User blog:Primussupremus/hyper notation.

The hyper notation uses the concept of the hyper operators  to achieve great results.

Before we can begin to write about the hyper notation we need to understand what the hyper operators are.

Hyper 0 is the Successor function f(x)=x+1,this is by far the simplest operator possible as all you are doing is adding one to a number.

Hyper 1 is addition x+x it can be thought of as repeated successorship,for example 5+5=10=1+1+1+1+1+1+1+1+1+1. Hyper 2 is multiplication which is also repeated addition of a number to itself a certain number of times,for example 5*3=5+5+5=15. Hyper 3 is exponentiation this is essentially repeated multiplication of a number to itself a certain number of times,for example 5^3=5*5*5=125. Hyper 4 is tetration this is where numbers really take off as this is the place where power towers are first found. Tetration is simply repeated exponentiation of a number a certain number of times,The easiest way to write it is by using the double up arrow. For example 4↑↑8=4^4^4^4^4^4^4^4. Hyper 5 is pentation this is repeated tetration of a number to itself a certain number of times. For example 5↑↑↑3=5↑↑(5↑↑5)=5^5^5^5^5^5^5...5(5↑↑5 5's in the stack). Hyper 6 is hexation this is repeated pentation of a number to itself a certain number of times. It is commonly denoted with four up arrows like so ↑↑↑↑. To give an example of a Hexation using small numbers we will use the example of 3↑↑↑↑3=3↑↑(3↑↑↑3). This means that the double up arrow operation(tetration) has been applied to 3 3↑↑↑3 times! Hyper 7 is heptation this is repeated hexation it is usually written as x↑↑↑↑↑y. An example of this is 3↑↑↑↑↑3. Hyper 8 is octation this is repeated heptation,x↑↑↑↑↑↑y where there are 6 up arrows between x and y. Hyper 9 is enneation this is repeated octation it is usually written as x↑↑↑↑↑↑↑y where there 7 up arrows between the x and y. Before we move on to hyper 10 we need to point something out about determining how many arrows are used in the notation. If we call the operator number p then there are p-2 up arrows needed to represent it,to make that a bit clearer here is an example:seeing as hexation is the 6th hyper operator it requires for up arrows as 6-2=4. We can write this as x↑(p-2)y where there are p-2 up arrows between the x and y. With that out of the way then lets move on. Hyper 10 is decation this is repeated enneation where there are 8 up arrows between x and y. For example x↑↑↑↑↑↑↑↑y is an example of decation. Seeing as we have defined the hyper operators it is now time to look at hyper notation a compact notation that can allow you to express higher order operators in a compact way. We write it as x[p](y)where x is the number you are using,p is the operator you are using and y is the number of times you apply the operator to that number. For example 3[4](3)=3↑↑3. 10[100]10=10↑(98)10 this means 10 followed by 98 up arrows then 10. 15[1500]89 means 15 followed by 1498 up arrows then 15.Moving onto things that are a bit bigger 10[10[100]10]10 means that you take 10 and apply the [10[100]10]th hyper operator to it 10 times. 10[10[10[100]10]10]10 is an even larger number as it is 10 with the [10[10[100]10]10]th hyper operator applied to it 10 times. An even larger number is 10[10[10[10[100]10]10]10]10,this ten with the [10[10[10[100]10]10]10]th hyper operator applied to it 10 times. But this isn't the end not by a long shot as you could even have something like 10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10 where there [10[10[10[10[10[10[100]10]10]10]10]10]10] numbers in the expansion. Or how about 10[10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10 where there [10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10] numbers in the expansion. [10...[10[10[10[10[10[10[10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10]10]10]10]10]10]...10] where there [10[10[10[10[10[10[10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10]10]10]10]10]10] numbers in the expansion. 10[10[10[10[10...[10[10[10[10[10[10[10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10]10]10]10]10]10]...10]10]10]10]10 where there are:[10[10[10[10...[10[10[10[10[10[10[10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10]10]10]10]10]10]...10]10]10]10] numbers in the expansion. But imagine instead of just having rows we had layers and then those layer hid more layers. 10[10[10[10[10[10...[10[10[10[10[10[10[10[10...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]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...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10]10]10]10]10]10]...10]10]10]10]10 where there [10[10[10[10[10...[10[10[10[10[10[10[10[10...[10[10[10[10[10[10[100]10]10]10]10]10]10]....10]10]10]10]10]10]10]10]...10]10]10]10] layers!