1.2. Usage of the -illions


Introduction

From the previous article, we are now familiar with the history of the -illions and the English numbers. However, we haven't gotten really into how and when they'll used, and in what kinds of situations we'd need usage of a name like nonillion (10^30). Here we will examine the -illions' usage in detail. But how exactly will we do that?

In July 2014, I decided to Google every -illion name up to a centillion (this includes a lot of unofficial Latin-based names that were devised to fill the gap between a vigintillion and centillion) and catalog the number of results. A centillion is a good stopping point for -illions in my opinion, partly because it's the largest canonical -illion, and partly because it's the 100th. This in turn is because reaching 100 is a classic stopping point for many things, ranging from top 100 lists to surveys of 100 people, and 100 works well with the base of our decimal system (aka 10) as it's reasonaly large in day-to-day context but not out of reach by any means.

In this article, I will share my results and then analyze them and see what they say about the usage of all those -illions.

The Google Results

Here is the table of the data I found through Google:

n

results for nth -illion

n

results

n

results

n

results

n

results

1

2,840,000,000

21

4,060

41

4,300

61

3,240

81

2,920

2

403,000,000

22

12,400

42

5,260

62

3,150

82

3,130

3

47,800,000

23

10,100

43

4,940

63

3,540

83

3,120

4

1,480,000

24

13,700

44

5,730

64

3,420

84

3,430

5

852,000

25

9,360

45

5,420

65

2,770

85

49,000

6

1,370,000

26

4,660

46

3,320

66

2,920

86

2,900

7

206,000

27

7,620

47

4,390

67

2,070

87

1,940

8

88,900

28

9,050

48

3,460

68

48,700

88

3,450

9

65,800

29

9,930

49

6,580

69

3,350

89

2,900

10

69,500

30

17,800

50

9,430

70

6,480

90

8,610

11

147,000

31

10,000

51

6,650

71

2,770

91

2,770

12

23,600

32

7,470

52

4,710

72

2,910

92

3,100

13

14,800

33

5,720

53

3,010

73

2,800

93

2,710

14

50,800

34

8,850

54

3,570

74

3,200

94

4,490

15

69,200

35

6,430

55

41,700

75

4,870

95

3,240

16

15,000

36

5,190

56

3,340

76

2,900

96

2,720

17

11,600

37

7,380

57

4,330

77

2,230

97

2,230

18

14,200

38

5,960

58

3,500

78

3,470

98

3,300

19

15,100

39

7,940

59

3,750

79

4,030

99

5,780

20

19,600

40

8,660

60

6,660

80

7,140

100

538,000


and a graph (logarithmic scale):


... and now we'll go through the quirks of the data.

Analyzing the Results

First off, what's up with the colors of the dots in the graph?

They're color-coded as follows:

The yellow dot is a million, the blue dot is a billion, the red dot is a trillion, the green dot is a quadrillion, the purple dot is a quintillion, the orange dot is a sextillion, the teal dot is a septillion, the brown dot is an octillion, the yellow-brown dot is a nonillion, and the dark green dot is a decillion. I felt like each of these first ten -illions should have its own unique color, so that's what I did.

Then the dark gray dots represent the other canonical -illions up to a vigintillion. In order they're undecillion, duodecillion, tredecillion, quattuordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion, and vigintillion.

The light gray dots represent non-canonical (not part of the English language) -illions whose names extrapolate from the canonical -illions. The best known system is formed like so:

The 21st thru 29th -illions are formed by applying the prefixes un-, duo-, tre-, quattuor-, quin-, sex-, septen-, octo-, and novem- to vigintillion, respectively, so the -illion after vigintillion is unvigintillion, followed by duovigintillion, trevigintillion, etc.

Then the 30th -illion is called a trigintillion, and the same prefixes un- through novem- are applied to it to form the 31st and 39th -illions.

The same process is done to quadragintillion (40th) for the 41st-49th -illions, so for example the 46th -illion would be a sexquadragintillion. Then you can do that for a quinquagintillion (50th -illion), sexagintillion (60th), septuagintillion (70th), octogintillion (80th), and nonagintillion (90th), so the 99th -illion, the last one before a centillion, is called a novemnonagintillion, equal to 10^(3*99+3) = 10^300.

Finally, the dark violet dot is a centillion, the 100th -illion.

Now let's look at the results:

The first -illion, a million, has an impresive 2.84 billion results. This isn't too surprising, considering how a million is an everyday occurrence in today's world. Millions show up in all kinds of contexts, even as hyperbole and in today's idioms (e.g. one in a million). This is also the only time the number has more results than the value of the number.

The second -illion, a billion, is a notable drop from a million in occurrence. It has about 403 million results, making it about 20% as common as a million. The biggest occurrence of billions is in economics dealing with billions of dollars. In fact, the economic occurrences largely crowd out other common occurrences when searching for a billion, such as hyperbole and statistics.

The third -illion, a trillion, is another, logarithmically bigger drop with 47.8 million results compared to 403 million results for a billion, making it about 11% as common as a billion, almost an order of magnitude (10x) less. Here, the vast majority of occurrences of a trillion are in economics, hugely crowding out statistical and other occurrences. Millions, billions, and trillions have a lot more results than the rest, because those are the three "familiar" -illions.

The fourth -illion, a quadrillion, is a still bigger drop from a trillion, with 1.48 million results compared to 47.8 results for a trillion. It's only 3% as common as a trillion. Unlike a trillion, a quadrillion doesn't have very much use in economics, other than listing values in the whole world or in currencies where the unit isn't very valuable (e.g. yens). Therefore, a good amount of the occurrences of quadrillions are instead in statistics or science. We can clearly see that quadrillions are a lot less common than trillions, but they're far from unheard of.

The fifth -illion, a quintillion (10^18), is not so sharp of a decrease from a quadrillion, with 852,000 occurrences, about 60% as many results as a quadrillion. By now, most of its occurrences are in science and statistics, since a quintillion exhausts economic usage. In general, a quintillion seems to be a cut-off point for large numbers as larger ones really don't get used much, but I find a decillion to be a bigger cutoff point.

The sixth -illion, a sextillion (10^21), oddly has more occurrences (60% more) than a quintillion, at 1.37 million occurrences. Perhaps that's because of it being part of two astronomical figures: Earth weighs 6 sextillion tons, and it's estimated that there are 300 sextillion stars in the observable universe.

The seventh -illion, a septillion (10^24), occurs mainly in astronomy and occasional statistics, but by now these occurrences are so rare that they're becoming outcrowded by stuff like dictionary entries. It's only about 20% as common as a sextillion.

Fun fact: When googling "septillion" the Googology Wiki entry is on the FIRST PAGE! This kind of googling of -illions was how I discovered Googology Wiki and the wonderful world of googology, and that may be true for other people.

The eighth -illion, an octillion (10^27), is pretty much the same situation as a septillion. It's 40% as common as a septillion.

The ninth -illion, a nonillion (10^30), is now rare outside of lists, but it's been occasionally used in outrageous statistics. It's about 80% as common as an octillion.

The tenth -illion, a decillion (10^33), seems to be a pretty big cutoff point for large numbers (see also a quintillion). Illions larger than this are very rarely heard of, as after that scientific notation is almost excluxsively preferred. A decillion is a few percent more common than a nonillion, and by now the number of results is in the ten-thousands order of magnitude.

The eleventh -illion, an undecillion (10^36), is actually more common than an octillion, nonillion, and decillion. This is only because of a news story of a New York man suing for two undecillion dolars bringing some sparks of mention of an undecillion on the Internet.

However, a duodecillion (10^39) through a novemdecillion (10^60) show a pretty regular pattern of "pretty scarce, more so than a decillion" (with a few exceptions). They mostly have between 10,000 and 20,000 results.

After a vigintillion (10^63), the last sequential canonical -illion, the numbers follow a pattern which is kind of irregular but shows a gradually becoming scarcer and scarcer trend. However, one thing to note (which isn't surprising) is that every tenth -illion (trigintillion, quadragintillion, quinquagintillion, etc) is more common than its neighbors.

After an untrigintillion (10^96, the 31st -illion), only three -illions have 10,000 or more results. In fact, they appear as three spikes in the graphs. They are a quinquinquagintillion (10^168, the 55th -illion (which has a nice ring to it in my opinion)), octosexagintillion (10^207, the 68th -illion), and a quinoctogintillion (10^258, the 85th -illion). I can't seem to figure out where quinquinquagintillion's popularity comes from (other than it's nice ring perhaps?), but the other two clearly come from calculations related to power levels in Dragon Ball Z; apparently Goku's power level is over an octosexagintillion from one calculation but over a quinoctogintillion in another. Weird, but at least the last two are plausible.

But that last value, a centillion (10^303, the 100th -illion), has more results than all -illions except for million through sextillion!! It's obvious why it has a large amount of results - it's the largest canonical -illion, therefore it has a similar spirit to the famed googol, said to be the quintessential large number. It's kinda weird but satisfying that it has that many results.

Also, here are a few additional large numbers' number of results for comparison:

googol (10^100) - 1,940,000 - more results than all -illions except for million, billion, and trillion
googolplex (10^googol) - 328,000 - more results than all -illions except for million through sextillion and centillion
googolduplex (10^googolplex) - 5,300 - still more than some of the -illions
googolplexian (other name for a googolduplex) - 35,100 - more than most of the -illions
googoltriplex (10^googolduplex) - 6,290

And some fake -illions that don't refer to any specific number as well:

jillion - 10,100,000 - more results than all except for million, billion, and trillion
zillion - 9,720,000 - ditto
gazillion - 4,100,000 - ditto
bazillion - 2,000,000 - ditto
bajillion - 761,000 - more results than all except for million through sextillion
gajillion - 182,800 - more than all except for million through septillion and centillion
kajillion - 88,700 - more results than all except for million through octillion, undecillion, and centillion

Conclusion

Now that we have a clear review of the usage of the -illions, it's time to examine how big they really are. They're a lot bigger than you probably think. In the next article we'll do just that, with a tour of the mind-blowing sizes of the -illions.

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