Large Numbers - Standardized System of Writing Very Large Numbers

Standardized System of Writing Very Large Numbers

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, the power towers of numbers 10, where denotes a functional power of the function (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 accurately, numbers in between can be expressed in the form, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g., a number between and (note that if ). (See also extension of tetration to real heights.)

Thus googolplex is

Another example:

(between and )

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 to get a number between 1 and 10. Thus, the number is between and . As explained, a more accurate 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

I.e., if a number x is too large for a representation 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. . If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.

Examples:

(between and )

(between and )

Similarly to the above, if the exponent of 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, add 1 to the exponent of, so we get e.g. .

If the exponent of 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, use the triple arrow operator, e.g. .

If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e.g. (between and ). 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 .

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

= ( 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): . For example:

= ( 10 → ( 10 → ( 10 → b → 2 ) → 2 ) → 2 )

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

= ( 10 → 3 → 3 )

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

For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example, and . Thus we have the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10x 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 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 in the superscript of the first arrow, etc., or we have a nested chain notation, e.g.:

(10 → 10 → (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 = (10 → 10 → n), these levels become functional powers of f, allowing us to write a number in the form where m is given exactly and n is an integer which may or may not be given exactly (for the example: . If n is large we can use any of the above for expressing it. The "roundest" of these numbers are those of the form fm(1) = (10→10→m→2). For example,

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, but also .

If m in 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 these levels become functional powers of g, allowing us to write a number in the form where m is given exactly and n is an integer which may or may not be given exactly. We have (10→10→m→3) = gm(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 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) = . If n is large we can use any of the above for expressing it. Thus we get a nesting of forms where going inward the k decreases, and with as inner argument a sequence of powers 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 =(10→10→10→n) with an approximate n. Note that the process of going from the sequence =(10→n) to the sequence =(10→10→n) is very similar to going from the latter to the sequence =(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, 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 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.