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Large numbers
Generally, large numbers are those greater than numbers in practical, everyday use. The line of demarcation varies by application. In the OEIS, integers with more than 200 base 10 digits are too large to show in the main sequence entry, but up to a thousand digits can be shown for a single term in the b-file.
For some scientific calculators, the largest number that can be shown with integer precision is
| (10 10 − 1) |
, or
| 9999999999 |
, as well as its negation
| − (10 10 − 1) |
(since there is usually a dedicated line segment on the display just for the purpose of providing a minus sign for 10-digit numbers). Those calculators generally have to resort to scientific notation for
| 13! |
or
| 14! |
and larger. The largest number they can show in scientific notation is usually
| 9999999999 × 10 99 |
. Computer algebra systems like Mathematica can show much greater numbers in full integer precision, like
| 100! |
and even greater in scientific notation. Symbolically, they can handle numbers that are much larger still, being able to give responses to queries like TrueQ[2^(2^127) > 3^(3^81)].
The standard system of Roman numerals can only cope with numbers up to 3999, but the Romans needed to count larger numbers for business and military applications. They invented various extensions to their system which allowed the representation of numbers in the millions, but none of those extensions remained standard. Today, a nation's debt tends to run in the trillions. Numbers greater than these are of no practical value in finance.
Physicists estimate the number of particles in the Universe is approximately
| 10 87 |
,[1] which is much more than "the number of raindrops that fell on Earth over the planet's lifetime!" Thus a googol,
| g = 10 100 |
, is about
| 10 13 |
times the estimated number of particles in the Universe. And then there's the googolplex
| 10 g |
.[2] RSA-129 is more than a googol but not as much as a googolplex. Also in this class is the smallest odd number not ruled out as a perfect number, which is more than
| 10 500 |
.
It is a well-known fact, a theorem proved in antiquity, that there are infinitely many prime numbers. Nevertheless, people delight in discovering larger and larger primes, as if to reassure themselves that they in fact don't run out. The Mersenne primes are a common example, being much easier to check for primality than general numbers.
The original Skewes number, Stanley Skewes' upper bound
| X |
, assuming the truth of RH, for the smallest number
| x |
for which the logarithmic integral
| li (x) |
is an underestimate of the prime counting function
| π (x) |
, is
| e e e 79 ≈ 10 10 10 34 |
. His contemporaries thought it the largest number in mathematics than ever served a purpose (rather than large numbers for their own sake). Numbers encountered in combinatorics and Ramsey theory often dwarf numbers encountered in other fields. The number
| 3 3 3 3 3 |
is quite large enough as it is. Graham's number is a much bigger multiple of that number.
The reciprocal of a large enough number is a small enough number, a concept used for limits and continuity in epsilon-delta analysis.
See also
[edit]Notes
[edit]References
[edit]- David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. Ed. London: Penguin Books (1997).
External links
[edit]- Bryan Clair, The Biggest Numbers in the Universe, Strange Horizons, 2001.
- Robert P. Munafo, Large Numbers, © 1996-2011.
- Robert P. Munafo, Hypercalc — The Calculator That Doesn't Overflow.
- Large numbers—Wikipedia.org.
- Googology Wiki—The large number encyclopedia.
- Names for Large Numbers — © 2001 by Russ Rowlett and the University of North Carolina at Chapel Hill.
