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 x^{n}^{–
1 }term (in this case x^{0}^{ }= 1)
by 1. Therefore we multiply 1 * 1 = 1 to obtain the 2^{nd} term.
And as this
equation (of degree 1) involves a 1step 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.
x^{2}
– 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 (x^{2} – 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. x^{3} –
3x^{2} + 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 x^{3} – 3x^{2} + 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. x^{4} – 4x^{3} + 6x^{2} – 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
2dimensional situation, whereby 1  as the solution for x  occurs twice.
Now if we think of this in geometrical terms as a 2dimensional
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).
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