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.
- 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].
- 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).- 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.
- 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
Main article: Numbers in computer arithmetic
127 · 128 · 256 · 32767 · 32768 · 65536 · 2147483647 · 4294967296 · 9007199254740991 · 9223372036854775807 · FRACTRAN catalogue numbersBignum 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
Concepts: Recursion