A New Arithmetic
Numbers Interact
Numbers Interact
2026-05-26
To my daughter, may she remember that interaction is exchange.
The first claim of this book is simple:
\[ 1 + 1 \text{ is not necessarily only } 2. \]
There is also an interaction term, and it need not always be zero.
This depends on a choice in mathematics that is not as obvious as it first appears.
If two real numbers are positive, with zero included, they can be written as square-forms:
\[ U=u^2,\qquad V=v^2. \]
If we consider the sum \(U+V\) and read only the visible square-values, the usual answer is immediate:
\[ U+V=u^2+v^2. \]
That is the standard visible reading. It counts the two square-values.
But the same terms also carry their square-form definition:
\[ U=u^2. \]
When the same \(+\) is read with the square-forms in view, \(U+V\) is again a positive number, and as such it can be written as a square-form. The inner expression of the sum is:
\[ u+v. \]
Therefore:
\[ U+V=(u+v)^2=uu+uv+vu+vv. \]
Since \(u^2=uu\) and \(v^2=vv\), the question becomes: what is
\[ uv+vu? \]
Those middle terms are the relation \(d(\cdot,\cdot)\) between the parts, the interaction term that the ordinary visible reading does not take into account. Numbers treated this way are ultrareals.
The standard arithmetic that we love is not wrong. It is recovered when the interaction term is not taken into account. This book describes a broader arithmetic in which a number is first written as a positive square-form:
\[ N=n^2. \]
The value \(N\) is always nonnegative. The lower-case \(n\) is the inner value. The upper-case \(N\) is the visible square-value.
Later, after the definition and examples are in place, the book will show what an ultrareal number is and why a real number from everyday context may be read as a density.
First, the book defines ultrareal numbers rigorously as positive square-forms, with zero included:
\[ U=u^2,\qquad u\ge0. \]
Second, it proves basic facts about them. In particular, it proves that the term-type-aware sum of two ultrareals is another ultrareal. The same operation \(+(\cdot,\cdot)\) is read through the terms supplied to it: lower-case terms add as inner states, while upper-case ultrareals return the square-form determined by those inner states:
\[ U+V=(u+v)^2. \]
Third, the book keeps algebraic assumptions explicit. Commutativity, associativity, and distributivity may be used when the arithmetic of the particular case supplies them. They are not imposed as extra requirements before the relevant terms have been specified. When the relevant inner product distributes, the square can be expanded:
\[ (u+v)^2=u^2+uv+vu+v^2. \]
The middle terms are not decoration. They are the relation \(d(\cdot,\cdot)\) between the parts:
\[ d(U,V)=uv+vu. \]
When that relation vanishes, ordinary arithmetic is recovered.
Finally, the book introduces orientation. To notate turn and opposition, we adjoin the symbol \(i\), with \(i^2=-1\), and use Euler’s identity. Here the symbol \(-1\) is not being used as in \(-1<0\), since there are no negative ultrareals. It is a notation convenience, a useful geometric trick on the map of presentations. In this role, \(i\) belongs on the presentation side of the notation: it records a turn. It is not itself an ultrareal. We know \(\sqrt{-1}\) is not ultrareal, since \(i^2=-1\) is not a positive number.
Once orientation is available, numbers themselves can be rotated. A half-turn looks the same from the square-form: front and back return the same value. A quarter-turn is different: it is sideways, orthogonal to the ordinary positive line.
That sideways orthogonality gives meaning to the return product:
\[ n n^*. \]
The symbol \(n^*\) is not an ad hoc complex conjugate. It is the reverse-oriented inner state, the return needed to recover the positive value from a rotated presentation. This is how signs, opposition, orthogonality, and ordinary arithmetic fit inside one positive square-form account.
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:
\[ u=\sqrt U. \]
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.
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.
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 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.
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 \(i^2=-1\).
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.
Ultrareal arithmetic uses one addition operation, \(+(\cdot,\cdot)\), read through the terms supplied to it.
Lower-case symbols name inner magnitudes, inner states, or presentations of inner states when relation data is present. Upper-case symbols name visible ultrareal values:
\[ U=u^2,\qquad V=v^2,\qquad u,v\ge0. \]
For natural scalar inner states, lower-case addition is ordinary inner-state addition:
\[ +(u,v)=u+v. \]
When upper-case ultrareals are added, the result is the ultrareal determined by the corresponding lower-case sum. Since \(U=u^2\) and \(V=v^2\):
\[ +(U,V)=U+V=(+(u,v))^2. \]
In the natural scalar case:
\[ +(U,V)=U+V=(u+v)^2. \]
Thus the printed sign is the same. The term type determines how the operation is read.
The natural scalar upper-case sum must remain inside \(\mathbb U\).
Let:
\[ U,V\in\mathbb U,\qquad U=u^2,\qquad V=v^2. \]
Then:
\[ u,v\in\mathbb R_{\ge0}. \]
The nonnegative reals are closed under ordinary addition, so the inner-state sum:
\[ x=+(u,v)=u+v \]
also lies in the nonnegative reals:
\[ x\in\mathbb R_{\ge0}. \]
By the definition of an ultrareal number, \(x^2\in\mathbb U\). Call this ultrareal \(X\):
\[ X=x^2. \]
This proves that adding two ultrareals gives another ultrareal. Since \(+(U,V)\) is the ultrareal determined by \(+(u,v)\):
\[ +(U,V)=U+V=X. \]
Therefore:
\[ U+V=X=x^2=(u+v)^2. \]
Nothing new is being added to the square-form. The upper-case result follows from the lower-case sum and the definition \(U=u^2\).
The operator is still \(+\). The operands determine which layer is being used.
Expanding this square gives the interaction terms. If the inner-state product distributes, then:
\[ (u+v)^2=(u+v)(u+v)=u^2+uv+vu+v^2. \]
The middle terms are ordered. They are the interaction descriptor:
\[ d(U,V):=uv+vu. \]
Thus:
\[ U+V=u^2+d(U,V)+v^2. \]
When the descriptor is being displayed, the same sum may also be written:
\[ U\,d\,V:=uu+uv+vu+vv. \]
Equivalently:
\[ U\,d\,V=U+V=u^2+d(U,V)+v^2. \]
The symbol \(d\) in this notation marks that the interaction descriptor is being included.
In general, there is no need to assume:
\[ uv=vu. \]
Nor is there a need to assume that every many-term expression is associative without stating the inner-state algebra that makes it so. Commutativity and associativity are available when the arithmetic of the particular case supplies them, but they are not imposed by the bare ultrareal definition.
Standard arithmetic is recovered when the descriptor vanishes:
\[ d(U,V)=0. \]
Then:
\[ U+V=u^2+v^2. \]
This is the non-interaction case. The visible values are counted together, and no interaction term remains.
This is not a second addition operation. It is the same upper-case sum read with zero relation data.
For unit values in this case:
\[ 1+1=2. \]
In the common commutative scalar case:
\[ uv=vu. \]
Full alignment gives:
\[ d(U,V)=uv+vu=2uv. \]
Then:
\[ U+V=u^2+v^2+2uv=(u+v)^2. \]
For unit values in the aligned case:
\[ 1+1=(1+1)^2=4. \]
This is why the opening claim is precise:
\[ 1+1 \text{ is not necessarily only } 2. \]
The printed expression is the same, but the term data and relation are not the same. Separated visible units recover \(2\). Aligned unit magnitudes produce \(4\).
Complete opposition in the same scalar setting gives:
\[ d(U,V)=-2uv. \]
Then:
\[ U+V=u^2+v^2-2uv=(u-v)^2. \]
Opposition can reduce a sum. It cannot create a negative ultrareal inside this bounded relation scale.
If \(u=v\), complete opposition gives:
\[ U+U=0 \qquad(d(U,U)=-2u^2). \]
This is cancellation to the zero boundary, not passage into negative ultrareal value.
An interaction descriptor may encode angular, hyperbolic, weighted, tangential, or otherwise structured relation data. It need not be commutative, scalar, or associative in advance. The formal requirement is admissibility: the result must remain an ultrareal.
The important order in the natural scalar case is the closure proof:
\[ u,v\in\mathbb R_{\ge0} \quad\Longrightarrow\quad x=u+v\in\mathbb R_{\ge0} \quad\Longrightarrow\quad X=x^2\in\mathbb U. \]
Then the upper-case sum can be written:
\[ U+V=X=(u+v)^2. \]
Only then, when the inner-state product distributes:
\[ d(U,V):=uv+vu. \]
Thus \(d(U,V)\) is not an extra number placed beside addition. It is the interaction term exposed by expanding the square of the added inner states.
For fully aligned addition in a commutative and associative inner-state setting, addition has the familiar laws:
\[ U+V=(u+v)^2=(v+u)^2=V+U. \]
For three aligned terms:
\[ (U+V)+W =U+(V+W) =(u+v+w)^2. \]
Without those properties, ordering and parentheses remain part of the data of the expression. The many-term form makes that structure explicit.
The previous chapter defined term-type-aware addition. Before the angular and exponential tools used in later chapters can rest on a rigorous foundation, scalar multiplication must be defined.
Let \(U=u^2\) and \(V=v^2\) be ultrareals with scalar natural inner states. Their product is:
\[ U \cdot V = (uv)^2. \]
The inner magnitude of a product is the product of the inner magnitudes. Since \(u,v\ge0\), the product \(uv\ge0\), so \((uv)^2\in\mathbb U\). Multiplication is closed.
In the ordinary scalar case, multiplication is commutative:
\[ U\cdot V=(uv)^2=(vu)^2=V\cdot U. \]
In the ordinary scalar case, multiplication is associative:
Let \(W=w^2\). Then:
\[ (U\cdot V)\cdot W=((uv)w)^2=(u(vw))^2=U\cdot(V\cdot W). \]
Multiplicative identity:
The ultrareal \(1=1^2\) satisfies:
\[ 1\cdot U=(1\cdot u)^2=u^2=U. \]
Absorption at zero:
\[ 0\cdot U=(0\cdot u)^2=0. \]
Whether multiplication distributes over addition depends on what the relevant addition does to inner magnitudes.
For the natural scalar ultrareal addition \(U+V=(u+v)^2\), the inner magnitude of a sum is the sum of inner magnitudes. Under this term-type rule the proof proceeds at the inner magnitude layer.
The inner magnitude of \(V+W\) is \(v+w\).
Therefore the inner magnitude of \(U\cdot(V+W)\) is:
\[ u\cdot(v+w)=uv+uw. \]
The inner magnitude of \(U\cdot V\) is \(uv\). The inner magnitude of \(U\cdot W\) is \(uw\). The inner magnitude of \((U\cdot V)+(U\cdot W)\) is:
\[ uv+uw. \]
Both sides square the same inner magnitude. Therefore, for the natural scalar addition:
\[ U\cdot(V+W)=U\cdot V+U\cdot W. \]
For a general descriptor \(d\), the inner magnitude of \(V+W\) is \(\sqrt{v^2+d(V,W)+w^2}\), which is not \(v+w\) unless \(d(V,W)=2vw\). In that case, the inner magnitude of \(U\cdot(V+W)\) is \(u\sqrt{v^2+d(V,W)+w^2}\), and the inner magnitude of \(U\cdot V+U\cdot W\) depends on \(d(UV,UW)\). Equality requires:
\[ d(UV,UW)=U\cdot d(V,W). \]
Here \(U\cdot d(V,W)\) is ordinary scalar scaling of the descriptor by the visible value \(U=u^2\). This is a compatibility condition on the descriptor. It holds for the angular descriptor \(d(V,W)=2vw\cos\Delta\) when the angle \(\Delta\) is preserved under scaling by \(U\). It is not automatic and should not be assumed without verification for a given \(d\).
For \(U=u^2\) and a nonnegative integer \(n\), define:
\[ U^n=(u^n)^2. \]
The inner magnitude of \(U^n\) is \(u^n\). Since \(u\ge0\), \(u^n\ge0\) for all \(n\ge0\), so \(U^n\in\mathbb U\).
Base cases: \(U^0=(u^0)^2=1\) and \(U^1=u^2=U\).
Power law:
\[ U^n\cdot U^m=(u^n)^2\cdot(u^m)^2=(u^n\cdot u^m)^2=(u^{n+m})^2=U^{n+m}. \]
Power of a product:
\[ (U\cdot V)^n=((uv)^2)^n=((uv)^n)^2=((u^n)(v^n))^2=U^n\cdot V^n. \]
Integer powers being defined, a power series in \(U\) is now meaningful:
\[ \sum_{n=0}^{\infty}a_n U^n=\sum_{n=0}^{\infty}a_n(u^n)^2, \]
provided the series converges. Applied to the standard exponential coefficients:
\[ e^U=\sum_{n=0}^{\infty}\frac{U^n}{n!}=\sum_{n=0}^{\infty}\frac{u^{2n}}{n!}=e^{u^2}. \]
This is the value-layer exponential: the standard real exponential evaluated at the visible value. Its output is an ultrareal.
A second exponential lives at the presentation layer. If the symbol \(i\) is adjoined with \(i^2=-1\), the power series may be evaluated at a purely imaginary argument \(i\theta\):
\[ e^{i\theta}=\sum_{n=0}^{\infty}\frac{(i\theta)^n}{n!}. \]
This series converges absolutely for every real \(\theta\). Its value is a complex number of modulus one. It is not an ultrareal. It is an orientation — a unit presentation carrying direction without inner magnitude other than one.
The two exponentials belong to different layers:
\[ \begin{aligned} \text{value layer:}\quad &e^U=e^{u^2},\qquad U\in\mathbb U,\\ \text{presentation layer:}\quad &e^{i\theta}=\cos\theta+i\sin\theta,\qquad\theta\in\mathbb R. \end{aligned} \]
The derivation of Euler’s formula — that \(e^{i\theta}=\cos\theta+i\sin\theta\) — is given in Chapter 004. Multiplication of ultrareals is not required for that derivation. What is required is multiplication under the power series with the single rule \(i^2=-1\). That rule was introduced as the definition of the adjoined symbol, not as a consequence of ultrareal arithmetic.
Relation-aware addition must remain inside the ultrareal domain, the positive real line with zero included:
\[ \mathbb U=[0,\infty). \]
The previous chapter proved closure for the term-type-aware ultrareal sum. If:
\[ U=u^2,\qquad V=v^2, \]
with \(u,v\in\mathbb R_{\ge0}\), then:
\[ u+v\in\mathbb R_{\ge0}. \]
Therefore:
\[ U+V=(u+v)^2\in\mathbb U. \]
That is the basic closure rule for ultrareal addition. If the inner-state product distributes, the same value can be read as the ordered expression:
\[ u^2+uv+vu+v^2\in\mathbb U. \]
Admissibility becomes explicit when the same sum is represented by descriptor data.
When the ordered cross terms are written as the descriptor,
\[ d(U,V)=uv+vu, \]
the sum becomes:
\[ U+V=u^2+d(U,V)+v^2. \]
In scalar cases, the descriptor is admissible for this addition when:
\[ u^2+d(U,V)+v^2\ge0. \]
Equivalently:
\[ U+V\in\mathbb U. \]
In the bounded angular or field-alignment scale, the descriptor satisfies:
\[ -2uv\le d(U,V)\le 2uv. \]
Closure is automatic:
\[ u^2+d(U,V)+v^2\ge(u-v)^2\ge0. \]
The smallest value occurs at complete opposition:
\[ d(U,V)=-2uv. \]
Then:
\[ U+V=(u-v)^2. \]
Thus opposition can cancel equal inner magnitudes to zero, but it cannot push an ultrareal result below zero while remaining in the bounded scale.
For nonzero ultrareals, exact cancellation in the bounded scale has only one form.
Let:
\[ A=a^2,\qquad B=b^2,\qquad a,b>0. \]
If:
\[ A+B=0, \]
then:
\[ a^2+d(A,B)+b^2=0. \]
Solving for the descriptor gives:
\[ d(A,B)=-(a^2+b^2). \]
By the arithmetic-geometric mean inequality,
\[ a^2+b^2\ge2ab, \]
with equality only when \(a=b\). Therefore, inside the bounded descriptor scale \(-2ab\le d(A,B)\le2ab\), exact cancellation requires:
\[ a=b,\qquad d(A,B)=-2ab. \]
Complete opposition is the only bounded relation that cancels two nonzero ultrareals, and it cancels them only when their inner magnitudes are equal.
For many ultrareals,
\[ U_i=u_i^2,\qquad i=1,\ldots,n, \]
the many-term inner-state sum is:
\[ x=u_1+\cdots+u_n. \]
Since every \(u_i\) is nonnegative, \(x\in\mathbb R_{\ge0}\), so:
\[ U_1+\cdots+U_n=(u_1+\cdots+u_n)^2\in\mathbb U. \]
When this value is reduced to scalar descriptor data, the pairwise descriptor table \(D=(d_{ij})\) records the interaction terms:
\[ \boxed{ U_1+\cdots+U_n = \sum_i u_i^2 +\sum_{i<j}d_{ij} } \]
The expansion is determined by the ordering, products, and parentheses of the chosen inner-state algebra.
When the inner-state product is distributive and the order is explicit, the pairwise descriptors are:
\[ d_{ij}=d(U_i,U_j):=u_i u_j+u_j u_i. \]
There is no separate interaction descriptor on the diagonal in this sum. The visible square terms are already present as \(u_i^2\). The off-diagonal entries record interactions between distinct parts:
\[ d_{ij}=d_{ji} \]
In matrix form, with
\[ \mathbf u=(u_1,\ldots,u_n)^{\mathsf T}, \]
the resulting value is:
\[ \sum_i u_i^2+\sum_{i<j}d_{ij}. \]
The descriptor table is admissible for the given addition when:
\[ \sum_i u_i^2+\sum_{i<j}d_{ij}\ge0. \]
In the common scalar matrix representation, this may be rewritten as a quadratic form \(\mathbf u^{\mathsf T}K\mathbf u\). That representation is admissible for every real inner state when:
\[ K\succeq0. \]
It is admissible for every nonnegative ultrareal inner state when it is copositive on the positive cone:
\[ \mathbf u\ge0 \quad\Longrightarrow\quad \mathbf u^{\mathsf T}K\mathbf u\ge0. \]
For a concrete sum, the state-specific condition is enough. For a whole class of sums, the table itself must satisfy the appropriate positivity condition.
Ordinary arithmetic is recovered when all off-diagonal relations vanish:
\[ d_{ij}=0\qquad(i\ne j). \]
Then:
\[ U_1+\cdots+U_n =u_1^2+\cdots+u_n^2. \]
Fully aligned addition is recovered when every off-diagonal descriptor is the commutative scalar descriptor:
\[ d_{ij}=2u_i u_j\qquad(i\ne j). \]
Then:
\[ U_1+\cdots+U_n=(u_1+\cdots+u_n)^2. \]
The arithmetic point is simple: addition is not only a rule for collecting visible values. It can also be a rule whose term data exposes relation through an admissible descriptor structure.
Angular relation is one important setting.
A lone ultrareal has natural inner state \(u\). When orientation is needed, an inner state may be presented as:
\[ a=ue^{i\alpha}. \]
Its reverse-oriented presentation is:
\[ a^*=ue^{-i\alpha}. \]
The returned density is \(aa^*=u^2\).
For two oriented presentations,
\[ a=ue^{i\alpha},\qquad b=ve^{i\beta}, \]
their reverse-oriented presentations are:
\[ a^*=ue^{-i\alpha},\qquad b^*=ve^{-i\beta}. \]
The relevant quantity is not either angle alone. It is their relative difference:
\[ \Delta=\alpha-\beta. \]
The angular descriptor is:
\[ d(U,V)=2uv\cos\Delta. \]
The resulting density is:
\[ |a+b|^2 = (a+b)(a^*+b^*) =aa^*+ab^*+ba^*+bb^* =u^2+v^2+2uv\cos\Delta. \]
In this angular case, the interaction descriptor is:
\[ d(U,V):=ab^*+ba^*=2uv\cos\Delta. \]
This is upper-case term-type-aware addition:
\[ U+V=u^2+d(U,V)+v^2. \]
When:
\[ \Delta=0, \]
then:
\[ \cos\Delta=1. \]
So:
\[ U+V=(u+v)^2. \]
This is aligned addition.
When:
\[ \Delta=\frac{\pi}{2}, \]
then:
\[ \cos\Delta=0. \]
So:
\[ U+V=u^2+v^2. \]
This is the Pythagorean case. The relation term vanishes.
If:
\[ c^2=u^2+v^2, \]
then:
\[ c=\sqrt{u^2+v^2}. \]
The Pythagorean theorem is therefore recovered as zero angular relation. In ordinary geometry that condition is called orthogonality.
When:
\[ \Delta=\pi, \]
then:
\[ \cos\Delta=-1. \]
So:
\[ U+V=(u-v)^2. \]
Opposition is a relative difference between positive inner magnitudes. It is not a negative ultrareal.
The same expression recovers the law of cosines. For the sum of oriented inner states:
\[ |u+ve^{i\Delta}|^2 =u^2+v^2+2uv\cos\Delta. \]
For the side opposite an angle \(C\), the conventional triangle formula is:
\[ c^2=u^2+v^2-2uv\cos C. \]
The sign difference is an orientation convention. The content is the same: the square-value depends on the relation between the parts.
The symbol \(i\) can be introduced rigorously by adjoining it to the reals with the rule:
\[ i^2=-1. \]
This gives the complex plane:
\[ \mathbb C=\{a+bi\mid a,b\in\mathbb R\}. \]
Define the exponential, cosine, and sine by their convergent power series:
\[ e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}, \]
\[ \cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}, \]
and:
\[ \sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}. \]
These series converge absolutely for every complex input, so separating the even and odd terms is legitimate.
For real \(\theta\):
\[ \begin{aligned} e^{i\theta} &=\sum_{n=0}^{\infty}\frac{(i\theta)^n}{n!}\\ &=\sum_{n=0}^{\infty}\frac{(i\theta)^{2n}}{(2n)!} +\sum_{n=0}^{\infty}\frac{(i\theta)^{2n+1}}{(2n+1)!}\\ &=\sum_{n=0}^{\infty}(-1)^n\frac{\theta^{2n}}{(2n)!} +i\sum_{n=0}^{\infty}(-1)^n \frac{\theta^{2n+1}}{(2n+1)!}\\ &=\cos\theta+i\sin\theta. \end{aligned} \]
This is Euler’s formula:
\[ e^{i\theta}=\cos\theta+i\sin\theta. \]
At \(\theta=\pi\):
\[ e^{i\pi}=-1, \]
which gives the standard notation for complete opposition. At \(\theta=\pi/2\):
\[ e^{i\pi/2}=i. \]
Thus trigonometry is compatible with the ultrareal reading: angles describe how inner presentations differ before a positive square-value is evaluated, and the adjoined symbol \(i\) gives a rigorous notation for those turns.
The natural inner state of an ultrareal is \(u\):
\[ U=u^2,\qquad u\ge0. \]
Rotation is not part of that basic definition. To notate opposition and turns, one may adjoin a symbol \(i\) to the reals with:
\[ i^2=-1. \]
Then orientation can be written as an optional presentation of the inner state:
\[ z=ue^{i\alpha}. \]
The reverse-oriented presentation is:
\[ z^*=ue^{-i\alpha}. \]
The star denotes return orientation. It is not introduced as an ad hoc complex conjugate. The ultrareal value recovered from the oriented presentation is:
\[ zz^*=u^2. \]
A single oriented presentation therefore returns to the same positive ultrareal. Orientation becomes operational when two or more presentations are added or compared.
The front presentation of an inner magnitude is:
\[ u. \]
The back presentation is the half-turn:
\[ -u=ue^{i\pi}. \]
Both give the same positive square-value:
\[ u^2=(-u)^2. \]
In that sense, the square-form looks the same from the front and the back.
The sideways presentation is the quarter-turn:
\[ iu=ue^{i\pi/2}. \]
Its raw square is:
\[ (iu)^2=-u^2. \]
This is not a negative ultrareal. It is the sideways square-presentation, the orthogonal direction exposed by adjoining \(i\). To recover density from a rotated presentation, use the return product:
\[ (iu)(iu)^*=u^2. \]
This is the meaning of \(n n^*\): an oriented inner state multiplied by its reverse-oriented return.
Two oriented inner states may differ by a half-turn:
\[ \Delta=\pi. \]
Then:
\[ \cos\Delta=-1. \]
Their sum value is:
\[ |u+ve^{i\pi}|^2 =u^2+v^2-2uv =(u-v)^2. \]
Both incoming ultrareals are positive. The opposition is the relative turn between their inner presentations.
If \(u=v\), the value cancels to zero:
\[ |u+ue^{i\pi}|^2=0. \]
The result is absence at the zero boundary, not a negative ultrareal.
The return product \(zz^*\) is not a convention for recovering a norm. Its structure is that of a standing wave.
A standing wave forms when two equal waves travel in exactly opposing directions. The propagation cancels. What remains is pure density at each point — energy without net flow, value without direction.
The oriented inner state \(z=ue^{i\alpha}\) presents inner magnitude \(u\) in direction \(\alpha\). Its return \(z^*=ue^{-i\alpha}\) presents the same inner magnitude in the opposing direction \(-\alpha\). Their product is:
\[ zz^*=(ue^{i\alpha})(ue^{-i\alpha})=u^2e^{i(\alpha-\alpha)}=u^2. \]
The orientations cancel. What remains is \(u^2\) — the ultrareal value, pure density, no net direction.
This is not cancellation to zero. The inner magnitude \(u\) is present in both \(z\) and \(z^*\). The opposed orientations cancel the directional component only. The value \(u^2\) survives precisely because it was never held in the orientation. It was always in the square-form.
The return partner of \(z\) is unique. For a given inner magnitude \(u\) and orientation \(\alpha\), there is exactly one presentation with the same magnitude and exactly opposing orientation: \(z^*=ue^{-i\alpha}\). Confronting \(z\) with its return is the unique way to recover \(u^2\) from an oriented presentation without further data.
A minus sign can describe a rotated presentation of a square-value. It does not name a negative member of \(\mathbb U\).
The adjoined symbol satisfies:
\[ i^2=-1. \]
In Euler notation, later derived from the power series, this is written:
\[ i=e^{i\pi/2}. \]
The symbol \(i\) marks the elementary quarter-turn whose square gives the minus-signed half-turn. Squaring doubles that turn:
\[ (iu)^2=(e^{i\pi/2}u)^2=e^{i\pi}u^2. \]
Ordinary notation writes the half-turn \(e^{i\pi}\) as \(-1\):
\[ (iu)^2=-u^2. \]
This expression is useful ordinary notation, but \(-u^2\) is not an ultrareal value. It is a half-turned square-presentation written with the conventional minus sign.
The ultrareal statement remains:
\[ u^2\in\mathbb U,\qquad -u^2\notin\mathbb U\quad(u>0). \]
A minus-signed presentation and opposed addition are different.
The expression:
\[ -u^2 \]
is a signed presentation of a single square-value outside \(\mathbb U\).
The expression:
\[ U+V=(u-v)^2 \qquad(d(U,V)=-2uv) \]
is an admissible addition of two positive ultrareals through opposition.
In the first case, the sign belongs to presentation. In the second case, the negative term belongs to the relation descriptor.
Neither case creates a negative ultrareal.
For an oriented presentation:
\[ z=ue^{i\alpha}, \]
the raw square is:
\[ z^2=u^2e^{i2\alpha}. \]
This raw square is generally an oriented square-presentation, not the ultrareal value itself. The positive ultrareal value is recovered by return:
\[ zz^*=u^2. \]
For two oriented presentations,
\[ a=ue^{i\alpha},\qquad b=ve^{i\beta}, \]
with reverse-oriented presentations:
\[ a^*=ue^{-i\alpha},\qquad b^*=ve^{-i\beta}, \]
their resulting positive value is:
\[ |a+b|^2 =(a+b)(a^*+b^*) =u^2+v^2+2uv\cos(\alpha-\beta). \]
The interaction descriptor in this oriented case is:
\[ d(U,V):=ab^*+ba^*. \]
The relative difference \(\Delta=\alpha-\beta\) determines the angular descriptor:
\[ d(U,V)=2uv\cos\Delta. \]
This is why \(\Delta\) is the right symbol for opposition and difference. It names the relation between presentations, not a hidden phase attached to every ultrareal.
Every ultrareal is a square-form: \(U=u^2\), \(u\ge0\). The square-form has a symmetry and a restriction.
The symmetry: the sign of the inner magnitude is invisible at the value layer.
\[ u^2=(-u)^2. \]
The two inner magnitudes \(+u\) and \(-u\) produce the same ultrareal. At the value layer, they cannot be distinguished. The square-form folds the real line: two points land on one.
The restriction: the natural inner state satisfies \(u\ge0\). Only the nonnegative sheet is in use.
Orientation resolves the fold. When the symbol \(i\) is adjoined with \(i^2=-1\), an inner state may be presented at any angle:
\[ z=ue^{i\alpha},\qquad u\ge0,\quad\alpha\in\mathbb R. \]
The return product always recovers the ultrareal value:
\[ zz^*=(ue^{i\alpha})(ue^{-i\alpha})=u^2=U. \]
Two successive unfoldings recover the familiar number systems from \(\mathbb U\).
Admit two orientations: front (\(\alpha=0\)) and back (\(\alpha=\pi\)).
Front presentation:
\[ z=ue^{i\cdot 0}=u. \]
Back presentation:
\[ z=ue^{i\pi}=-u. \]
Both present the same ultrareal \(U=u^2\). Their return products confirm this:
\[ u\cdot u^*=u^2,\qquad(-u)\cdot(-u)^*=(-u)(-u)=u^2. \]
The real line \(\mathbb R\) is \(\mathbb U\) unfolded into two orientation sheets, meeting at zero. Positive reals are front presentations of ultrareals. Negative reals are back presentations of the same ultrareal values viewed from the opposing direction. Zero folds to itself; it is the fixed point of the unfolding.
The standard set-theoretic containment \([0,\infty)\subset\mathbb R\) remains true. It describes position in the signed number line. It does not describe priority. In the ultrareal reading, \(\mathbb U\) is prior: the negative reals add no new values. They are \(\mathbb U\) viewed from the back.
Admit all orientations: \(\alpha\in[0,2\pi)\).
Each ultrareal \(U=u^2\) has a full circle of presentations:
\[ \{ue^{i\alpha}:\alpha\in[0,2\pi)\}. \]
These are exactly the complex numbers of modulus \(u\). The complex plane \(\mathbb C\) is \(\mathbb U\) unfolded into full orientation. Every complex number \(z=ue^{i\alpha}\) is an oriented presentation of the ultrareal \(u^2\). The origin is the fixed point of all unfoldings.
The standard account places \([0,\infty)\subset\mathbb R\subset\mathbb C\), each a subset of the next extension. The ultrareal account reverses the priority: \(\mathbb U\) is the ground, \(\mathbb R\) is the two-sheet unfolding of \(\mathbb U\), and \(\mathbb C\) is the full-circle unfolding. The containments as sets remain. The priority is inverted.
Each unfolding adds orientational richness. No unfolding adds new values.
A negative number does not represent a negative quantity at the ultrareal level. It is a back-facing presentation of a positive value. The minus sign is an orientation instruction written as if it were a value. The square-form absorbs it:
\[ (-u)^2=u^2. \]
The symbol \(i\) is the tool of the unfolding. Adjoining it does not enlarge the value domain. It records turns.
Mathematics is the map. \(\mathbb U\) is the closest the map gets to the territory.
This note does not belong at the beginning of the book. The formal construction comes first: ultrareals are positive square-forms, and the same symbol \(+\) is overloaded by layer. Lower-case operands name inner states. Upper-case operands name visible ultrareals.
The pedagogical consequence is simpler:
count the parts
notice the relation
Counting asks how many visible units are present. Relation-aware arithmetic asks what value is produced when the relation between the parts is kept visible.
Early arithmetic often treats addition as union:
put this group with that group
count the new group
That lesson is useful, but it is not the whole operation. Union says that groups are considered together. Relation asks what happens between the parts.
The distinction can be taught without making arithmetic harder. A learner can hold both ideas:
\[ 1+1=2 \]
for non-interacting units, and:
\[ 1+1=4 \]
for fully aligned unit magnitudes.
The first counts separated visible units. The second measures the square-value with full alignment included.
The same care applies to minus signs. A minus sign may mean removal, opposite direction, cancellation, comparison, or bookkeeping. Those are different uses, and they should not be collapsed into the claim that a negative object exists in the ultrareal layer.
In ultrareal notation, value remains positive:
\[ U=u^2,\qquad U\in\mathbb U. \]
Opposition belongs to relation:
\[ U+V=(u-v)^2 \qquad(d(U,V)=-2uv). \]
This lets subtraction, opposition, and cancellation be introduced honestly as operations or relations, not as mysterious negative things.
If arithmetic is learned only as inventory, the relation between parts becomes invisible. But relation is often the point: parts may align, interfere, cancel, or form a structure whose value is not captured by counting alone.
Arithmetic can count parts and still keep relation visible.
The book began with the claim:
\[ 1+1 \text{ is not necessarily only } 2. \]
That claim now has a formal meaning. If two unit ultrareals are added with zero relation, ordinary arithmetic is recovered:
\[ 1+1=2. \]
If they are added in full alignment, the result is:
\[ 1+1=4. \]
The difference is not a contradiction. It is a difference in term data and relation.
An ultrareal number is a positive square-form:
\[ U=u^2,\qquad u\ge0,\qquad U\in\mathbb U. \]
In everyday terms, it is a density-value:
\[ N=n^2. \]
When only visible values are counted, the recovered standard case gives:
\[ U+V=u^2+v^2. \]
That remains in \(\mathbb U\) because \(u^2+v^2\ge0\). With the square-form terms kept in view, the term-type-aware result remains in \(\mathbb U\): since \(u,v\in\mathbb R_{\ge0}\), the inner-state sum \(x=u+v\) is nonnegative, so \(X=x^2\) is an ultrareal. With \(X=U+V\):
\[ U+V=X=x^2=(u+v)^2. \]
When the inner-state product distributes:
\[ (u+v)^2=u^2+uv+vu+v^2. \]
The interaction descriptor is:
\[ d(U,V)=uv+vu. \]
When the descriptor is being emphasized, the same sum can be written:
\[ U\,d\,V=uu+uv+vu+vv. \]
Equivalently:
\[ U\,d\,V=U+V=u^2+d(U,V)+v^2. \]
Ordinary arithmetic is the recovered non-interaction case where the cross terms vanish:
\[ d(U,V)=0. \]
Aligned addition in a commutative scalar setting has \(d(U,V)=2uv\). Opposition has \(d(U,V)=-2uv\).
Angular relation is one way to supply the descriptor:
\[ d(U,V)=2uv\cos\Delta. \]
Here \(\Delta\) is relative difference, not an intrinsic phase required by every ultrareal. A lone ultrareal has natural inner state \(u\). Orientation enters only when the situation calls for oriented presentation or relation between parts.
There are no negative ultrareals. The ultrareal layer is the positive real line with zero included:
\[ \mathbb U=[0,\infty). \]
Minus signs can record presentation, direction, bookkeeping, comparison, cancellation, or relation. The symbol \(i\) may be adjoined to the real notation with \(i^2=-1\) to notate turn and opposition without adding negative values to \(\mathbb U\).
Once orientation is admitted, a rotated inner state has a reverse-oriented return \(n^*\). The density is recovered by:
\[ n n^*. \]
This is why the square-form can look the same from the front and the back while still distinguishing the sideways, orthogonal presentation.
The program is conservative in its algebra and radical in its organization: keep positive value positive, keep relation explicit, and recover standard arithmetic as the case where the relation term vanishes.