Notations

A reference for the primary notations that define large numbers.

1 - Exponents

This is something most people should be familiar with. The notation is rendered as:

a^b

(In proper mathematical syntax, the b would be on the top right of the a. The ^ between them would be omitted.)

What this means is simple:

a×a×...b times...×a×a

The popularity of exponents in googology is because 10^x is always a 1 followed by x 0s, eg.:

10^8 = 100,000,000

Repeating superscripts is also acceptable if the first one gets too big, eg.:

10^10^10

2^4^32

(This is evaluated right-to-left, since in proper syntax the rightmost number would be furthest to the top-right. i should add an image)

However, this eventually gets too tedious:

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

This where we introduce our next notation,

2 - Knuth's Up-Arrow Notation

This notation is rendered as

a[variable number of ↑s]b

a↑b is just a^b, eg.:

10↑12 = 10^12 = 1,000,000,000,000

a↑↑b is a↑a↑a↑...b times...↑a↑a↑a:

10↑↑10 = 10↑10↑10↑10↑10↑10↑10↑10↑10↑10

This is called tetration.

(tritation - exponentiation, bitation - multiplication, untation - addition)

a↑↑↑b is a↑↑a↑↑...b times...↑↑a↑↑a, pentation:

10↑↑↑5 = 10↑↑10↑↑10↑↑10↑↑10

And so on.

Repeated arrows are evaluated right-to-left, so 10↑↑10↑↑10 should be treated as

10↑↑(10↑↑10),

NOT

(10↑↑10)↑↑10

Often the arrows in this notation are rendered as circumflex accents (superscript indicators) with 10↑↑↑10 being written as 10^^^10.

Once there are too many arrows, like

10↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑10,

we introduce

3 - Operator Notation

This is rendered as

a{c}b

(number of curly brackets/braces vary)

a{c}b counts the arrows in Up-Arrow Notation:

10{5}10 = 10↑↑↑↑↑10

Eventually the arrow counter becomes heavily nested:

10{10{10{10{10}10}10}10}10

So we introduce expansion:

10{{1}}3 = 10{10{10}10}10

Repeated expansion is called multiexpansion:

10{{2}}5 = 10{{1}}10{{1}}10{{1}}10{{1}}10

Like Up-Arrow Notation before it, repetition of multiexpansion produces powerexpansion:

10{{3}}4 = 10{{2}}10{{2}}10{{2}}10

and then expandotetration:

10{{4}}3 = 10{{3}}10{{3}}10

Nested expansion is explosion:

10{{{1}}}3 = 10{{10{{10}}10}}10

Repeating gives us multiexplosion:

10{{{2}}}3 = 10{{{1}}}10{{{1}}}10

And the pattern repeats to produce detonation:

10{{{{1}}}}3 = 10{{{10{{{10}}}10}}}10

and pentonation:

10{{{{{1}}}}}3 = 10{{{{10{{{{10}}}}10}}}}10

And when the curly brackets or braces or whatever get too numerous:

10{{{{{{{{{{{{{{{{{{{{10}}}}}}}}}}}}}}}}}}}}10

we introduce one of two notations: BEAF and BAN.

4 - BEAF/BAN