The concept of “invisible numbers” has existed in many forms over the years. It would not be a surprise if you recognize what we are about to show you, but know it under a different term. We have looked online for what this rule is actually called in the mathematical lexicon and have come up empty. Thus, we here at D.E.C.I.D.E. present to you (our understanding of) The Rules of Invisible Numbers (RIN).
At it’s most basic RIN accepts that each and every written numerical digit has multiple other numbers connected to it that are not written. These unwritten numbers are never shown because they have no effect on the written number. How is this? Well let me ask you, if you add Zero to a number, what is the new value? If you multiply a number by One what is the new value? As it turns out, each of the basic mathematical functions has an “invisible number”, or a number which when put through that function maintains the original value of the other numerical value shown.
Lets take a look at some examples…
As is shown, any number (N) with the appropriate Invisible Number to the function applied, will equal the original number (N). So let’s take a look at why these are never written...
The above equation is the fundamental basis of RIN. It shows each of the six most basic functions in mathematics being used in conjunction, where each individual function and the equation as a whole both have no effect on the value of the number (N). Due to this, even though these invisible numbers exist, they are never written. This can be taken a step further (as shown below) and each of the “invisible numbers” would have their own set of “invisible numbers” (ad infinitum). As can be seen, writing the invisible numbers can quickly get out of hand.
We do not write these “invisible numbers” because doing so would be redundant and a waste of writing space as well as time. So what is the point of this rule? Why is it important? Well, the RIN isn’t completed yet, the above is just the basics, it’s only the First Rule of RIN. The applicability of RIN and it’s effects are seen more clearly in the higher powered mathematical functions. In all following examples the invisible number will be bracketed [] to show it would not normally be written.
First Rule of Invisible Numbers: If writing a number and an applicable function would have no effect on the numerical value of the formula, DO NOT WRITE IT.
Now for the second rule of RIN, let’s start simple… Have you ever wondered why N^0 = 1? This is a mathematical fact, and there are numerous “explanations” as to why. RIN gives a very clear-cut reason why. Let’s go backwards (or forwards) a few steps… why does N^1 = N?
N^3 is saying “take the number (N) and multiply it by itself and by itself again” N^3 = NxNxN
N^2 is saying “take the number (N) and multiply it by itself” N^2 = NxN
N^1 Is saying “take the number (N) and leave it by itself)” N^1 = N
So what about N^0? This is where RIN is important… the above description of how exponents works does so because we were taught to ignore the invisible numbers. But for higher powered mathematics the invisible numbers are very important. Let’s look at the above examples adding in the invisible number for multiplication (1)…
N^3 is saying “take the invisible number (1) and multiply it by the numerical value (N) 3 times”
N^3 = [1]xNxNxN
N^2 is saying “take the invisible number (1) and multiply it by the numerical value (N) 2 times”
N^2 = [1]xNxN
N^1 Is saying “take the invisible number (1) and multiply it by the numerical value (N) 1 times”
N^1 = [1]xN
So…
N^0 is saying “take the number (1) and multiply it by the numerical value (N) 0 times”
N^0 = 1
Do you see the pattern? Notice how in writing it out in english using RIN we now have both N and the number it is being raised to a power to, whereas originally trying to write it without RIN the numbers didn’t match up until you wrote it as a formula. Math is all about patterns, and once you use the invisible numbers as part of the explanation the reasoning behind why N^0=1 becomes clear. So why does this work? Because…
Second Rule of Invisible Numbers: If a number (M) using a mathematical function is put to a numerical value (N), the effects of that function would be similar to using the next lower function’s “invisible number” and then using that function to propagate (M) (N) number of times to determine the answer (ans). Inverted functions follow the same methodology, but invert (M) and (ans) positions in the propagation formula.
So what about division and roots? They still work, but are inverted functions. They still use the Second Rule of Invisible Numbers, but use the inversion clause…
Now what about other higher powered mathematical functions? Do they still follow these rules? Yes, they do. For example let’s look at the Factorial and our Summorial (discussed in our article: Missing Functionality: Summorials and Factoriation Notation)
A Factorial (!) of N! is often described as “multiply all numbers from 1 to N” seems straight-forward right? Ok, so N! with N=3 means N!= 1x2x3. nice and simple! But what if we were to say N=0? well that would be 1. Unfortunately, this is where the kerfuffle between the mathematical lexicon and our understanding of math may differ… there are TWO possibilities, depending on which side of belief you belong to determines which method you will use…
Method B; Using RIN: N! would be described as “multiply all numbers from 1 to N including the invisible number”. So in this method N! with N=3 means N! = [1]x1x2x3, and if N=0 then N!=1
Method A; Using RIN: N! would be described as “multiply all numbers from the invisible number to N”. So in this method N! with N=3 means N!=[1]x2x3, and if N=0 then N!=1.
A Summorial (@) isn’t much different. It would be described as “add all numbers from 1 to N” seems straight-forward right? Ok, so N@ with N=3 means N@= 1x2x3. nice and simple! And if you were to say N=0? well that would be 0. Do you see the problem yet? The Summorial, if written as expected as it is merely an additive version of the Factorial, works perfectly fine with RIN. It only has ONE possible RIN interpretation.
Method A; Using RIN: N@ would be described as “add all numbers from the invisible number to N”. so this method N@ with N=3 means N@=[0]+1+2+3
So if Method B for Factorials is to be the method chosen, then describing A Summorial without RIN would have to say “add all numbers from 0 to N” this allows for the possibility of Method B.
MethodB; (adjusted) Using RIN: N@ would be described as “add all numbers from 0 to N including the invisible number”. So in this method N@ with N=3 means N3 = [0]+0+1+2+3, and if N=0 then N@=0
So what is the difference in how these work? Absolutely no difference (according to the current mathematical lexicon), Except we here at D.E.C.I.D.E. have discovered a “missing” mathematical function that would affect the internal formulae (the propagated formula) of Closed Functions such as the Factorial and Summorial. This “new” mathematical function’s interaction with these specific closed functions depends entirely on which method is used… This will be discussed in greater detail in: Missing Functionality: Simultaneous Operations, but for now just know that there are two potential methods for it Factorials (and Summorials) to use RIN that both work equally well (so far).
The final rule of RIN is probably the simplest.
Third Rule of Invisible Numbers: There are no invisible numbers that precede an inverse function (subtraction, division, root).
This rule exists because the inverse functions, when preceded by what would normally be an invisible number, generate non equatable answers. i.e. 0-N = negative N, 1/N equals a value between 0 and 1, and so on… However, there is some contention about negative numbers and invisible numbers.
Negative numbers (i.e. -5, -2, -N) can be claimed to be saying “0-5, 0-2, 0-N and simply having the 0 as an “invisible number”. The validity of which we explore more in the article: , part of the Problems with the Current Base Mathematics System series
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