prerequisite

Basic definitions of fraction is a prerequisite here. This concept can be covered at a Kindergarten level. We are not requiring any fraction arithmetic yet. The terms "numerator" and "denominator" can be a mouthful, so we opt for "tops" and "bottoms" instead. This adds more context to the term "improper fractions" (now called "top heavy") and "proper fractions" (now called "bottom heavy").

nouns, verbs, adjectives

nouns

simple nouns are numbers and variables.

variables: are "blank numbers", or containers for amounts such as your age or test score. Variables are written as letters or shapes, such as

${\displaystyle {\begin{aligned}x&={\textrm {Heather's\ age}}\\\heartsuit &={\textrm {Chandler's\ age\ }}=x+3\\\square &={\textrm {Chandler's\ testscore}}\end{aligned}}}$

compound nouns, aka layers, are combinations of verbs and nouns, and they are usually written with parentheses or shapes. Examples of nouns (simple and compound) are: 1, x, ${\displaystyle \heartsuit }$, x + 1, y * ( x + 1 ).

anti-nouns are negative, reciprocal, log, and root

phrases are mathematical expressions. We prefer this term for syllabic brevity.

forward-nouns are positive numbers, integers (not reciprocals), and leaves

verbs

verbs are operations like +, -, *, /, ^, 0--, --0, cos(), etc.

anti-verbs are minus, divide, log() and root(,)

forward-verbs are plus, times, powers

seed & repeater verbs: in the context of self-plus and self-times, the primitive operation is called the seed verb and the resulting operation from the repetition is the repeater verb. For instance, in 3x = x + x + x, the seed verb is "+" and the repeater verb is times. Likewise, x^3 = x * x * x has the seed verb "*" and repeater verb as "^".

The table below shows that most arithmetic traces back to the + verb.

symbol	verb	common names	example
−	anti-plus	minus, subtract	5 − 3 = 2
*	self-plus	times, multiply	5 * 3 = 15
/	santi-plus, short for "self-anti"	divide	15 / 3 = 5
^	self-times, or self-self-plus	power, exponential	5 ^ 3 = 125 the symbol ^ is only seen in computer programs. In math, this is written as 53 = 125
${\displaystyle \circ \!{\frac {\quad }{\quad }}}$	left-divide, or anti-self-self-plus left (no acronyms :-)	logarithm	${\displaystyle \circ \!{\frac {125}{5}}=3}$
${\displaystyle {\frac {\quad }{\quad }}\!\circ }$	right-divide, or anti-self-self-plus right	root	${\displaystyle {\frac {125}{3}}\!\circ =5}$

The Blead section of this book explains these verbs in detail.

adjectives

adjectives are the negative & reciprocal noun modifier. They can be introduced as a reflection of a noun about the verb's center. (the center of a verb + or * is the group-theoretic identity 0 or 1 respectively)