Large Numbers

### Large Numbers

Allen Klinger, © 4/1/2009

People today can handle huge numbers because of words and signs. As examples of extremely big amounts think about answering these questions. How many grains of sand are on a beach? How many different card hands are possible? How many electrons are in the universe? In an era where counting meant using fingers and toes nobody could have come up with an answer. But today words like hundred, thousand, and million let us express how much, how many, and compare very big quantities..

Words and signs are both short ways of writing about very big quantities. Signs that express multiplication or times are at the heart of taking ordinary numbers, symbols like 1, 2, 3, or 7, 8, 9, and writing about huge values. One multiplication usually uses a number like 10 or 2, times itself as in the statement a hundred is ten times ten. But we can also benefit from things like four times three times two is twenty-four.

In 1808 Christian Kramp [12] first used a sign to show the product of increasing integer numbers from 1 to n, including all the intermediate values. He wrote that by the symbol n!. Today it goes by the name factorial. (You also can look at this as multiplying numbers from the largest in the product to the smallest.)

Another way multiplies the same number times itself. That number of times of multiplications (two, three, ... , etc.) is at the heart of many scientific expressions. Two is a useful number for computers; with money ten has that role today. Words like ten cents (pennies) is the same as a dime, ten dimes is a dollar can be written this way using same-number expressions:

10 cents = 1 dime; 10 dimes = 1 dollar so 100 cents = 10 times 10 cents = 1 dollar

Likewise three cuts can take a pie into eight exactly equal parts, or as a same number expression,

2 parts times 2 parts times 2 parts = 8 parts

Instead of writing out the same number each time it makes sense to write how many times it appears. The name word exponent - means number of repeats of a certain number in the multiplication; it shortens the last two expressions to:

102 = 100 and 23 = 8

Question 1.What value is 7! ?
Question 2.What do we call ! ?
Question 3.Which words match up with the symbols 103 and 106 (Hint: some possibilities are hundred, thousand, million, billion.)?

Example 1. A comparison of 3(2)1 and 4(3)(2)1 yields a ratio of 4.
Example 2. What is the size of 17! = (17)(16)(15)(14)(13)(12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)?

Factorial grows very rapidly. For n small here are the computations and the first few values, showing its growth:
 n n(n-1)(n-2)...(2)1 = n! 1 1 = 1 2 2(1) = 2 3 3(2)(1) = 6 4 4(3)(2)(1) = 24 5 5(4)(3)(2)(1) = 120 6 6(5)(4)(3)(2)(1) = 720

There are three ways to represent size. Scientific work uses factorial and expontents because they make concise descriptions of very large quantities. A third way uses language: special words such as undecennary. Scientists also use the word notation to describe such short writing techniques. The next two items show how to get an idea about size from these nonword, numerical ways to jot things down.

If we replace a number in a product by a smaller one then we know that we'll get a smaller result.
"(A) > (B)" means that (A) is bigger than (B). With 17 > 10, 16 > 10, 15 > 10, 14 > 10, 13 > 10, 12 > 10 and 11 > 10,

17! = 17(16)(15)(14)(13)(12)(11)(10!) > 10(10)(10)(10)(10)(10)(10)(10!) . (If the equality is unclear please see Detail.)

The last line in the table leads to:

17! > 10(10)(10)(10)(10)(10)(10)(10)(9)(8)(7)(6!) = 107(72)(7)(720)

9(8)(7) = 72(7) = 504 > 500 and 720 > 500 mean that

17! > 25(1011)

There are more than 100,000 million, in fact more than 2.5 million million in 17!.

To find the exact number in 17! do the multiplication. For instance:

17! = 17(16)(15)(14)(13)(12)(11!) = (8,910,720)(11!) = (98,017,920)(10!) = (980,179,200)(9!) = (8,821,612,800)(8!) = (572,902,400)(7!) = (494,010,316,800)(6!) = (2,964,061,091)(5!) = 2.964061901x1012(5!) = 1.48203095x1013(4!) = 5.928123802x1013(3!) = 1.7784371406x1014(2!) = 3.5568742812x1014 = 355,687,428,120,000.

What does the number 8 mean in the expression 108?

Just the number of tens (10's) multiplied together. A Wide Range of Sizes.

### Notation

In an address recognizing his contributions to computing, K. Iverson, the originator of the APL programming language, explicitly pointed to "Notation as a Tool for Thought" [8]. M. Gardner [9], unlike Iverson, not a mathematician but a writer, suggested a reasonable expansion of the "!" symbol. He suggests that for an even number m = 2n we should write m!! to indicate the product of all the evens until 2, and then 1:

m!! = (2n)!! = (2n)(2n-2)(2n-4)...(2)1

Here is a brief table to show how rapidly !! increases compared with !:

 Value n n! n!! 2 2 2 4 24 8 6 720 48 8 40320 384 10 3628800 3840 12 479001600 46080 14 87178291200 645120 16 20922789888000 10321920

### History

Working with large numbers has been part of civilization for thousands of years. Ancient Babylonia, knowledge in the general Mesopotamian geographical area, and the Sumerian civilization from before 3500 BC dealt with such things as the 20th power of 20: 104,857,600,000,000,000,000,000 (or in words "104 quintillions, 857 quadrillions and 600 trillions"). Excerpts above are from a web site by Ernest Moyer - see [11] - as is the image of the cuneiform representing that number::

The Sumerians used 60 as a base or unit of numbering. This the reason that we divide a minute into 60 seconds and an hour into 60 minutes. They were aware of the precession of the equinoxes, the phenomenon that leads to an annual lag of ' ... 50 seconds (something that is) ... in 25,920 years ... 360 degrees, one complete cycle of the zodiac, or ... one "Great" or Platonic Year".' Other 'number relationships were known in ancient times ... 25,920 divided by 60 ... yields ..."432." ... "540 times 800 is 432,000, which is the number given by Berossos (learned Babylonian priest) for the sum of years of the antediluvian kings. ... (A) number in Europe ... India .. and in Mesopotamia ... (used as a) measure of a cosmic eon."' (Both quotations from [10, pp. 116-118 (1962)].) The Sumerian year without the annual five festival days consisted of 72 five-day weeks: the product of 72 and 360 is 25,920. The fourth power of 60 or 60x60x60x60 is 12,960,000; twice 12,960 is 25,920.

### Theory

Factorial is defined only on integers. The Gamma Function has the same values at integer arguments but also exists numerically in between them. Approximating values is another option; see Stirling's Formula and [1, p. 257]. There is a wealth of information on Real and Complex arguments, along with identities, approximations, and references at [3]. The derivation of Stirling's Formula and the relationship between the Gamma Function and factorial is through infinite series, calculus, and exponential and logarthmic functions. See other factorial definitions. Much more is known about factorial.

Factorial occurs in Combinatoric areas - problems where counting must take place. For instance, the fifty-two card game Hearts can be played with three or more. When there are three, each has a seventeen-card hand. How many ways can one's hand be arranged? The answer is 17! - we found that value above, so there are 355,687,428,120,000 arrangements. The first move in a game of hearts is to select three cards to pass: the number of ways to do that comes from partitioning the seventeen into two groups, one of three, the other fourteen cards. The arrangements of the three, and those of the fourteen are respectively, 3! and 14! many. If we divide 17! by these two expressions we get exactly the number of ways we can choose three from the seventeen. It is fairly easy to see that this number is
(17)(16)(15)/(3)(2)(1) = (17)(8)(5) = (17)(40) = 680. For more on counting in both solid objects and computer codes see [4a].

### Applications

1. Large numbers provide keys to encode transactions. The analysis of that topic is part of a field of computer communications. Transmitting images from space to the earth's surface so they are not corrupted by noise, involves using ideas from this area. Secure transfer of funds depends on being unable to factor a large number rapidly. A systematic development of the underlying topics appears in [4b] which gives another number in the range of the quantity 63.2 * 1063, namely 52! A Stirling's formula approximation to that quantity is given as 8.053 * 1067 [4b, p. 47].

2. Two large numbers satisfy a simply-stated property in a situation where so far we don't know of any others that do so:

4700063497 . The much larger quantity below was recently found. Begin below (discusses newest large number satisfying that property). Go to the preceding link second (electronic mail about current research on this issue).

63130707451134435989380140059866138830623361447484274774099906755 ... or in Words (please click).

The size of 4700063497 can be stated two ways. The ordinary approach uses commas. Every three-digits group is then seen visually. Such words as thousand, million help us say things. This number, namely 4,700,063,497, is more than four thousand seven hundred million. Put another way, it is more than four billion. The other method applies exponents. It says that the quantity is 4.700063497*109 in scientific or exponential notation.

The following quotation (from Frank Pilhofer's web page Googolplex which cites the American Heritage Dictionary) expresses another way to describe very large numbers.

"Edward Kasner once asked his nine-year-old nephew (Milton Sirotta) to invent a name for a very large number, ten to the power of one hundred; and the boy called it a googol. He thought this was a number to overflow people's minds, being bigger than anything that can ever be put into words. Another mathematician then shot back with Googolplex, and defined it to be 10 to the power of Googol" (quotes from Googolplex).

3. From the discussion above we know that this is a true statement:

4,700,063,497 < 17!

4. Here is a problem in factorials. Find the value of m so that this is a true statement:

m! < 4,700,063,497 < (m+1)!

The answer can be found by Approximating Factorials or Exact Calculation.

5. This is a difficult problem in factorials. Find the value of m so that this is a true statement:

m! < 63,130,707,451,134,435,989,380,140,059,866,138,830,623,361,447,484,274,774,099,906,755
< 63.2(1063) < (m+1)! Answer

6. One approach to the preceding is to use the method for estimating the size of 17! by replacing its factors by 10.

7. Use the two tables 6.5, 6.6 [1, p. 276] to find n! for these values of n: 100, 500, 1000.

8. Instead of bounding the two values in 3. and 4. by factorials one can get closer using the Gamma function. Seek values r so each is between Gamma(r) and Gamma(r + 10^-6) [the Gamma function at an argument a millionth above the lower one, r]. Calculated.

9. Use the quantity googol here: i.e., a 1 followed by 100 zeros or 10100. Express or bound the numbers:

63,130,707,451,134,435,989,380,140,059,866,138,830,623,361,447,484,274,774,099,906,755!

and

4,700,063,497!

in terms of googol notation.

10. Ackermann's function [2, Dictionary of Algorithms and Data Structures]

(algorithm)

Definition: A function of two parameters whose value grows very fast.

Formal Definition:

* A(1, j)=2j
* A(i, 1)=A(i-1, 2)
* A(i, j)=A(i-1, A(i, j-1))

Note: In 1928, Wilhelm Ackermann (1896 - 1962) observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive. A(x,y,z) was simplified to a function of 2 variables by Rosza Peter whose initial condition was simplified by Raphael Robinson. Ackermann was a student of David Hilbert.

11. Mathematica gives these values for the factorial of the numbers 1, 2, ..., 20:

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000

Choice of 17 was arbitrary, but it is the natural result from dividing 52 in three. This happens when three people play the card game hearts. For the hearts rules see Board and Card Games.

The material continues with two urls that begin with words about size: the first provides material about big and small numbers. The second relates to aspects of the human body's size. Other continuations, emphasize scientific notation. There are exercises and explanations Math & Physics, an interactive tool for number conversion, Practice Converting, and an Online Encyclopedia.

### References

[1] Abramowitz, M. and Stegan, I., eds., Handbook of Mathematical Functions - With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, 55, June 1964, Tenth Printing, December 1972, with corrections.

[2] Black, P., Dictionary of Algorithms and Data Structures, National Institute of Standards and Technology, 1998.

[3] Weisstein, E., World of Mathematics, CRC Press LLC, 1999; 1999-2002, Wolfram Research, Inc.

[4] Golomb, S., Polyominoes [with more than 190 diagrams by Warren Lushbaugh, NY: Charles Scribner's Sons, 1965], [4a] "Chapter V: Some Theorems About Counting," [4b] Second Edition, Princeton NJ: Princeton Univ. Press, 1994, 1965.

[5] Knuth, D., Fundamental Algorithms, Second Edition, Reading MA: Addison-Wesley, 1973, 1968, pp. 44-73.

[6] Pilhofer, Frank, email fp@fpx.de, http://www.fpx.de/fp/Fun/Googolplex/, March 20, 2002.

[7] Rules, http://www.everyrule.com .

[8] Iverson, Kenneth E., "Notation as a Tool of Thought," Communications of the Association for Computer Machinery, 23(8): 444-465, 1980.

[9] Gardner, Martin, The Numerology of Dr. Matrix.

[10] Campbell, Joseph, The Masks of God: Oriental Mythology.

[11] Moyer, Ernest, PO Box 1206, Hanover, PA 17331, USA; web site http://www.egyptorigins.org/; email address given there.

[12] Davis, Philip J. and Hersh, Reuben, The Mathematical Experience, Boston, MA: Birkhauser, 1981.
 1 April 2009 Version http://www.cs.ucla.edu/~klinger/size.html ©2009 Allen Klinger