A student asked me:
> What is the value of the digit 7 in 789,354,199,033?
To decide place values, we look at all the positions, starting at the decimal point, going right to left.
ones------------------------------------\
tens----------------------------------\ \
hundreds----------------------------\ \ \
\ \ \
thousands------------------------\ \ \ \ D
10 thousands-------------------\ \ \ \ \ e
100 thousands----------------\ \ \ \ \ \ c
\ \ \ \ \ \ i
millions------------------\ \ \ \ \ \ \ m
10 millions-------------\ \ \ \ \ \ \ \ a
100 millions----------\ \ \ \ \ \ \ \ \ l
\ \ \ \ \ \ \ \ \
billions-----------\ \ \ \ \ \ \ \ \ \ P
10 billions------\ \ \ \ \ \ \ \ \ \ \ o
100 billions---\ \ \ \ \ \ \ \ \ \ \ \ i
\ \ \ \ \ \ \ \ \ \ \ \ n
trillions---\ \ \ \ \ \ \ \ \ \ \ \ \ t
10 \ \ \ \ \ \ \ \ \ \ \ \ \
trillions--\ \ \ \ \ \ \ \ \ \ \ \ \ \ |
100 \ \ \ \ \ \ \ \ \ \ \ \ \ \ |
trillions-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |*
+--+--+--+,+--+--+--+,+--+--+--+,+--+--+--+,+--+--+--+
Write your number into the boxes above. Make sure the last digit of your number goes just before the decimal point in the chart.
What if you have positions after the decimal point? You count from tenths until you get to the position you are trying to determine the value of. Write the number you are trying to evaluate into the boxes below, starting at the decimal point on the left.
decimal +--+--+---+--+--+---+--+--+---+--+--+---+--+--+
point | | | | | | | | | | | | | | |
--------> * | | | | | | | | | | | | | | |
+--+--+---+--+--+---+--+--+---+--+--+---+--+--+
/ / / / / / / / / / / / / /
tenths----------/ / / / / / / / / / / / / /
hundredths--------/ / / / / / / / / / / / /
/ / / / / / / / / / / /
thousandths---------/ / / / / / / / / / / /
ten-thousandths-------/ / / / / / / / / / /
hundred-thousandths-----/ / / / / / / / / /
/ / / / / / / / /
millionths----------------/ / / / / / / / /
ten-millionths--------------/ / / / / / / /
hundred-millionths------------/ / / / / / /
/ / / / / /
billionths----------------------/ / / / / /
ten-billionths--------------------/ / / / /
hundred-billionths------------------/ / / /
/ / /
trillionths---------------------------/ / /
ten-trillionths-------------------------/ /
hundred-trillionths-----------------------/
If you study this pattern, and the list of larger number names further down, you should be able to create your own chart going as high, and as small, as you'd like to go.
Here is the entire chart of all the number names (I know of) in the American Numbering System. Note that the British Numbering System does not match the American system after the millions. Thousands and Millions match, but then the British system has milliards, billions, billiards, trillions, trilliards, etc.
Since our numbering system is based on LATIN, for the number of sets of zeroes after 1,000. I have listed it by the number of ',000' sets after one thousand. Notice that BI=2 (bicycle), TRI=3 (tricycle), QUAD=4 (quadruplets). So, the latin numbers for 2 up to 100 are approximately represented in the names of the number groups.
number
of
0's Name samples of some of them
==--======---==================
thousand 1,000
1 million 1,000,000
2 billion 1,000,000,000
3 trillion 1,000,000,000,000
4 quadrillion 1,000,000,000,000,000
5 quintillion 1,000,000,000,000,000,000
6 sextillion 1,000,000,000,000,000,000,000
7 septillion 1,000,000,000,000,000,000,000,000
8 octillion 1,000,000,000,000,000,000,000,000,000
9 nonillion 1,000,000,000,000,000,000,000,000,000,000
10 decillion 1,000,000,000,000,000,000,000,000,000,000,000
11 undecillion 1,000,000,000,000,000,000,000,000,000,000,000,000
12 duodecillion
13 tredecillion The latin roots of the numbers show the
14 quattuordecillion # of groups of 3-zeroes that follow 1000.
15 quindecillion 2=bi, 3=tri, 4=quad, 5=quin, 6=sex, 7=sept
16 sexdecillion 8=oct, 9=non, 10=decem, 20=vigin.
17 septendecillion
18 octodecillion Names for groups 3 to 63 come from several
19 novemdecillion sources, both encyclopedic and dictionary.
20 vigintillion
21 unvigintillion Prefixes un-, duo-, tre-, quattuor, quin-,
22 duovigintillion sex-, sept-, octo, and novem-, appear to
23 trevigintillion modify TEN (decem) to make 11 to 19.
24 quattuorvigintillion
25 quinvigintillion I followed the same pattern before
26 sexvigintillion vigintillion to create 21 to 29 sets
27 septenvigintillion of 000's after 1,000.
28 octovigintillion
29 novemvigintillion decem=10, viginti=20, centi=100.
30 trigintillion
31 untrigintillion
32 duotrigintillion
33 tretrigintillion
34 quattuortrigintillion
35 quintrigintillion
36 sextrigintillion
37 septtrigintillion
38 octotrigintillion
39 novemtrigintillion
40 quardragintillion
41 unquardragintillion
42 duoquardragintillion
43 trequardragintillion
44 quattuorquardragintillion
45 quinquardragintillion
46 sexquardragintillion
47 septquardragintillion
48 octoquardragintillion
49 novemquardragintillion
50 quinquagintillion
51 unquinquagintillion
52 duoquinquagintillion
53 trequinquagintillion
54 quattuorquinquagintillion
55 quinquinquagintillion
56 sexquinquagintillion
57 septquinquagintillion
58 octoquinquagintillion
59 novemquinquagintillion
60 sexagintillion
61 unsexagintillion
62 duosexagintillion
63 tresexagintillion
64 quattuorsexagintillion
65 quinsexagintillion
66 sexsexagintillion
67 septsexagintillion
68 octosexagintillion
69 novemsexagintillion
70 septuagintillion
71 unseptuagintillion
72 duoseptuagintillion
73 treseptuagintillion
74 quattuorseptuagintillion
75 quinseptuagintillion
76 sexseptuagintillion
77 septseptuagintillion
78 octoseptuagintillion
79 novemseptuagintillion
80 octogintillion
81 unoctogintillion
82 duooctogintillion
83 treoctogintillion
84 quattuoroctogintillion
85 quinoctogintillion
86 sexoctogintillion
87 septoctogintillion
88 octooctogintillion
89 novemoctogintillion
90 nonagintillion
91 unnonagintillion
92 duononagintillion
93 trenonagintillion
94 quattuornonagintillion
95 quinnonagintillion
96 sexnonagintillion
97 septnonagintillion
98 octononagintillion
99 novemnonagintillion
100 centillion 100 sets of 000 after 1,000.
One GOOGOL is a ONE followed by 100 zeroes.
One GOOGOLPLEX is a ONE followed by one googol zeroes.
(you can't write a googolplex, even in a million years)
What is a googol?
one googol is equal to ten, raised to the one hundredth power. It is written as a one followed by one hundred zeroes. This is a googol:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
What is a googolplex?
one "googolplex" is equal to ten raised to the googol power. If it were to be written out, it would be a one followed by one googol zeroes.
One googolplex equals 10 raised to the 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 power.
One googolplex is written as a one followed by 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 zeroes.
There is a fascinating article in Compton's Living Encyclopedia (AOL keyword: ENCYCLOPEDIA) that compares the American Numbering System with the British Numbering System. Once you get past eight zeroes, the names do not match. For example, a number followed by 9 zeroes is in the British "milliards" but the American "trillions".
In other words, the British Numbering System is completely different from the American Numbering System after the thousands and millions. Thousand and Million are the same, but then you see Milliard, Billion, Billiard, Trillion, Trilliard, etc.
TRIVIA QUESTION: If a "Billiard" is a one followed by 15 zeroes in the British System, then does this mean that is where we got the name of the game BILLIARDS, which is played with a cue ball and 15 numbered balls? If someone knows, please let me know.
Special thanks to Søren Gjellerup Christiansen in Denmark for this improvement on the paragraph above:
TRIVIA ANSWER:
I think it more likely that it has this name "billiard" because it is played with a stick (1) and 15 balls (0) - don't count the cue ball, it isn't numbered. Line them op and you get 1000000000000000 = one billiard.I just find this more intuitive myself.
Of course, other people have written in to say that there is no relationship between the name of the game and the name of the number. But it is fun to think about.
number
of 0's Name samples of some of them ==--======---================== 3 thousand 1,000 6 million 1,000,000 9 milliard 1,000,000,000 12 billion 1,000,000,000,000 15 billiard 1,000,000,000,000,000 18 trillion 1,000,000,000,000,000,000 21 trilliard 1,000,000,000,000,000,000,000 24 quadrillion 1,000,000,000,000,000,000,000,000 27 quadrilliard 1,000,000,000,000,000,000,000,000,000 30 quintillion 1,000,000,000,000,000,000,000,000,000,000 33 quintilliard 1,000,000,000,000,000,000,000,000,000,000,000 36 sextillion 1,000,000,000,000,000,000,000,000,000,000,000,000 39 sextilliard 42 septillion NOTE that if you divide the # of zeroes by six, 45 septilliard you have the number that relates to the word of 48 octillion the group. 2=bi, 3=tri, 4=quad, 5=quin, 6=sex, 51 octilliard 7=sept, 8=oct, 9=non, 10=decem, 20=viginti. 54 nonillion 57 nonilliard the names for number groups 60 decillion (TEN) 3, 6, 9, 12, and every multiple 63 decilliard of 6 up to 120 came from 66 undecillion several sources, both dictionary 69 undecilliard and encyclopedic. 72 duodecillion 75 duodecilliard I followed the pattern of 78 tredecillion "million" to "milliard" to list 81 tredecilliard all other groups up to 123. 84 quattuordecillion 87 quattuordecilliard The prefixes un-, duo-, tre-, quattuor-, 90 quindecillion quin-, sex-, septen, octo-, and novem- 93 quindecilliard appear to modify the latin TEN (decem), 96 sexdecillion to make 11, 12, up to 19. 99 sexdecilliard 135 septendecillion 105 septendecilliard 108 octodecillion 111 octodecilliard 114 novemdecillion 117 novemdecilliard 120 vigintillion (TWENTY) 123 vigintilliard 126 unvigintillion I followed the prefix-pattern 129 unvigintilliard used before -decillion and 132 duovigintillion -decilliard to create the names 135 duovigintilliard for 126 to 177, 21 through 29 138 trevigintillion groups of six. 141 trevigintilliard 144 quattuorvigintillion 147 quattuorvigintilliard 150 quinvigintillion 153 quinvigintilliard 156 sexvigintillion 159 sexvigintilliard 162 septenvigintillion 165 septenvigintilliard 168 octovigintillion 171 octovigintilliard 174 novemvigintillion 177 novemvigintilliard you can figure out the names are for 180 through 597 zeroes by looking at the list for American Numbering System. 180, 186, 192 ...: Divide the number by 6, use that name. 183, 189, 195 ...: Change the name for 3 less. 600 centillion 603 centilliard? (absolute guess, not found in any source).
I don't know if the terms Googol and Googolplex are used in the British Numbering System. Since there were coined by American Mathematicians in 1929 and 1935, it is likely that they are not officially used in the U.K., but unofficially, I'm sure that some of them use it for fun.
You can find some of his puzzles on his web page at www.MAZES.com or at ftp:// www.mazes.com. More puzzles can be found at http://www.mazes.com/buymazes. John has written materials that can help people get ready for the Driver's License exam. You can find these materials at www.DriverExam.org or at ftp:// www.driverexam.org.
John is a computer programmer, consultant, web page designer, teacher, tutor and puzzle creator. John telecommutes throughout the world and is available to work for you. Please e-mail: John (at) Mazes.com.
If you use information from a source like this in a report, you should give proper credit, citing the author as a source. In other words, you must acknowledge where you found the information and who supplied it. This is a matter of professional courtesy. Here is an example of a footnote:
Footnote Example:
* John Knoderer, [webmaster(at)Mazes.com], "American Numbering System and
Place Values," http://www.mazes.com/numberingsystem.html, 10 September 1997.