MathNovatory/Pure Maths/Numerical Structure/Power-Root


Power-Root

3. The third stage - POWER

In this exponential system :
(a) the point of departure cannot be the unit (1) (for reasons which will become clearer shortly) because it would be impossible to find an increment which would produce a series, and we must use the number 2 as the simplest outset ;
(b) however, we can also use 2 as the simplest increment ;
(c) the process is that of power
, as compared to multiplication, and addition in the two previous stages.

This system produces a series in the following manner :
1) the point of departure 2 (= 2)
2) the preceding number (2) raised to the power of the increment (also 2)
     or the point of departure raised once: 22 (= 4)
3) The preceding number (4) raised to the power of the increment (2)
     or the point of departure raised twice: (22)2 (= 16)
4) The preceding number (16) raised to the power of the increment (2)
     or the point of departure raised three times: ((22)2)2 (= 256)
          and so on...

Root

Producing roots
If we repeated the inverse procedure, we would take the root (in this case, the square root) using the same increment at each operation, indicated as index rather than as exponential.
1) Once back at the point of departure 2 (= 2) :
2) the square root of the preceding number (2)
     or the point of departure once : 22 (= 1.4142)
3) the square root of the preceding number (1.4142)
     or of the point of departure twice : (22)2 (= 1.1892)
4) the square root of the preceding number (1.1892)
     or of the point of departure three times: ((22)2)2 (= 1.0905)
          and so on, these values always approaching the unit (1) which is the lower limit of this series.
We have here a series whose central point is 2 and whose increment (the power) is also 2; the limits are the unit (1) at one end and positive infinity at the other.
It is now quite evident why we could not use the unit as a point of departure; this value, being at the lower limit of the series, is not really part of the exponential system.

Comparisons
In this system, the central point is not a sterile increment, contrary to the other two systems. On the other hand, the central point of the preceding system (the unit, centre of the multiplicative system) is once again the inferior limit of this exponential stage.

Qualitative
Here is an excellent example of Qualitative Mathematics -
     The same quantitative value of "1" can be -
          the simplest increment in Addition,
          the central point of departure in Multiplication, and
          the lower limit in Power.
Let's inverse the situation -
     The same qualitative central point of departure can be -
          the quantitative value "0" in Addition,
          the quantitative value "1" in Multiplication, and
          the quantitative value "2" in Power.
Understanding is impossible without Qualitative Mathematics.
     It seems evident that multiplying by "1" has the same effect as adding "0".

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