# Properties of Ultrareals
An ultrareal number is a positive square-form:
```text
U = u^2
```
with:
```text
u >= 0
```
The basic ultrareal domain is:
```text
UR = {u^2 | u >= 0}
```
## Positive Definiteness
Every ultrareal is nonnegative:
```text
U >= 0
```
There are no negative ultrareals.
The only ultrareal that is neither positive nor negative is zero:
```text
0 = 0^2
```
## Square Representation
Every positive ordinary number can be represented as an ultrareal:
```text
X = x^2
```
This does not make `X` unreal. It gives `X` an inner value `x`.
The distinction is:
```text
visible value: X
inner value: x
```
## Equality
Two ultrareals are equal when their positive square-forms are equal:
```text
u^2 = v^2
```
Since the ultrareal layer uses `u >= 0` and `v >= 0`, this also means:
```text
u = v
```
If signs or rotations are introduced, they belong to the inner layer, not to the ultrareal value itself.
## Zero
Zero is the additive identity:
```text
0^2 +_{k} u^2 = u^2
```
for any relation coefficient `k`, because the relation term vanishes:
```text
2k(0)u = 0
```
## Relation-Aware Addition
The general two-term addition is:
```text
u^2 +_{k} v^2 = u^2 + v^2 + 2kuv
```
The coefficient `k` records relation.
If `k` comes from an angle, then:
```text
k = cos(theta)
```
and:
```text
u^2 +_{theta} v^2 = u^2 + v^2 + 2uv cos(theta)
```
## Closure
For:
```text
-1 <= k <= 1
```
the result remains ultrareal:
```text
u^2 + v^2 + 2kuv >= 0
```
The smallest case is opposition:
```text
k = -1
```
which gives:
```text
(u - v)^2 >= 0
```
So relation-aware addition does not require negative ultrareals.
## Standard Arithmetic
Standard arithmetic is recovered when:
```text
k = 0
```
Then:
```text
u^2 +_{0} v^2 = u^2 + v^2
```
So standard addition is the non-interaction case.
## Aligned Addition
Aligned ultrareal addition is:
```text
u^2 +_{1} v^2 = (u + v)^2
```
This operation is commutative:
```text
u^2 +_{1} v^2 = v^2 +_{1} u^2
```
and associative:
```text
(u^2 +_{1} v^2) +_{1} w^2 = u^2 +_{1} (v^2 +_{1} w^2)
```
because both sides equal:
```text
(u + v + w)^2
```
## Many-Term Addition
For many terms, the relation-aware form is:
```text
(u_1 + u_2 + ... + u_n)^2
```
Expanded:
```text
u_1^2 + u_2^2 + ... + u_n^2
+ 2 sum_{i
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(built: 2026-04-25 22:39 EDT UTC-4)