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Large numbers

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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
10g
.[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
eee 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

Notes

  1. Wells, p. 209.
  2. Wells, p. 203.

References

  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. Ed. London: Penguin Books (1997).

External links