Large numbers

Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 10⁹, for example, means one billion, a 1 followed by nine zeros: 1 000 000 000, and 1.0 × 10⁻⁹ means one billionth, or 0.000 000 001. Writing 10⁹ instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is.

Examples of large numbers describing everyday real-world objects include:

• The number of cells in the human body (estimated at 3.72 × 10¹³)[3]
• The number of bits on a computer hard disk (as of 2020, typically about 10¹³, 1–2 TB)
• The number of neuronal connections in the human brain (estimated at 10¹⁴)
• The Avogadro constant is the number of “elementary entities” (usually atoms or molecules) in one mole; the number of atoms in 12 grams of carbon-12 – approximately 6.022×10²³.
• The total number of DNA base pairs within the entire biomass on Earth, as a possible approximation of global biodiversity, is estimated at (5.3±3.6)×10³⁷[4][5]
• The mass of Earth consists of about 4x10⁵¹ nucleons
• The estimated number of atoms in the observable universe (10⁸⁰)
• The lower bound on the game-tree complexity of chess, also known as the “Shannon number” (estimated at around 10¹²⁰)[6]

Other large numbers, as regards length and time, are found in astronomy and cosmology. For example, the current Big Bang model suggests that the universe is 13.8 billion years (4.355 × 10¹⁷ seconds) old, and that the observable universe is 93 billion light years across (8.8 × 10²⁶ metres), and contains about 5 × 10²² stars, organized into around 125 billion (1.25 × 10¹¹) galaxies, according to Hubble Space Telescope observations. There are about 10⁸⁰ atoms in the observable universe, by rough estimation.[7]

According to Don Page, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is

${\displaystyle 10^{10^{10^{10^{10^{1.1}}}}}{\mbox{ years}}}$

which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10⁻⁶ Planck masses.[8][9] This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to properties of statistical mechanics; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.

Combinatorial processes rapidly generate even larger numbers. The factorial function, which defines the number of permutations on a set of fixed objects, grows very rapidly with the number of objects. Stirling's formula gives a precise asymptotic expression for this rate of growth.

Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their logarithms.

Gödel numbers, and similar numbers used to represent bit-strings in algorithmic information theory, are very large, even for mathematical statements of reasonable length. However, some pathological numbers are even larger than the Gödel numbers of typical mathematical propositions.

Logician Harvey Friedman has done work related to very large numbers, such as with Kruskal's tree theorem and the Robertson–Seymour theorem.

• googol = ${\displaystyle 10^{100}}$
• centillion = ${\displaystyle 10^{303}}$ or ${\displaystyle 10^{600}}$, depending on number naming system
• millinillion = ${\displaystyle 10^{3003}}$ or ${\displaystyle 10^{6000}}$, depending on number naming system
• millinillinillion = ${\displaystyle 10^{3000003}}$ or ${\displaystyle 10^{6000000}}$, depending on number naming system
• The largest known Smith number = (10¹⁰³¹−1) × (10⁴⁵⁹⁴ + 3×10²²⁹⁷ + 1)¹⁴⁷⁶ ×10^3913210
• The largest known Mersenne prime = ${\displaystyle 2^{82,589,933}-1}$ (as of December 21, 2018)
• googolplex = ${\displaystyle 10^{\text{googol}}=10^{10^{100}}}$
• Skewes' numbers: the first is approximately ${\displaystyle 10^{10^{10^{34}}}}$, the second ${\displaystyle 10^{10^{10^{964}}}}$
• Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using Knuth's up-arrow notation
• Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at MIT on 26 January 2007

A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.

To compare numbers in scientific notation, say 5×10⁴ and 2×10⁵, compare the exponents first, in this case 5 > 4, so 2×10⁵ > 5×10⁴. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×10⁴ > 2×10⁴ because 5 > 2.

Tetration with base 10 gives the sequence ${\displaystyle 10\uparrow \uparrow n=10\to n\to 2=(10\uparrow )^{n}1}$, the power towers of numbers 10, where ${\displaystyle (10\uparrow )^{n}}$ denotes a functional power of the function ${\displaystyle f(n)=10^{n}}$ (the function also expressed by the suffix "-plex" as in googolplex, see the Googol family).

These are very round numbers, each representing an order of magnitude in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.

More precisely, numbers in between can be expressed in the form ${\displaystyle (10\uparrow )^{n}a}$, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g. ${\displaystyle 10^{10^{10^{10^{10^{4.829}}}}}=(10\uparrow )^{5}4.829}$, a number between ${\displaystyle 10\uparrow \uparrow 5}$ and ${\displaystyle 10\uparrow \uparrow 6}$ (note that ${\displaystyle 10\uparrow \uparrow n<(10\uparrow )^{n}a<10\uparrow \uparrow (n+1)}$ if ${\displaystyle 1<a<10}$). (See also extension of tetration to real heights.)

Thus googolplex is ${\displaystyle 10^{10^{100}}=(10\uparrow )^{2}100=(10\uparrow )^{3}2}$

Another example:

${\displaystyle 2\uparrow \uparrow \uparrow 4={\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\qquad \quad \ \ \ 65,536{\mbox{ copies of }}2\end{matrix}}\approx (10\uparrow )^{65,531}(6\times 10^{19,728})\approx (10\uparrow )^{65,533}4.3}$ (between ${\displaystyle 10\uparrow \uparrow 65,533}$ and ${\displaystyle 10\uparrow \uparrow 65,534}$)

Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (n) one has to take the ${\displaystyle log_{10}}$ to get a number between 1 and 10. Thus, the number is between ${\displaystyle 10\uparrow \uparrow n}$ and ${\displaystyle 10\uparrow \uparrow (n+1)}$. As explained, a more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 10¹⁰, or the next, between 0 and 1.

Note that

${\displaystyle 10^{(10\uparrow )^{n}x}=(10\uparrow )^{n}10^{x}}$

I.e., if a number x is too large for a representation ${\displaystyle (10\uparrow )^{n}x}$ we can make the power tower one higher, replacing x by log₁₀x, or find x from the lower-tower representation of the log₁₀ of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from 10).

If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so we can use the double-arrow notation, e.g. ${\displaystyle 10\uparrow \uparrow (7.21\times 10^{8})}$. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.

Examples:

${\displaystyle 10\uparrow \uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}}$ (between ${\displaystyle 10\uparrow \uparrow \uparrow 2}$ and ${\displaystyle 10\uparrow \uparrow \uparrow 3}$)

${\displaystyle 10\uparrow \uparrow 10\uparrow \uparrow (10\uparrow )^{497}(9.73\times 10^{32})=(10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})}$ (between ${\displaystyle 10\uparrow \uparrow \uparrow 4}$ and ${\displaystyle 10\uparrow \uparrow \uparrow 5}$)

Similarly to the above, if the exponent of ${\displaystyle (10\uparrow )}$ is not exactly given then giving a value at the right does not make sense, and we can, instead of using the power notation of ${\displaystyle (10\uparrow )}$, add 1 to the exponent of ${\displaystyle (10\uparrow \uparrow )}$, so we get e.g. ${\displaystyle (10\uparrow \uparrow )^{3}(2.8\times 10^{12})}$.

If the exponent of ${\displaystyle (10\uparrow \uparrow )}$ is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and we can, instead of using the power notation of ${\displaystyle (10\uparrow \uparrow )}$, use the triple arrow operator, e.g. ${\displaystyle 10\uparrow \uparrow \uparrow (7.3\times 10^{6})}$.

If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e.g. ${\displaystyle 10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})}$ (between ${\displaystyle 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 4}$ and ${\displaystyle 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 5}$). This can be done recursively, so we can have a power of the triple arrow operator.

We can proceed with operators with higher numbers of arrows, written ${\displaystyle \uparrow ^{n}}$.

Compare this notation with the hyper operator and the Conway chained arrow notation:

${\displaystyle a\uparrow ^{n}b}$ = ( a → b → n ) = hyper(a, n + 2, b)

An advantage of the first is that when considered as function of b, there is a natural notation for powers of this function (just like when writing out the n arrows): ${\displaystyle (a\uparrow ^{n})^{k}b}$. For example:

${\displaystyle (10\uparrow ^{2})^{3}b}$ = ( 10 → ( 10 → ( 10 → b → 2 ) → 2 ) → 2 )

and only in special cases the long nested chain notation is reduced; for b = 1 we get:

${\displaystyle 10\uparrow ^{3}3=(10\uparrow ^{2})^{3}1}$ = ( 10 → 3 → 3 )

Since the b can also be very large, in general we write a number with a sequence of powers ${\displaystyle (10\uparrow ^{n})^{k_{n}}}$ with decreasing values of n (with exactly given integer exponents ${\displaystyle {k_{n}}}$) with at the end a number in ordinary scientific notation. Whenever a ${\displaystyle {k_{n}}}$ is too large to be given exactly, the value of ${\displaystyle {k_{n+1}}}$ is increased by 1 and everything to the right of ${\displaystyle ({n+1})^{k_{n+1}}}$ is rewritten.

For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example, ${\displaystyle 10\uparrow (10\uparrow \uparrow )^{5}a=(10\uparrow \uparrow )^{6}a}$, and ${\displaystyle 10\uparrow (10\uparrow \uparrow \uparrow 3)=10\uparrow \uparrow (10\uparrow \uparrow 10+1)\approx 10\uparrow \uparrow \uparrow 3}$. Thus we have the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10^x are "almost equal" (for arithmetic of large numbers see also below).

If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it acts. We can simply use a standard value at the right, say 10, and the expression reduces to ${\displaystyle 10\uparrow ^{n}10=(10\to 10\to n)}$ with an approximate n. For such numbers the advantage of using the upward arrow notation no longer applies, and we can also use the chain notation.

The above can be applied recursively for this n, so we get the notation ${\displaystyle \uparrow ^{n}}$ in the superscript of the first arrow, etc., or we have a nested chain notation, e.g.:

(10 → 10 → (10 → 10 → ${\displaystyle 3\times 10^{5}}$) ) = ${\displaystyle 10\uparrow ^{10\uparrow ^{3\times 10^{5}}10}10}$

If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function ${\displaystyle f(n)=10\uparrow ^{n}10}$ = (10 → 10 → n), these levels become functional powers of f, allowing us to write a number in the form ${\displaystyle f^{m}(n)}$ where m is given exactly and n is an integer which may or may not be given exactly (for example: ${\displaystyle f^{2}(3\times 10^{5})}$). If n is large we can use any of the above for expressing it. The "roundest" of these numbers are those of the form f^m(1) = (10→10→m→2). For example, ${\displaystyle (10\to 10\to 3\to 2)=10\uparrow ^{10\uparrow ^{10^{10}}10}10}$

Compare the definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus ${\displaystyle G<3\rightarrow 3\rightarrow 65\rightarrow 2<(10\to 10\to 65\to 2)=f^{65}(1)}$, but also ${\displaystyle G<f^{64}(4)<f^{65}(1)}$.

If m in ${\displaystyle f^{m}(n)}$ is too large to give exactly we can use a fixed n, e.g. n = 1, and apply the above recursively to m, i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of f this gives multiple levels of f. Introducing a function ${\displaystyle g(n)=f^{n}(1)}$ these levels become functional powers of g, allowing us to write a number in the form ${\displaystyle g^{m}(n)}$ where m is given exactly and n is an integer which may or may not be given exactly. We have (10→10→m→3) = g^m(1). If n is large we can use any of the above for expressing it. Similarly we can introduce a function h, etc. If we need many such functions we can better number them instead of using a new letter every time, e.g. as a subscript, so we get numbers of the form ${\displaystyle f_{k}^{m}(n)}$ where k and m are given exactly and n is an integer which may or may not be given exactly. Using k=1 for the f above, k=2 for g, etc., we have (10→10→n→k) = ${\displaystyle f_{k}(n)=f_{k-1}^{n}(1)}$. If n is large we can use any of the above for expressing it. Thus we get a nesting of forms ${\displaystyle {f_{k}}^{m_{k}}}$ where going inward the k decreases, and with as inner argument a sequence of powers ${\displaystyle (10\uparrow ^{n})^{p_{n}}}$ with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.

When k is too large to be given exactly, the number concerned can be expressed as ${\displaystyle {f_{n}}(10)}$=(10→10→10→n) with an approximate n. Note that the process of going from the sequence ${\displaystyle 10^{n}}$=(10→n) to the sequence ${\displaystyle 10\uparrow ^{n}10}$=(10→10→n) is very similar to going from the latter to the sequence ${\displaystyle {f_{n}}(10)}$=(10→10→10→n): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions ${\displaystyle {f_{qk}}^{m_{qk}}}$, nested in lexicographical order with q the most significant number, but with decreasing order for q and for k; as inner argument we have a sequence of powers ${\displaystyle (10\uparrow ^{n})^{p_{n}}}$ with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.

For a number too large to write down in the Conway chained arrow notation we can describe how large it is by the length of that chain, for example only using elements 10 in the chain; in other words, we specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number we can apply the same techniques again for that.

The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and the arithmetic.

${\displaystyle 100^{12}=10^{24}}$, with base 10 the exponent is doubled.

${\displaystyle 100^{100^{12}}=10^{2*10^{24}}}$, ditto.

${\displaystyle 100^{100^{100^{12}}}\approx 10^{10^{2*10^{24}+0.30103}}}$, the highest exponent is very little more than doubled (increased by log₁₀2).

• ${\displaystyle 100\uparrow \uparrow 2=10^{200}}$
• ${\displaystyle 100\uparrow \uparrow 3=10^{2\times 10^{200}}}$
• ${\displaystyle 100\uparrow \uparrow 4=(10\uparrow )^{2}(2\times 10^{200}+0.3)=(10\uparrow )^{2}(2\times 10^{200})=(10\uparrow )^{3}200.3=(10\uparrow )^{4}2.3}$
• ${\displaystyle 100\uparrow \uparrow n=(10\uparrow )^{n-2}(2\times 10^{200})=(10\uparrow )^{n-1}200.3=(10\uparrow )^{n}2.3<10\uparrow \uparrow (n+1)}$ (thus if n is large it seems fair to say that ${\displaystyle 100\uparrow \uparrow n}$ is "approximately equal to" ${\displaystyle 10\uparrow \uparrow n}$)
• ${\displaystyle 100\uparrow \uparrow \uparrow 2=(10\uparrow )^{98}(2\times 10^{200})=(10\uparrow )^{100}2.3}$
• ${\displaystyle 100\uparrow \uparrow \uparrow 3=10\uparrow \uparrow (10\uparrow )^{98}(2\times 10^{200})=10\uparrow \uparrow (10\uparrow )^{100}2.3}$
• ${\displaystyle 100\uparrow \uparrow \uparrow n=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{98}(2\times 10^{200})=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{100}2.3<10\uparrow \uparrow \uparrow (n+1)}$ (compare ${\displaystyle 10\uparrow \uparrow \uparrow n=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{10}1<10\uparrow \uparrow \uparrow (n+1)}$; thus if n is large it seems fair to say that ${\displaystyle 100\uparrow \uparrow \uparrow n}$ is "approximately equal to" ${\displaystyle 10\uparrow \uparrow \uparrow n}$)
• ${\displaystyle 100\uparrow \uparrow \uparrow \uparrow 2=(10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3}$ (compare ${\displaystyle 10\uparrow \uparrow \uparrow \uparrow 2=(10\uparrow \uparrow )^{8}(10\uparrow )^{10}1}$)
• ${\displaystyle 100\uparrow \uparrow \uparrow \uparrow 3=10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3}$ (compare ${\displaystyle 10\uparrow \uparrow \uparrow \uparrow 3=10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{8}(10\uparrow )^{10}1}$)
• ${\displaystyle 100\uparrow \uparrow \uparrow \uparrow n=(10\uparrow \uparrow \uparrow )^{n-2}(10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3}$ (compare ${\displaystyle 10\uparrow \uparrow \uparrow \uparrow n=(10\uparrow \uparrow \uparrow )^{n-2}(10\uparrow \uparrow )^{8}(10\uparrow )^{10}1}$; if n is large this is "approximately" equal)

For a number ${\displaystyle 10^{n}}$, one unit change in n changes the result by a factor 10. In a number like ${\displaystyle 10^{\,\!6.2\times 10^{3}}}$, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor ${\displaystyle 10^{50}}$ too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).

Given a strictly increasing integer sequence/function ${\displaystyle f_{0}(n)}$ (n≥1) we can produce a faster-growing sequence ${\displaystyle f_{1}(n)=f_{0}^{n}(n)}$ (where the superscript n denotes the n^th functional power). This can be repeated any number of times by letting ${\displaystyle f_{k}(n)=f_{k-1}^{n}(n)}$, each sequence growing much faster than the one before it. Then we could define ${\displaystyle f_{\omega }(n)=f_{n}(n)}$, which grows much faster than any ${\displaystyle f_{k}}$ for finite k (here ω is the first infinite ordinal number, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.

For example, starting with f₀(n) = n + 1:

• f₁(n) = f₀ⁿ(n) = n + n = 2n
• f₂(n) = f₁ⁿ(n) = 2ⁿn > (2 ↑) n for n ≥ 2 (using Knuth up-arrow notation)
• f₃(n) = f₂ⁿ(n) > (2 ↑)ⁿ n ≥ 2 ↑² n for n ≥ 2
• f_k+1(n) > 2 ↑^k n for n ≥ 2, k < ω
• f_ω(n) = f_n(n) > 2 ↑^{n – 1} n > 2 ↑^{n − 2} (n + 3) − 3 = A(n, n) for n ≥ 2, where A is the Ackermann function (of which f_ω is a unary version)
• f_ω+1(64) > f_ω⁶⁴(6) > Graham's number (= g₆₄ in the sequence defined by g₀ = 4, g_k+1 = 3 ↑^g_k 3)
  • This follows by noting f_ω(n) > 2 ↑^{n – 1} n > 3 ↑^{n – 2} 3 + 2, and hence f_ω(g_k + 2) > g_k+1 + 2
• f_ω(n) > 2 ↑^{n – 1} n = (2 → n → n-1) = (2 → n → n-1 → 1) (using Conway chained arrow notation)
• f_ω+1(n) = f_ωⁿ(n) > (2 → n → n-1 → 2) (because if g_k(n) = X → n → k then X → n → k+1 = g_kⁿ(1))
• f_ω+k(n) > (2 → n → n-1 → k+1) > (n → n → k)
• f_ω2(n) = f_ω+n(n) > (n → n → n) = (n → n → n→ 1)
• f_ω2+k(n) > (n → n → n → k)
• f_ω3(n) > (n → n → n → n)
• f_ωk(n) > (n → n → ... → n → n) (Chain of k+1 n's)
• f_ω²(n) = f_ωn(n) > (n → n → ... → n → n) (Chain of n+1 n's)

The busy beaver function Σ is an example of a function which grows faster than any computable function. Its value for even relatively small input is huge. The values of Σ(n) for n = 1, 2, 3, 4 are 1, 4, 6, 13 (sequence A028444 in the OEIS). Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 3.5×10¹⁸²⁶⁷.

Although all the numbers discussed above are very large, they are all still decidedly finite. Certain fields of mathematics define infinite and transfinite numbers. For example, aleph-null is the cardinality of the infinite set of natural numbers, and aleph-one is the next greatest cardinal number. ${\displaystyle {\mathfrak {c}}}$ is the cardinality of the reals. The proposition that ${\displaystyle {\mathfrak {c}}=\aleph _{1}}$ is known as the continuum hypothesis.