| a primer to the basic algebraic notion of mathematics |
In counting, we develop the sub-notions of generative (comparison), -unit (one), cumulative (summation, addition), total (count), equal (equivalent), group (associativity);- and extend these to iterative (repetition, multiplication) and reďterative (distributivity); and to zero (none: not-one), and reörder (commutativity), and negative (minus).
Unit (one), is written, 1, or often, l, from the linguistic definite article, el/le/la, meaning, The.
The first operation, incremental cumulation (summation, addition), is written, +, eg. 1+1+1, for implicit sequential unit associativity.
Total (count), is an unstructured summary written efficiently, eg. 3 (eg. decimal notation).
(A structured summary is a generic tradename: an It.)
Equal (equivalent), written, = , means, Both having or being the same total,
eg. 1+1+1 = 3.
Identical, written, ==, means, Both have the same structure and total,
eg. 1+1+1 == 1+1+1.
The unit (one) is generative: Every summary count is a cumulation of units (ones).
Cumulation implies (requires) the constancy (commutivity) of the unit (one), eg. 1+(1) = (1)+1, either "1" first, whence is derived general reörder (commutativity), eg. 1+(2) = (2)+1, and group (associativity):--
Group (associativity), is written within-and-by pairs of parentheses, ( ), to explicitly order (pick) operation sub-totals (sub-counts), eg. (1+1)+2 = 1+(1+2).
Iteration (repetition), requires (implies) identical groups, the same size (sub-total-count) and structure, eg. (1+3)+(1+3), neg. (1+3)+(2+2) where total and subtotals are equal but structure is not; And is written as a summary count of repetitions-of, adjacently left before the group, eg. (1+3)+(1+3) = 2(1+3): Iteration takes the group as a constant, unit-instance, which therefor equals its subtotal (subcount), eg. 2(1+3) = 2(4), or which =2(2+2).
Reďterate (distributivity), regroups so as to apply the iteration to each component, of the same repeated group, and is written as an equality (equivalent), eg. 3(1+2) = 3(1)+3(2).
Zero (none: not-one), remains, when counting is done: It is written, 0, -(like the "O", open, empty, and wasn't used already)-.
It was discerned that reörder (commutativity) has no effect on sum and total. It is written in the manner of an equality (equivalent), eg. 1+2 = 2+1. It may be proven by grouping (associativity).
And we append Negative totals, to count the Remainder (the not-yet-counted), as, minus, written, - , eg. 3-1 = 2 because 3 = 1+2.
Iteration (repetition) we discover quickly, is a generally useful tool for quickly writing sums, as mathematics becomes an arena for proof, demonstration, discernment, discovery, -utilizing linguistic means for recognition:- Iteration becomes multiplication with its assigned, written incremental operation, × , times, eg. 2×3 = 2(3), left-to-right (linguistical order).
Multiplication is reörderable (commutative), eg. 3×5 = 5×3, because addition (upon which multiplication is built) is total (countable), reďterable (distributive), iterable (repetitive), and has a generative unit, eg. 3×5 = 3(5) = 5+5+5 = 5(1+1+1) = 5(3) = 5×3.
And iterations of multiplication, are further written superponentially (exponentially),
eg. 3×3 = 23.
(We see also, 23 = 32, that is =3(3) in operation or function
form, for the base, 3, a constant function.)
We note that the exponentiation line for a common base, is additive, 23×33 = (2+3=5)3; which has iteration and thus multiplication;
Observe, we do not co-operate exponentiation, because iterations are equivalent to multiplication in the exponent itself, which is reörderable (commutative), eg. 3(25) = (5×5)×(5×5)×(5×5) = ... = 3×25 = ... = 2(35) by iteration.
And exponentiation is not reörderable, neg. 23 > 32, because multiplication has no generative unit, itself.
(We also note that multiplication has an 'exterior-log-property' mapping one-to-one-and-'bounding'-onto addition: the idempotent multiplicative unit "identity element" maps to the additive null "identity"; but the idempotent "zero element" log 0 = -∞ "negative infinity-bound" is singly-exceptional not included in the additive.)
This is what we've learned from addition: and deduced its operational structures (later eg. prime factors in multiplication).
We find this standard 'Left Multiplication', means, repetitions of additions: This remains true even in higher mathematics where the addition or multiplication processes are defined specially for an application: The consecutivity discovered by incremental additions, and repetitivity discovered for multiplications, being iterations firstly, remain intact under specific larger inclusive applications:--
Whenever an additive or multiplicative property is invented on a set, a grouped universe, the elements of the set may be discerned by method to include the same possible totals (counts) we've discovered here, eg. later the gut (matrix) [5] = 5; and our basic operations remain as superimposable co-operations, and our basic numbers remain as superset thereto, eg. modulo numbers, or extensible sets, eg. {R+R*√2} .... Or it may remain implicitly true but undiscernably, and only our possible totals (counts) applicable multiplicatively (to indicate the count of repetitions defined on the set) ... thus always defining multiplication on any (every) additive-ruled group, as, the multiplication-by (iteration-per) our totals (counts), whether or not multiplication be defined completely for the elements of the set, itself, eg. sets of algebraic operations not identified as counting-numbers.
The reason for this statement is that, the addition intrinsic to our set of totals (counts) is the same effectual method as the addition ruling any other such set; and co-operates consistently, whether or not our totals (counts) are individually and all and distinguishably included in that set. And thus multiplication is also already defined if addition is so defined on a set, (NB. this is at odds for order in teaching a type of sets called, rings, having purportedly two, operations: when we've here already shown one extends from the other); whence any explicit (to-be) definition of multiplication must fit this, same, rule, even as to potentially identify elements of the set as our totals (counts),- and is not a second operation but the iterativity extension of the first already extant....
Thus we produce the basic arithmetic of integers, which co-applies to every subsequent group having an additive operation: An integer times such a group element means repeated additions (the operation, it supports), eg. 3(a) = a+a+a, where a+a is its own a-addition.
And for exception a counterexample, not disproof: Consider the finite set of Modulo-7 (N7 aka Z7) where multiplication sort-of-has a generative unit: -for all but zero it does:- 3, 23 = 2, 33 = 6, 43 = 4, 53 = 5, 63 = 1 mod 7, having skipped 0; Then compare 32 = 1 mod 7, which likewise = 3(32), which = 3×23 = 3(23), but still 32 ≠ 23 directly, but in some underlying sense appear as co-roots meeting above.
Mathematics has natural-language order to its concepts;-- and conversely.
A premise discovery under the title,