# What We Want To Calculate The aim of this book is not to make arithmetic vague. It is to make clear what arithmetic is calculating. Standard arithmetic calculates the case where units do not interact: ```text k = 0 ``` Ultrareal arithmetic asks what changes when the relation is not zero. ## The Basic Objects We begin with positive square-forms: ```text U = u^2 V = v^2 ``` The visible values are `U` and `V`. The inner values are `u` and `v`. ## The Basic Question The central question is: ```text what is the value of U with V? ``` Not merely: ```text how many units are present? ``` but: ```text what does their relation contribute? ``` ## The Relation Term The general two-term calculation is: ```text U +_{k} V = u^2 + v^2 + 2kuv ``` where `k` records the relation. Important cases: ```text k = 1 aligned joining k = 0 non-interaction or orthogonality k = -1 opposition ``` So: ```text aligned: U +_{1} V = (u + v)^2 standard: U +_{0} V = u^2 + v^2 opposed: U +_{-1} V = (u - v)^2 ``` ## What This Lets Us Calculate This framework lets us calculate: ```text standard arithmetic as non-interaction joined addition as aligned interaction opposition and cancellation Pythagorean addition angle-dependent addition trigonometric addition laws repeated units split-and-rejoin cases rotated negative values rotated infinity ``` The point is not that every situation uses the same `k`. The point is that ordinary arithmetic silently sets `k = 0`. Ultrareal arithmetic makes the relation visible.