MathNovatory/Pure Maths/Numerical Structure/Addition-Subtraction


Addition-Subtraction

1. The first stage - ADDITION

(a) the point of departure - the central point (0) would be the preferred point of departure but any other point would also be possible ;
(b) the increment - the unit (1) would be the simplest increment, being the smallest integer which produces a change when it is added, but any number other than zero would also be possible ;
(c) the procedure - addition.
The system of addition operates as follows :
1) the point of departure : zero (= 0)
2) the increment (1) added to the preceding number (0)
     or to the point of departure once : 0 + 1 (= 1)
3) the increment (1) added to the preceding number (1)
     or to the point of departure twice : 0 + 1 + 1 (= 2)
4) the increment (1) added to the preceding number (2)
     or to the point of departure three times : 0 + 1 + 1 + 1 (= 3)
          and so on, to produce the series 0, 1, 2, 3, 4 ...

Sum of the first members
If the members of this Additive series are defined as A0, A1, A2, A3...
     A0 would always have the value of "0",
     A1 would be the Additive increment, and
     the others, A2, A3 ... would follow the normal Additive process.
The sum of the members to An would be defined as AnAn+1 / 2A1.
     EX - the sum of the first 7 members with A1 = 1 (0, 1, 2, 3, 4, 5, 6, 7) would be (7 x 8) / (2 x 1) = 28.
     EX - the sum of the first 7 members with A1 = 2 (0, 2, 4, 6, 8, 10, 12, 14) would be (14 x 16) / (2 x 2) = 56.

Subtraction

Producing negatives
If, at any given point, we applied the inverse procedure, we would subtract the increment from the preceding number at each operation until we return to zero, our original point of departure. Beyond this point, we
1) subtract the increment (1) from the preceding number (0)
     or from the point of departure once : 0 - 1 (= -1)
2) subtract the increment (1) from the preceding number (-1)
     or from the point of departure twice : 0 - 1 - 1 (= -2)
3) subtract the increment (1) from the preceding number (-2)
     or from the point of departure three times : 0 - 1 - 1 - 1 (= -3)
          and so on...
These procedures generate a series of numbers which has zero as its centre and which runs from negative infinity (-inf) on one side to positive infinity (inf) on the other : -inf ... -3 -2 -1 0 1 2 3 ... inf.
It is interesting to note that the value of the central point (0) would be sterile as an increment and cannot be used in this function (adding zero to any value would leave it unchanged).

Looking forward

The law of REGULARITY in abbreviations
It is now possible to establish a link between the first stage of addition and the second stage of multiplication by observing that the process of multiplication is only the abbreviation of several additions with a constant increment ; all we do is note, in an abbreviated form, the number of times that we have added and the size of the constant increment.
It is most important to notice that, without a constant increment, multiplication would not be possible. We have here one of the natural laws, that of REGULARITY, operating within a mathematical structure ; we shall come back to this law as we discuss the specific laws themselves and their application to musical systems.

What we used to indicate (0) + 1 + 1 + 1 in the first stage (= 3)
     we can now indicate in an abbreviated form (0+) 3 x 1 (= 3)
If the increment of addition were 2 instead of 1, we would have the series of even numbers;
     -inf ... -6 -4 -2 0 2 4 6 ... inf
and what we used to indicate in the first stage (0) + 2 + 2 + 2 (= 6)
     could now be abbreviated to (0+) 3 x 2 (= 6).

To Multiplication-Division