Array Notation


As I've been studying large numbers, I developed various notations to represent them. The three main ones I use are: extended operater notation, array notation, and extended array notation. The extended operator notation and the array notation are two ways of writing the same thing, the array notation is cleaner looking, but the extended operator notation can give one a feel of how large the numbers get. The extended array notation leads to unspeakably enormous numbers, so huge that the other notations cant even come close. Following are the 5 rules to solve the array notation  first use rule 1 if it applies, if not then use rule 2, etc. If rules 1 through 4 doesn't apply then rule 5 will.


Rule 1: Condition  only 1 or 2 entries  {a} = a, {a,b} = a+b. (in other words take the sum of the entries).
Rule 2: Condition  last entry is 1  {a,b,c,...,k,1} = {a,b,c,...,k} (in other words remove trailing 1's).
Rule 3: Condition  2nd entry is 1  {a,1,c,d,..,k} = a.
Rule 4: Condition  3rd entry is 1  {a,b,1,..,1,d,e,..,k} = {a,a,a,..,{a,b1,1,..,1,d,e,..,k},d1,e,..,k}  the ".." between the 1's represent 1's  there can be any number of ones, from 1 "1" (3rd entry alone) to a string of 1s  the last 1 of this string is what becomes {a,b1,1,..,1,d,e,..,k} all entries prior becomes "a". For an array like this {3,2,1,1,1,5,1,2} the last 1 in the string is the one before the 5 (not the one after  since it is a different string of 1s).
Rule 5: Condition  Rules 14 doesn't apply  {a,b,c,d,...,k} = {a,{a,b1,c,d,...,k},c1,d,..,k}.
I usually use angle brackets instead of {}, but angle brackets dont like to show up right on the web site.


I also have names for the various entries.
Consider {a,b,c,d,e,f}
a is the base
b is the extender
c is the operator
d is the level of operators
e is the level of level of operators
f is the level of level of level of operators.

Here are some results:
{a,b,1} = a+b
{a,b,2} = a*b
{a,b,3} = a^b
{a,b,4} = a {4} b (a tetrated to b) = a^(a^(a^(....a^(a^a)..)) or a to the power of itself b times.
{a,b,5} = a tetrated to itself b times = a {4} (a {4} (a {4} ....a)) = a {5} b (a pentated to b). etc...


My extended operator notation  is related to my array notation in this way:
a {c} b = {a,b,c}, where c=1,2,3,4,5 etc represents adding, multiplying, exponentiation, tetration, pentation, etc. (I first encountered the names tetration, pentation, etc. in a book which I no longer remember the name).
a {{c}} b = {a,b,c,2}
a {{{c}}} b = {a,b,c,3} etc.
a {{1}} b = a { a { a....a { a { a } a } a.... a } a } a (b a's from center out)  I call this a expanded to b
a {{2}} b = a expanded to itself b times (or a multiexpanded to b) = a {{1}} (a {{1}} (a {{1}} (a .... (b times)
a {{3}} b = a multiexpanded to itself b times (or a powerexpanded to b)
a {{4}} b = a powerexpanded to itself b times (or a expandotetrated to b)
etc.
a {{{1}}} b = a {{ a {{ a ... a {{ a {{ a }} a }} a ... a }} a }} a ( b a's from center out)  I call this a exploded to b.
a {{{2}}} b = a exploded to itself b times (a multiexploded to b)
a {{{3}}} b = a multiexploded to itself b times (a powerexploded to b)
a {{{4}}} b = a powerexploded to itself b times (a explodotetrated to b)
etc.
a {{{{1}}}} b = a detonated to b = {a,b,1,4}
a {{{{{1}}}}} b = a pentonated to b = {a,b,1,5}
{a,b,1,1,2} = {a,a,a,{a,b1,1,1,2}} = a {{{{{{{....{{{a}}}....}}}}}}} a (where there are a {{{{{...{{a}}...}}}}} a angle brackets (where there are a {{{{..{{a}}..}}}} a angle brackets ( where there are .......b times ..... (where there are a angle brackets))))))  YIKES!!
I have operator notations for up to 8 entry arrays ( i.e. {a,b,c,d,e,f,g,h}), however they will be quite cumbersome to type out  a,b,and c are shown in numeric form, d is represented by angle brackets (as seen above), e is shown by [ ] like brackets, but rotated 90 degrees, where the brackets are above and below (uses e1 bracket sets), f is shown by drawing f1 vertical Saturn like rings around it, g is shown by drawing g1 Xwing brackets around it, while h is shown by sandwiching all this in between h1 3D versions of [ ] brackets (above and below) which look like square plates with short side walls facing inwards.


Notice that my array notation uses a 1D array, I later came up with an extended array notation which involves any number of dimensions in the arrays.
/a,b\ = {a,a,a,a,a,....,a} (ba's)
\2 /
/a,b\ = /a,a,a,a,......,a\ (b a's)
\k / \k1 /
{a,b} /a,a,a,......,a\
 2  = a,a,a,.......,a (bxb array of a's)
{ } a,a,a,.......,a (the 2 is going into 3rd dimension  if it was a k, then it would be reduced to k1
.................. and still be in the 2nd plane.
..................
\a,a,a,.....,a/


The extended array notation has 7 rules (use to be 8 rules, but two of them could fuse together), the first 5 rules are similar to the array notation, the 6th rule is similar to what's shown above  these rules will be quite combersome to type out  due to the dimensionality of the brackets. I'll soon provide a jpeg of these 7 rules to display them better.


Here is a brief description of the 7 rules of extended array notation:
Rule 1: If there are only one or two entries on the main row and nothing beyond the main row  then take the sum of the entries to solve the array (an extention of rule 1 of the array notation).
Rule 2: If rule 1 doesn't apply and if there are 1's on the end of any rows, remove them (but leave a placeholder 1 to keep the row from being deleted)  if there are any rows on the end of each plane that consists of a single one, then remove them (leave a place holder 1 for the plane if needed to keep from deleting it)  ... if there are any nspaces on the end of an n+1space that consists of a single one, then remove them (leave place holder 1 for the n+1 space to keep from deleting it),...  this rule removes insignificate 1s and is an expansion of rule 2 of array notation.
Rule 3: If second entry is a 1 and first two rules doesn't apply  then take the base entry as the solution to the array (similar to rule 3 of array notation).
Rule 4: If third "significate entry" (could be third entry, or an entry on another row, plane,..nspace) is a 1 and rules 13 don't apply, then define k as the largest number that the following is true: the pth significate entry is 1 for all p (3<=p<=k) and the k+1th entry is in the same row as the kth entry  then the kth significate entry becomes entire array with second entry reduced by 1, k+1th significate entry is reduced by 1, entries beyond k+1 remain the same, all significate entries before k becomes the base entry. (extention of rule 4 of array notation).
Rule 5: If rules 14 don't apply and the main row has atleast 3 entries  then the second entry becomes the entire array with the second entry reduced by 1, the third entry is reduced by 1, the base and all entries after the third remain the same.
Rule 6: If rules 15 don't apply and if main row contains only first two entries, and the next significate entry is not a 1  then the next significate entry would be in the next dspace  d>0 (if d=1, then entry in next row, if d=2 then in next plane, etc.)  solve array as such, the third (next) significant entry is reduced by 1, all entries beyond remain the same, and the main dspace becomes a b^d array of a's (a=base, b=2nd entry).
Rule 7: If other rules don't apply and the main row has only first two enties, and the third significate entry is a 1 then define k just like in rule 4, but since rule 4 doesn't apply then the kth and k+1th significate entry is not on the same row, so it must be in the next dspace, the kth significate entry would then be the only entry in it's dspace  if not it would of been an insignificate 1 and would of been removed, so it must be a significate place holder 1. The k+1th significate entry is reduced by 1, entries beyond remain the same, the first k1 significate entries become the base and the entire dspace where the kth significate entry is becomes a b^d array of a's (a and b are the base and second entries).
Entries are read in "book" order  read first row from left to right, then read next row left to right, until you read the first plane (or page), then read next page, until you finish the "book" (representing realm = 3space)  then read the next book until the shelf of books are done (representing 4space)  then read the next shelf, until book case is finished (5space)  etc. The main row is the very first row to read, the main nspace is the first nspace to read.
Significance Rules: Any non1 entry is significate (entries can be any positive integer)  any row with a non1 entry is significate, any nspace with a non1 entry is significate. Any entry between two significate entries on same row is significate, any row between significate rows in same plane is a significate row, any nspace between two significate nspaces in the same n+1space is significate. First entry on any significate nspace is significate. First two entries on main row are significate. Significate 1's are not removed by rule 2, they act as place holders.


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