Dual Percoxy Notation – Mathematical Axiom

Dual Percoxy Notation is a method of notation for extremely large numbers. Dual Peroxy Notation is a form of tetration that can be written in short expressions but can be used to express both small and large integers. This notation was invented by Matthew Tainsh in April 2016.


Dual Peroxy Notation uses subscript, superscript, brackets and backtick(s).

First, let’s examine the simple operations:


That’s quite straight forward, but Dual Percoxy Notation is more complex.

The superscript of the base shows the power(s), the subscript shows how tall the power tower is or how many sets of power towers there are.

Note: Always write super and subscript in between brackets to show that you are using this notation.

First Example

In this example we can see that the power integer is given by the superscript and the magnitude of the tower is given by the subscript.


The usual use of the notation is:


The tower starts at ‘b’ and terminates at ‘c’.

Second Example

The backtick (`) in the superscript (10`2) tells us that the power integers are from 10 to 2. As shown in the example, the powers decrease from 10 to 2, again, shown by the backtick.


To simplify this further:


One set is given by the superscript, and the copies of the sets are given by the subscript, as shown in the example above.

Unfortunately, ‘d’ always equals one when ‘c’ equal one. Shows in the examples, ‘c=2’ because of the fact that a power tower terminates when an exponent equals 1.


There can be more than one backtick, the amount of backticks represent the how much you subtract from ‘b’. The examples show one backtick, because of that the powers decrease by one. e.g. (10`2) means the powers are 10, 9, 8, 7, 6, 5, 4, 3, 2. If we use two backticks; (10“2), this means the powers decrease in twos, 10, 8, 6, 4, 2.


This expression works because ‘b’ (10) and ‘c’ (2) are divisible by the amount of backticks there are, 2. If  the two superscript integers aren’t divisible by the amount of backticks, the expression does not exist.


If there are more than several backticks insisted, the expression can be show as:


The amount of backticks ‘e’ is written in Curly Brackets. {e}

Therefore, ‘b’ and ‘a’ aren’t divisible by ‘e’, than it doesn’t exist.


This example shows that there are twenty backticks between 100 and 20.

Note: When noting the amount of backticks in curly brackets, always have a backtick before and after to display that you are using this notation.


Subscript integers can be written in any form. In the example below, one set has 999,999 exponents, and it has 1 million sets. That’s a 1 billion number tall power tower.

There is a special number that can be expressed more simply using this notation:


When all values are the same, you take away the subscript and replace the superscript value for the Greek Symbol ‘equiv’. Equiv tells us that each value is equal.

To get a bit more complicated, we can have two equiv’s within the superscript. If the base is ten, then the subscript and superscript are both an Equinoxal. This means that there are and Equinoxal amount of Equinoxals in the power tower.


This is a gigantic number is named an Equiduoxal, and we can go a lot higher using the same notation.

If we had three Equivs for example, then the sub and superscript values would be equal to an Equiduoxal, and so on and so on…

If we add a backtick and a two into the superscript, the number suddenly becomes unimaginably times more massive.




This is very hard to comprehend, but we can go a lot larger:


The last example above tell us that there are an Equinoxal amount of equiv’s in the superscript.

We must remember that no matter how big the superscript value is, if it contains an equiv, the subscript value will always equal to the superscript value.


…and you can go are big as you like:



That’s the complete notation. Below will be a list of some names and equinoxes for large numbers using this notation. This notation has no important use besides expressing significantly huge numbers but please help and try to make this notation noticed as its only an invented axiom. If you have any interests in large numbers, I suggest exploring the website ‘List of googologisms – Googology Wikia‘.

Dual Percoxy Notation

Second Equinox:


Third Equinox:


Fourth Equinox:


Fifth Equinox:





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