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This page contains numbers appearing in computer arithmetic.

List of numbers appearing in computer arithmetic

  • 2,040 (two thousand forty) is the smallest number n, such that 2n cannot be stored on the TI-89 exact mode.
    • It is also the number of pips in a double-15 domino set.
  • 32,767 is a positive integer equal to \(2^{15} - 1 = 2^{2^4 - 1} - 1\). It is notable in computer science for being the maximum value of a 16-bit signed integer, which spans the range [-32768, 32767]. In English, its full name is "thirty-two thousand seven hundred sixty-seven." Its prime factorization is 7 × 31 × 151.
  • There are 2 × 192 × 9 × 106 + 2 × 106 - 1 = 3,457,999,999 different finite numbers, which can be represented exactly in the 32-bit decimal floating point format.
    • Its prime factorization is 53 × 73 × 107 × 8,353.
  • There are 232 - 224 - 1 = 4,278,190,079 different finite numbers, which can be represented exactly in the 32-bit floating point format.
    • This number is a prime number.
  • 9,007,199,254,740,991 is a positive integer equal to \(2^{53} - 1\). It is notable in computer science for being the largest odd number which can be represented exactly in the double floating-point format (which has a 53-bit significand).
    • Its prime factorization is 6,361 × 69,431 × 20,394,401.
  • 9,223,372,036,854,775,807 is a positive integer equal to \(2^{63} - 1 = 2^{2^6 - 1} - 1\). It is notable in computer science for being the maximum value of a 64-bit signed integer, which has the range [-9223372036854775808, 9223372036854775807].[1]
    • Its full name in English in the short scale is "nine quintillion two hundred twenty-three quadrillion three hundred seventy-two trillion thirty-six billion eight hundred fifty-four million seven hundred seventy-five thousand eight hundred seven".
    • Its prime factorization is 72 × 73 × 127 × 337 × 92,737 × 649,657.
  • There are 2 × 768 × 9 × 1015 + 2 × 1015 - 1 = 13,825,999,999,999,999,999 different finite numbers, which can be represented exactly in the 64-bit decimal floating point format.
    • Its prime factorization is 11 × 1,256,909,090,909,090,909.
  • There are 264 - 253 - 1 = 18,437,736,874,454,810,623 different finite numbers, which can be represented exactly in the 64-bit floating point format.
    • Its prime factorization is 230,999 × 79,817,388,276,377.
  • 1,124,000,727,777,607,680,000 is a positive integer equal to \(22!\). It is notable in computer science for being the largest factorial number which can be represented exactly in the double floating-point format (which has a 53-bit significand).
  • 1022 is a positive integer equal to ten sextillion. It is notable in computer science for being the largest power of ten which can be represented exactly in the double floating-point format (which has a 53-bit significand). This number is called goonrol.
  • Many handheld calculators have an overflow error after reaching \(10^{100}\), also called a googol.

Approximations in other notations

For 32,767:

Notation Lower bound Upper bound
Scientific notation \(3.2767\times10^4\)
Arrow notation \(181\uparrow2\) \(8\uparrow5\)
Steinhaus-Moser Notation 5[3] 6[3]
Copy notation 2[5] 3[5]
Chained arrow notation \(181\rightarrow2\) \(8\rightarrow5\)
Taro's multivariable Ackermann function A(3,12) A(3,13)
Pound-Star Notation #*(127)*2 #*(128)*2
PlantStar's Debut Notation [2] [3]
BEAF {181,2} {8,5}
Hyper-E notation 32E3 E[8]5
Bashicu matrix system (0)[181] (0)[182]
Hyperfactorial array notation 7! 8!
Bird's array notation {181,2} {8,5}
Strong array notation s(181,2) s(8,5)
Fast-growing hierarchy \(f_2(11)\) \(f_2(12)\)
Hardy hierarchy \(H_{\omega^2}(11)\) \(H_{\omega^2}(12)\)
Slow-growing hierarchy \(g_{\omega^2}(181)\) \(g_{\omega^5}(8)\)

For 1,124,000,727,777,607,680,000:

Notation Lower bound Upper bound
Scientific notation \(1.124\times10^{21}\) \(1.125\times10^{21}\)
Arrow notation \(10↑21\) \(2\uparrow70\)
Steinhaus-Moser Notation 17[3] 18[3]
Strong array notation s(10,21) s(2,70)
Copy notation 1[22] 2[[3]]
Taro's multivariable Ackermann function A(3,67) A(3,68)
Pound-Star Notation #*(2)*63 #*(2)*64
BEAF {10,21} {2,70}
Hyperfactorial array notation 22!
Fast-growing hierarchy \(f_2(63)\) \(f_2(64)\)
Hardy hierarchy \(H_{\omega^2}(63)\) \(H_{\omega^2}(64)\)
Slow-growing hierarchy \(g_{\omega^{\omega 2+1}}(10)\) \(g_{\omega^{\omega^22+3}}(4)\)

For 1022:

Notation Lower bound Upper bound
Scientific notation \(1\times10^{22}\)
Arrow notation \(10\uparrow22\)
Steinhaus-Moser Notation 17[3] 18[3]
Copy notation 9[22] 1[23]
Taro's multivariable Ackermann function A(3,70) A(3,71)
Pound-Star Notation #*(2,0,7,7)*4 #*(2,6,3)*7
BEAF {10,22}
Hyper-E notation E22
Hyperfactorial array notation 22! 23!
Fast-growing hierarchy \(f_2(67)\) \(f_2(68)\)
Hardy hierarchy \(H_{\omega^2}(67)\) \(H_{\omega^2}(68)\)
Slow-growing hierarchy \(g_{\omega^{\omega2+2}}(10)\)

See also

Large numbers in computers

Sources

  1. Free Pascal team, Int64 - 64-bit, signed integer - Reference for unit 'System', Free Pascal
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