POWERS OF 10 AND SOME LARGE NUMBERS
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Before we examine some powers of 10, let's consider a basic premise about whole numbers in general. There are three basic aspects of a whole number: its name, its numeral, and its elemental nature. Consider the number ten. Its name is "ten". Its numeral is "10". Its elemental nature is found in a collection of things, such as, in the figure of ten balls to the right. |
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Consider the number one thousand. Its name is "one thousand". Its numeral is "1000". Its elemental nature is found in a collection of things, such as, in the figure to the right of one thousand red boxes (1000 is 24 short of 322 = 1024). Seeing a Million Pixels The elemental nature of the number one million, 106, one followed by 6 zeros, 1 000 000, probably cannot be totally discernable to the human eye. The reason is that if you are close enough to discern the discrete particles in the collection, then you cannot take in the entire field of view. If you back up far enough to take in the field of view, you cannot discern the discrete particles. They tend to blur together, like pixels in a computer graphic. The graphic below, for instance, contains 128 x 128 red pixels and 129 x 129 white pixels interspersed between the red pixels. That is only 16384 red pixels, nowhere near one million, yet it gets to the crux of the matter in what the human eye can discern. (There would have to be about 61 patches like the one shown below to approach one million red pixels. If the screen resolution on your monitor is 1200 x 800 pixels, and it was filled with the same red pixel pattern as shown below, there would be almost one-quarter million red pixels showing, but not quite. The full screen could display only about 15 patches. If you could squeeze the red pixels closer together in order to get 61 patches showing, then it would be that much harder to discern the red pixels.) |
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In the next section, we see the dimensional aspects of the powers 0, 1, 2, and 3. However, you do not really see 1000 cubes in the figure for 103 because some are hidden behind other cubes. You can see 3 faces of one of the cubes, but only 1 or 2 faces of others. (How many cubes can you see? If you can see any part of a cube, you can count it. You can win many bets with this question, because most people will miscount the number that can be seen.) |
DIMENSIONAL ASPECTS OF POWERS OF 10
AND SOME SMALL NUMBERS
The prefixes in the table above are used when a number represents a quantity rather than a count, such as,
kilowatt, nanosecond or centimeter.
10! = 1×2×3×4×5×6×7×8×9×10
= 3,628,800
~ 3.6 x 106 (approx. 3.6 million)
If you put a different letter, a thru j, on each one of someone's thumbs and fingers, there would be 10!=3,628,800 ways to do that.
Ten Trillion = 10x(10x103)3 (see the discussion of Centillion below on this page.)
The age of the universe is estimated to be approx. 1.37 x 1013 years
(as of the early 21st century.)
16! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16
= 20,922,789,888,000
~ 2.1 x 1013 (approx. 21 trillion)
A billion trillion = one sextillion = (10x106)3 (see the discussion of Centillion below on this page.)
22! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22
= 1,124,000,727,777,607,680,000
~ 1.124 x 1021 (approx. 1.124 billion trillion or 1.124 sextillion)
Factorial Facts
1023
Avogadro's number (constant): NA = 6.022 141 79 × 1023 = 602 214 179 000 000 000 000 000
That is about 602 billion trillion.
( 6.022 141 79(30)×1023 mol–1 CODATA-06, number of atoms in 12 grams of carbon-12)
1038 = one hundred undecillion = 100x(10x1011)3
(see the discussion of Centillion below on this page.)
Breaking a 128 bit SSL encryption key, commonly used in web browsers, using brute force computation is a problem
requiring up to 2128 = 3.4 x 1038 operations. Even allowing 240 (approx. 1 trillion) operations per second
(teraflop), it would require 288 seconds or about 9.8 x 1018 years to carry out that many operations.
Since the age of the universe is estimated to be approx. 1.37 x 1013 years, the
brute force method of computing a solution is simply impossible
(as of the early 21st century).
(Even if your computer was twice as fast (241 (approx. 2 trillion)
operations per second) and you were able to break the key in
half the number of operations (2127 operations),
it will still take about 2.45 x 1018 years.)
(see the discussion of Centillion below on this page.)
52! = 1×2×3×4×5×6×7×8×9×10×...x20x21x22x...x30x31x32x...x40x41x42x...×50×51×52
= 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
~ 81 x 1066 (approx. 8.1x1067 or 81 unvigintillion)
52! is a number with 68 digits! If you shuffled a deck of 52 playing cards, it is the number of different
shuffles (orderings) that would be possible. Needless to say, no one could ever perform that many shuffles. Even if you
could shuffle the deck a trillion times a second (and get a different arrangement every time), it would take
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824 seconds, or approx. 2.6x1048 years, a period
of time way longer (2x1035 times longer) than the estimated age of the universe (as of the early 21st century.)
1080 = 100x(10x1025)3
(one hundred quinvigintillion)
The edge of the observable universe is now located about 46.5 billion light-years (4.399×1026 meters) away in any direction.
Estimates (early 21st century) of the matter content of the observable universe indicate that it contains on the order of 1080 atoms.
(Ref: Eddington number)
70! is the smallest factorial with over a googol of digits! 70! > 10100
More links for "googol".
CENTILLION = 10303 = (10x10100)3
Million, Billion, Trillion, Quadrillion,...,Centillion,...
These are numbers that start with one thousand and append 3 zeros each time.
A Trillion, 1,000,000,000,000, can be described as 1000 with 3 (tri) sets of 3 zeros appended.
One decillion is 1000 followed by 10 sets of 3 zeros or 30 zeros appended to 1000.
A Centillion is 1000 with 100 sets of 3 zeros appended.
A "Kilillion" would be 1000 with 1000 sets of 3 zeros appended or 103003.
In between these last two numbers is 1000!, one thousand factorial, a number containing 2568 digits.
Click here to see this large number. 1000!.
Click below for an Educational site, which can name any numbers put into it (up to centillion).
mathcats.com
Naming large numbers is based on the formula (10x10n)3.
The name of a number of this form is constructed by creating a prefix (usually Latin, bi-, tri-, quad-, quint-, sex-, sept-,
oct-, non-, dec-, undec-, dodec-,...,vigin-, unvigin-, dovigin-, ..., trigin-,...,cen-, etc.) for the number n and a suffix,
either -llion, -illion, or -tillion.
Thus, (10x10100)3 = cen-tillion, and you will find other examples in the discussions above.
Click below for another site, which can provide a name for any number. How high can you count? by Landon Curt Noll
How High Can You Count?
109,999,999
Great Internet Mersenne Prime Search - GIMPS
The number 109,999,999 is the smallest 10 million digit number.
On August 23, 2008, a UCLA computer in the GIMPS PrimeNet network discovered the 45th known (at that time) Mersenne prime number,
243,112,609 - 1,
a mammoth 12,978,189 digit number (almost 13 million digits!)
The prime number qualifies for the Electronic Frontier Foundation's $100,000 award for discovery of the
first 10 million digit (or more) prime number.
The Great Internet Mersenne Prime Search (GIMPS), founded in 1996, is a distributed computing project dedicated
to finding Mersenne primes. Their website is www.mersenne.org/ .
In 1999 the first Mersenne prime number with over 1 million digits was found.
In 2008 two Mersenne primes, with over 10 million digits, were found.
The largest prime number found to date is the one cited above, M43112609 = 243,112,609 - 1.
This number starts with the digits 316 and ends with 511. A link to a listing of this number, 316...511, in text form (16.3 megabytes),
is M43112609
Mersenne Prime, M43112609 = 243,112,609 - 1 (12,978,189 digits)Really, really big, yet finite, numbers.
For discussions and explorations of numbers larger than even those mentioned above
you can look up Skewes' number · Moser's number · Graham's number and notational methods
used to represent them Knuth's up-arrow notation · Conway chained arrow notation.
Graham's number is so large than it cannot be conveniently expressed in conventional number notation.
If all the matter in the observable universe could somehow be converted to ink and paper, there would
not be enough ink and paper to write the number down in conventional number notation, that is, the notation
used above to write a googol in base ten notation. Needless to say, the elemental nature of these numbers
only exists in the abstract.
Find out more about large numbers.
http://www.absoluteastronomy.com/topics/Large_numbers
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send comments to Thomas M. Green c/o CCC Math Dept. hwalters@contracosta.edu