## Friday, August 25, 2017

### A Special Case (1)

If  x = 1, then we can thereby obtain the simple polynomial equation x – 1 = 0.

The corresponding unique sequence of digits associated with this equation can then be obtained as follows.

Starting with 1, we multiply the negative of the coefficient of the xn– 1 term (in this case x0 = 1) by 1. Therefore we multiply 1 * 1 = 1 to obtain the 2nd term.

And as this equation (of degree 1) involves a 1-step procedure, we keep in turn multiplying 1 by 1 to obtain each additional term.

So the unique infinite digit sequence associated with the equation (i.e. x – 1 = 0) is

1, 1, 1, 1, 1,…

Now if we obtain the square of our expression then (x – 1)2 = 0, i.e.

x2 – 2x + 1 = 0.

Now to calculate the unique digit sequence associated with this equation, we start with

0, 1 and then obtain (2 * 1) – (1 * 0) = 2 + 0 = 2.

So we now have 0, 1, 2.

Then the next term = (2 * 2) – (1 * 0) = 3.

And by continuing in this manner we find that the unique digit sequence associated with the equation (x2 – 2x + 1 = 0) is the set of natural numbers i.e.

1, 2, 3, 4, 5, 6,….

However there is an alternative way of obtaining this sequence from the previous sequence, whereby the nth term in the latter represents the sum of the first n terms in the previous sequence.

And this procedure can be extended indefinitely.

Thus when we cube (x – 1)3 = 0, i.e. x3 – 3x2 + 3x – 1 = 0, the unique digit sequence associated with this equation can be directly obtained from the previous sequence, whereby once again the nth term of the latest represents the sum of the first n terms of the previous sequence.

So the unique digit sequence associated with the equation x3 – 3x2 + 3x – 1 = 0 is

1, 3, 6, 10, 15, 21,…

And to give just one more illustration the unique digit sequence for (x – 1)4  = 0

i.e. x4 – 4x3 + 6x2 – 4x + 1 = 0 is

1, 4, 10, 20, 35, 56,…

In the following grid, I show the first 9 terms in the respective unique number sequences for the 9 equations from x – 1 = 0 to (x – 1)9 = 0 .

 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 3 6 10 15 21 28 36 45 1 4 10 20 35 56 84 120 165 1 5 15 35 70 126 210 330 495 1 6 21 56 126 252 462 792 1287 1 7 28 84 210 462 924 1716 3003 1 8 36 120 330 792 1716 3432 6435 1 9 45 165 495 1287 3003 6435 12870

Now the remarkable feature regarding these number sequences is that there is a direct horizontal/vertical type correspondence as between entries.

In other words, the sequence of numbers in each row (read from left to right) exactly matches the sequence of numbers (read from top to bottom) in the corresponding column.

And when interpreted properly there is a fascinating explanation for this significant finding.

As we have seen, 1 represents the solution to all these equations.

However whereas in the first case (i.e. for x – 1 = 0), this solution occurs just once, in all other cases it does so on multiple occasions, so that for example for (x – 1)9 = 0, it repeats 9 times.

Now whereas in conventional mathematical terms, there is no distinction as between these separate solutions, in fact this is not quite the case (with each root or solution for x carrying a unique meaning).

This can best be illustrated with respect to the 2-dimensional situation, whereby 1 - as the solution for x - occurs twice.

Now if we think of this in geometrical terms as a 2-dimensional figure e.g. a square, it has two sides i.e. length and width.

However when length and width = 1, there is a distinction in that these two measurements are horizontal and vertical with respect to each other.

Thus if we identify the length with the horizontal measurement, then the width represents the vertical (and vice versa).

This is equally true in 3 dimensions. So if we now add in the height (in a cube with each side = 1), it is now vertical with respect to each of the other measurements (considered as horizontal). And though this cannot be visualised with respect to higher linear dimensions, the same principle holds with each new dimension distinguished as vertical in relation to any other dimension considered as horizontal.

And we can see that this property is intrinsic to the very nature of the number sequences associated, with the equations in the various dimensions having a distinctive horizontal/vertical correspondence in evidence.

So this enables one to distinguish each new unitary solution to x (as the degree of the equation increases by 1) in a distinctive manner (as relatively vertical to any previous root representing a horizontal measurement).