Numbers in computer arithmetic

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 in a double-15 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].Free Pascal team, Int64 - 64-bit, signed integer - Reference for unit 'System', Free Pascal
**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 	\(3.2767\times10^4\) 	\(3.2767\times10^4\)
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! 	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 	\(1\times10^{22}\) 	\(1\times10^{22}\)
Arrow notation 	\(10\uparrow22\) 	\(10\uparrow22\)
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} 	{10,22}
Hyper-E notation 	E22 	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 	\(g_{\omega^{\omega2+2}}(10)\)	\(g_{\omega^{\omega2+2}}(10)\)
== See also ==
== Sources ==
Category:Numbers
Category:Computers
Category:Lists
Category:Class 1
Category:Class 2
Category:Numbers with 4 to 6 digits
Category:Numbers with 7 to 29 digits