This page contains numbers appearing in computer arithmetic.

## Contents

## List of numbers appearing in computer arithmetic

**2,040**(two thousand forty) is the smallest number*n*, such that 2^{n}cannot be stored on the TI-89 exact mode.**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 × 10
^{6}+ 2 × 10^{6}- 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 2
^{32}- 2^{24}- 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 7
^{2}× 73 × 127 × 337 × 92,737 × 649,657.

- There are 2 × 768 × 9 × 10
^{15}+ 2 × 10^{15}- 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 2
^{64}- 2^{53}- 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).- In the short scale, this number is written as 1 sextillion 124 quintillion 727 trillion 777 billion 607 million 680 thousand.
- In the long scale, this number is written as 1 trilliard 124 trillion 727 billion 777 milliard 607 million 680 thousand.

**10**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^{22}`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 10^{22}:

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

Main article:

127 · 128 · 256 · 32767 · 32768 · 65536 · 2147483647 · 4294967296 · 9007199254740991 · 9223372036854775807 · FRACTRAN catalogue numbers**Numbers in computer arithmetic****Bignum Bakeoff contestants**: pete-3.c · pete-9.c · pete-8.c · harper.c · ioannis.c · chan-2.c · chan-3.c · pete-4.c · chan.c · pete-5.c · pete-6.c · pete-7.c · marxen.c · loader.c

**Channel systems:**lossy channel system · priority channel system

**Uncomputable functions:**Busy beaver function · Maximum shifts function · Doodle function · Betti number · Xi function · ITTM busy beaver · Rayo(n) · FOOT(n)

**Concepts:**Recursion

## Sources

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

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