Background ~
Our numerical system, base digits 0 through 9 is believed to have originated from Mesopotamia [an area where the ancient city of Babylon once stood], around 3-4,000 B.C. Still today, Babylon remains the model, a standard for modern day decadent themes [as far as Hollywood is concerned]. 🙂 The Arabic numeral system, originating from what is now Iraq has got to be the only redeemable thing I can [ever] think of that originated from the Middle East. This “base-10” numeric system has been adopted across the world since then; it comprises the entire set of real numbers and is said to include all whole numbers except zero, without fractions or negative numbers.

Here I seek to uncover what many may never have considered, which is that unique relationships exist between multiples of whole numbers. For example, any multiples of 9 and 0 will reduce to nothing but 9 and 0 respectively. My wish is that everyone is already aware “nothing from nothing leaves nothing” as the song goes, but anything multiplied by 9 will also leave nothing but 9 as its terminal value. Kind of funny how all that works. So I want to re-visit numbers in a new way using this concept and expound on a universal truth [not theory] regarding numerals 0 through 9, that being 1,2,3,4,5,6,7 and 8 for how they might behave in the real world, inside the world known as “numbers”. I’m here to report that relationships do exist between these root values; whole numbers 0 thru 9 make up the basis for all possible values all the way to infinity, or at least [any number] that can be tested, so we need not discuss extremely large sequences of numbers when we can already determine any outcomes from examining the base numerals of 0 thru 9. For my previous writing on isolating the number 9 please type in “For Geeks Only” in the search bar on my home page. In the case where I already lost you? This previous article on the number 9 will provide a good primer before attempting to digest this latest work. Thanks in advance!

First Things First ~
Let’s get something out of the way, any whole number multiplied by [or combinations of] 9 will reduce to 9, just as any number multiplied by 0 will terminate in zero. So a unique relationship exists between these two single numeric values as these are the only two digits in possession of this quality. So for me this warrants recognizing that a unique relationship exists between these two numerals. I look on multiplication operations using either a 0 or 9 as if to say this is where number sequences go to die. But how about the other digits, namely numbers 1 through 8, is anything strange happening among these worthy of even reporting on? The answer is a resounding yes! There are individual couplings [or relationships] that also exist between other single digits in this numerical sequence.

Let’s Investigate ~
I found that distinct relationships exist between the numerals 1 and 8, 2 and 7, 3 and 6, and 4 and 5. I cannot decipher the meaning behind these relationships at this writing but it can be expressed graphically, or just by doing some simple calculations. Please see below, where I use a clockface diagram to show graphically the value results of multiplying consecutive numerals by 2 or by 7 then reducing those values to a single digit. Example, 2 x 1 = 2, 2 x 2 = 4, 2 x 3 = 6, etc. through 2 x 9. The resulting values happen to mirror those whenever 7 is multiplied by the same consecutive numerical sequence, [numbers 1 through 9]. The only difference being that the string of values for number 2 runs in a clockwise direction while the same results for number 7 will run in a counter clockwise direction. So looking at the diagram [and following the red line] we see starting with 2 on the clock [2 x 1], the next stop on the clock is 4, representing 2 x 2, etc. as it makes its way around to 6, 8, then back to 1, 3, 5, 7, 9 before the cycle begins again at 2. Whereas an operation on the numeral 7 goes like this, 7 x 1 = 7, x 2 = 5, then 3, then 1, then 8, then 6, then 4, then 2, then 9 before returning to 7 for a repeat of the cycle. In other words the same set or sequence of values as when we reduced the numeral 2 calculations to a single digit only moving in the opposite direction.

Below I uncover the relationship between numerals 4 and 5 as this pattern is one of the more unique in the set of root numbers between 1 – 9. Let’s write out the number stops you see from the graphical presentation below in yellow . In the case of numeral 4, we have –
4 x 1 = 4
4 x 2 = 8
4 x 3 = 12 = 3
4 x 4 = 16 = 7
4 x 5 = 20 = 2
4 x 6 = 24 = 6
4 x 7 = 28 = 1
4 x 8 = 32 = 5
4 x 9 = 36 = 9, before it repeats this same sequence [to infinity] beginning with 4 x 10 = 4, etc.

Now 5:
5 x 1 = 5
5 x 2 = 1
5 x 3 = 6
5 x 4 = 2
5 x 5 = 7
5 x 6 = 3
5 x 7 = 8
5 x 8 = 4
5 x 9 = 9, same results as in the case of the numeral 4, however, this time moving in a counter-clockwise, or opposite direction.

What I Found ~
By calculating single digit reductions on base numerals 1, 2, 3, and 4, they will always move in a clockwise direction while the same results using numerals 5, 6, 7, and 8 will move in a counter-clockwise direction. As mentioned, the numerical sequence of single digit values between 1 and 8 are identical just as the results for 2 and 7, 3 and 6, and 4 and 5 are identical. What is more strange than this is the fact that these operations reveal bedfellows here in the cases of 1 and 8, 2 and 7, 3 and 6, and 4 and 5. Now guess what all these paired values add up to? 9! Which I find quite fascinating, strange really. We know from this work and before that any number multiplied by 9 will also reduce only to 9. Therefore, no reason to build a clockface diagram for operations on the numeral 9, nor the number 0 for that matter.

More ~
I do have something more to share with you, the strange case of numerals 3 and 6 above, these two numbers are joined at the hip as well, or it appears that way. “Nikola Tesla believed that the numbers 3, 6, and 9 hold the key to understanding the Universe”. Apparently the genius Tesla was fascinated by the behavior of 3, 6, and 9, referring to it as his “369 Theory”. He looked at his idea as key to understanding some problems in physics. It may have had something to do with this – any whole number multiplied by 3 or 6 and then reduced to a single digit will yield only three possible results, those being 3, 6, or 9. That’s what this diagram is revealing. However, in the case of operations on the number 3, the resulting values will move in a clockwise direction, from 3 to 6 to 9 before returning to 3, while consecutive multiples of 6 will move in the opposite direction of 3, but still maintain only three possible results no matter what whole number is used as a multiplier; rotating from 6 to 3 to 9 then back to 6.

Conclusion ~
Conclusion? This is a tough one to make any conclusions on though I would have to say my opinions expressed here are beyond “assumptive” at this point. We’re traveling down a highway of knowledge here [at least presently] however, void of any final destination. I haven’t a clue as to how this information can be applied to aid in the preservation of mankind, or even build a homemade peanut-butter sandwich for that matter. Not yet anyways, but something tells me, something is tugging on my sleeve and keeps saying, “it will, it will”. So looks like another “wait and see” kind of moment huh? I will declare this in the definitive – this information has been hiding right in front of our eyes now and for what, 3-4,000 years? We only thought we knew how to count! 🙂