User:Numiri/KinderCalculus/Like Terms, Distributive Identities

Combining Like Terms (Nevadab)
In traditional algebra, "like terms" refer to addition of terms with similar powers like 3x^2 + 4x^2 = 7x^2. Here we consider a more general "like terms" criteria to include those with different powers but can be combined via multiplication instead of addition. We call this principle "Nevadab", which is an initialism for "neighboring-verbs, add depth, alike bases". The "b" in Nevadab is silent. The odd word is a poor attempt at forcing the initialism into something pronounceable like "nevada".

Under the Nevadab criteria, we can combine a^3 * a^4 into a^7. Combining these terms involve 3 steps:   Check that the verbs are neighboring: the verbs +, *, ^ form a ascending complexity sequence of plus, self-plus, self-self-plus. Each successive pair are neighboring verbs. + and ^ are distant verbs, not neighboring.  Check that the bases are alike: the bases in the 2 terms must be the same.  Add depth: to combine the terms, simply add the depth  The diagram below illustrate the generalization in an operator precedence diagram:

.                          3a + 4a = 7a. |                                             .                                |                                              .                                |                                              .                          +-+-+                                        .                          | \       / |                                        .                          |  o--*--o  |. | /      \ |       3    3                           .            3a * 4a  -+  equals   +- a  + b. (not neighbors)     |           |      (irreducible). | *--o--   |. +-+-+                                       .                                |                                              .                                |                                              .                                |                                              .                            3    4    7                                        .                           a  * a  = a.

Distributive Identities (Evert)
Everts are distributive rules. They are called evert for their inside-out effects on the layering. Below we generalize the distributive rule beyond a(b+c) = ab + ac. This generalization simplifies the many distributive rules into one -- greatly simplifying algebra!

Mathematical expressions have a syntactic structure that can be drawn as a tree. For example, .                          3             3                                     .            3  3            \           /                                      .           a  b      is      ^ -- * -- ^. /          \                                      .                           a             b. . and. .                           a. 3           \                                                 .            (ab)      is      * -- ^ -- 3. /                                                .                            b. . In the picture below, the center box represents this syntactic parse tree of the distributive expression: .                                                                              .                       3(a + b) = 3a + 3b. .                               |                                              .                                |         u + v     u   v. same as -->         |         -  =  - + -. right             |           d       d   d. |                                             .                       \        |        /                                     .                        \       |       /                                      .                         \      |      /                                       .                          +-+-+                                        .                          | \       / |                                        .                          |  o--*--o  |. | /      \ |                                        .                          |           |       3    3                           .            3a * 3b  -+  equals   +- a  + b. (uninteresting)     |           |      (irreducible). | \        |                                        .                          |   *--o--  |. | /        |                                        .                          +-+-+                                        .                         /      |      \                                       .                        /       |       \                                      .                       /        |        \                                     .                                |                                              .                    uv          |           uv       d__   d__. log u + log v =  o         |          o  =  V u * V v. b      b        b           |           d. |                                             .                                                                               .                              3     3   3                                      .                          (ab)  =  a   b. .                                                                              . notes:. 1. the diagonals are distributive relations against the leaf & lattice (u v). 2. log is nearly evertable, but not quite, with + replacing *. . in the box,. o  =  repeater verb. . u,v = leaf, lattice. a,b = bases. d  =  depth. .
 * = seed verb.

Here is the same information in tabular form: