# Ultrareal Numbers An ultrareal number is a positive square-form: $$ U=u^2,\qquad u\ge0. $$ The number is $U$. Its inner magnitude, or natural inner state, is $u$. The ultrareal domain is the positive real line with zero included: $$ \mathbb U=\{u^2\mid u\in\mathbb R_{\ge0}\}=[0,\infty). $$ The map from inner magnitude to visible value is: $$ q:\mathbb R_{\ge0}\to\mathbb U,\qquad q(u)=u^2. $$ Because the inner magnitude is constrained by $u\ge0$, this representation is unique. For every $U\in\mathbb U$ there is exactly one natural inner state: $$ u=\sqrt U. $$ ## Value And Inner Magnitude The notation separates two roles: $$ \begin{aligned} \text{visible value:}\quad &U,\\ \text{inner magnitude:}\quad &u. \end{aligned} $$ The visible value is the square-value handled by arithmetic. The inner magnitude is the lower-case value through which relation terms are formed. This separation is the structural move of the book. Ordinary arithmetic normally works directly with visible values. Ultrareal arithmetic keeps the inner magnitude available, so a sum can expose terms that depend on how the parts meet. ## Positivity Every ultrareal lies in the positive real layer with zero included: $$ U=u^2\ge0. $$ Its modulus is the value itself: $$ |U|=U. $$ It vanishes only at zero: $$ |U|=0\quad\Longleftrightarrow\quad U=0. $$ In this square-form sense, the ultrareal layer is positive definite with zero included: no member of $\mathbb U$ is below zero, and only zero has zero modulus. ## Equality Two ultrareals are equal exactly when their visible square-forms are equal: $$ U=V \quad\Longleftrightarrow\quad u^2=v^2. $$ Since $u,v\ge0$, this is equivalent to equality of their natural inner states: $$ U=V \quad\Longleftrightarrow\quad u=v. $$ ## Zero Zero is the ultrareal whose inner magnitude is zero: $$ 0=0^2. $$ It is the additive identity for ordinary visible addition and for relation-aware addition, because any relation term containing its inner magnitude vanishes. ## No Negative Ultrareals There are no negative ultrareals. Let $U$ be a nonzero ultrareal: $$ U=u^2,\qquad u>0. $$ For every allowed inner magnitude $r\in\mathbb R_{\ge0}$, $$ r^2\ge0. $$ Therefore no allowed inner magnitude can produce $-U$, because $-U<0$ in ordinary signed notation. The symbol $-U$ may still be useful as ordinary notation, but it is not a member of $\mathbb U$. The conclusion is: $$ \mathbb U=[0,\infty),\qquad \mathbb U\cap(-\infty,0)=\varnothing. $$ Signs belong to presentation, comparison, direction, cancellation, or relation. They do not name negative ultrareal values. When opposition must be notated, the symbol $i$ may be adjoined to the real notation with the rule $i^2=-1$. ## Oriented Presentations A lone ultrareal does not require orientation. Its natural inner state is $u$. In problems where orientation matters, one may introduce an oriented inner presentation: $$ z=ue^{i\alpha}. $$ This is not a new ultrareal value. It is a presentation of the same inner magnitude with an added orientation parameter. Its reverse-oriented presentation is: $$ z^*=ue^{-i\alpha}. $$ The star does not mean an unexplained extra operation. It means return: the same inner magnitude with the opposite orientation. The ultrareal value recovered from the oriented presentation is: $$ zz^* =(ue^{i\alpha})(ue^{-i\alpha}) =u^2. $$ Thus self-orientation cancels in a single ultrareal. Relative orientation matters only when two or more inner states are added or compared.