One of the biggest issues that we here at D.E.C.I.D.E. have with the current accepted Mathematical Lexicon is the fact that we have some mathematical functions that should have equivalents for other functions but don’t. The two most prominent examples come from Factorials “!” and
Summation Notation :
To understand what we mean by this let’s look at the Factorial first.
Factorials
Factorials are a mathematical function given the designation “!” where: N! = 1x2x3x…xN. In other words, any number (N) that is put to a Factorial, will equate to the total factors of 1 through the given number.
There, that’s pretty straightforward. So, what’s the problem with this? Absolutely nothing. Factorials are a useful tool… the problem doesn’t stem from what a Factorial is, but rather from the fact that there are no equivalents for other functions, such as addition. Basically, if I asked you what function would I use to describe ‘1x2x3x…xN’ you’d be able to answer with ‘N!’, but if I were to instead ask you the function that would describe ‘1+2+3+…+N’ there would be no answer. Sure, some would say Summation Notation covers that missing function but they’re only partially right.
Summation Notation
means i=1 : the lower limit of the Summation string; xi: the equative element; n: the upper limit of the Summation string.” In other words, any number (N) that is put to a Summation Notation, will equate to the total sums of “i ” through the given number.
In this case with i=1… xi= 1+2+3+…+N. Wait, isn’t this what I was looking for with the Factorial? No. The reason being? You can change the value of “i ”… if you have “i=5” then the Summation Notation equates to: i=5… xi= 5+6+7+…+N. This is something that you can’t do with a Factorial, thus making the two non equivalent.
The Factorial Equivalent, Introducing the Summorial
The Summorial is a mathematical function the D.E.C.I.D.E. crew has agreed would be the additive equivalent of the Factorial. In our works the Summorial will be represented with the designation “@”, where N@= 1+2+3+…+N. In other words, any number (N) that is put to a
Summorial, will equate to the total sums of 1 through the given number.
As can be easily seen this is the additive equivalent of the Factorial. For those who do not find this clear enough, look at image 1a below.
Factorial Summorial
3! = 1x2x3 3@ = 1+2+3
6! = 1x2x3x4x5x6 6@ = 1+2+3+4+5+6
8! = 1x2x3x4x5x6x7x8 8@ = 1+2+3+4+5+6+7+8
Aside from using addition as it’s base mathematical function, a Summorial works in every other regard as a Factorial is currently noted as working.
The Summation Notation Equivalent, Introducing Factoriation Notation
As the Summorial is to Factorials, Factoriation Notation is the mathematical function the D.E.C.I.D.E. crew has agreed would be the multiplicative equivalent of the Summation Notation. As with Factorial -> Summorial, the only difference between Factoriation Notation and Summation Notation is the base mathematical function used. In this case, we are exchanging the addition property for a multiplication property.
means i=1 : the lower limit of the Factoriation string; xi: the equative element; n: the upper limit of the Factoriation string.” In other words, any number (N) that is put to a Factoriation Notation, will equate to the total sums of “i ” through the given number. In this case with i=1… xi= 1*2*3*…*N.
Aside from multiplication as it’s base mathematical function, a Factoriation Notation works in every other regard as a Summation Notation is currently noted as working.
Where do we go from here?
As we assume is clear the only differences between the two types of notations (and the two “***ials”) is simply what basic mathematical function is being used. The difference between one of the “***ials” and it’s corresponding notation however is simply that the notation allows for a starting point other than 1, whereas the “***ials” must always start at 1
Now, due to the similarities between a Factorial and a Summation Notation, we recognize that there will be dispute about adding in “unnecessary functions”. While those of us here at D.E.C.I.D.E. do not view our additions as being unnecessary, we do recognize this potential view… as such we recommend one of the following remedies…
Remedy 1: Use either Factorials and Summorials, or use Factoriation Notaion and Summation Notation… removing the other from the mathematical lexicon entirely
Remedy 2: Keep all forms of each and denote them in Order Of Operations differently. As of right now these functions aren’t given a specific spot in the order of operations, though we here at D.E.C.I.D.E. believe they all belong to the closed function group (meaning they must be completed at the same time as parenthesis). This concept is talked about in greater depth in the article Missing Functionality: Advancing PEMDAS/BODMAS
As we are but a small group, we do not have the power to unilaterally change the mathematical lexicon. While this is the reason our website even exists, it is up to our readers to decide if what we say holds enough merit to compel a change. Decide for yourself, and next time you meet a leader of knowledge in the mathematics field, bring it up with them and question them. Change may start with a few radical ideas, but it takes more than one voice to make that change become reality.
コメント