Numbers in computer arithmetic

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This page contains numbers appearing in computer arithmetic.

List of numbers appearing in computer arithmetic

• 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.
• 2,147,483,647 is a positive integer equal to \(2^{31} - 1\). It is the maximum positive value for a 32-bit signed integer in computing.
  • It causes Year 2038 problem, an issue that arises due to the way Unix time is represented in systems that use a signed 32-bit integer to store time. Unix time, also called POSIX time or Epoch time, is a system for tracking time by counting the number of seconds that have elapsed since January 1, 1970, 00:00:00 UTC (the Unix epoch). The maximum value that a 32-bit signed integer can hold corresponds to January 19, 2038, at 03:14:07 UTC. Without action, the old 32-bit time system could cause widespread errors and system failures starting in January 2038. However, many systems have already begun transitioning to 64-bit time.
• 4,294,967,295 is a positive integer equal to \(2^{32} - 1\). It is the maximum positive value for a 32-bit unsigned integer in computing.
• 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-precision floating-point format. The number is expressed with all 1 in the 52-bit fraction.
  • 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].
  • Due to Year 2038 problem, 64-bit signed integer is the current standard of Unix time representation. It can represent 292 billion years in either direction of the 1970 epoch. In some systems such as JavaScript, "Unix time in milliseconds" is used, because it can still represent sufficiently long period of time (292 million years), but the Unix time standard is the representation in seconds.
  • 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.
• 18,446,744,073,709,551,615 is a positive integer equal to \(2^{64} - 1\). It is the maximum positive value for a 64-bit unsigned integer in computing.
• The maximum number which can be expressed in the single-precision floating-point format is, in the binary expression, 0 for the sign, 11111110 for the exponent, and 11111111111111111111111 for the fraction. It expresses 2¹²⁷ × (2 − 2⁻²³) = 2¹²⁸ - 2¹⁰⁴ = 340282346638528859811704183484516925440.
• The maximum number which can be expressed in the double-precision floating-point format is, in the binary expression, 0 for the sign, 11111111110 for the exponent, and 1111111111111111111111111111111111111111111111111111 for the fraction. It expresses 2¹⁰²³ × (1 + (1 − 2⁻⁵²)) = 2¹⁰²⁴ - 2⁹⁷¹. It is approximately 1.7976931348623157 × 10³⁰⁸ and the full decimal expansion is
  179769313486231570814527423731704356798070567525844996598917476803157260780028538760589558632766878171540458953514382464234321326889464182768467546703537516986049910576551282076245490090389328944075868508455133942304583236903222948165808559332123348274797826204144723168738177180919299881250404026184124858368
  .
• Sbiis Saibian says 2²⁰⁴⁰ - 1 is the largest number that can be stored on the TI-89 exact mode. It can be simulated online. Full decimal expansion is
  126238304966058622268417487065116999845484776053576109500509161826268184136202698801551568013761380717534054534851164138648904527931605160527688095259563605939964364716019515983399209962459578542172100149937763938581219604072733422507180056009672540900709554109516816573779593326332288314873251559077853068444977864803391962580800682760017849589281937637993445539366428356761821065267423102149447628375691862210717202025241630303118559188678304314076943801692528246980959705901641444238894928620825482303431806955690226308773426829503900930529395181208739591967195841536053143145775307050594328881077553168201547775

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	\(g_{\omega^2}(181)\)	\(g_{\omega^5}(8)\)

See also

Large numbers in computers

Main article: Numbers in computer arithmetic

127 · 128 · 256 · 32767 · 32768 · 65536 · 2147483647 · 4294967296 · 9007199254740991 · 9223372036854775807 · FRACTRAN catalogue numbers
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