous-numbers/ : A interesting blog started in 2007 which talks about the smallest no hitters.