# Rotation and Opposition
An ultrareal number is a positive square-form:
```text
U = u^2
```
with `u` real. But the inner value that exposes it can carry orientation.
The simplest orientation is sign:
```text
u
-u
```
Both expose the same ultrareal:
```text
u^2 = (-u)^2
```
So sign is not visible as a negative magnitude. It is visible only when inner values are joined.
## Opposition
If two inner values are opposed, their joined value is:
```text
u^2 + (-v)^2 := (u - v)^2
```
Expansion gives:
```text
(u - v)^2 = u^2 + v^2 - 2uv
```
The negative sign appears in the relation term. It does not create a negative ultrareal.
The special case of perfect opposition is cancellation:
```text
u^2 + (-u)^2 := (u - u)^2 = 0
```
The result is not less than zero. It is absence after opposition.
## Negative Values
A negative value requires a rotation out of the ultrareal layer:
```text
-U = (iu)^2
```
because:
```text
(iu)^2 = i^2 u^2 = -u^2
```
So a negative number is not a negative ultrareal. It is a rotated square-value.
The symbol `i` marks that rotation. It does not mean the positive value has disappeared. It means the square-value is returning through the inner layer from another direction:
```text
U = u^2
-U = (iu)^2
```
## Rotated Infinity
The same rule applies at infinity.
Positive infinity is the unbounded limit of positive square-forms:
```text
U = u^2
u -> infinity
U -> infinity
```
Negative infinity is not a different kind of negative substance. It is the same unbounded positive square-form seen through the rotated branch:
```text
-U = (iu)^2
u -> infinity
-U -> -infinity
```
So:
```text
-infinity = rotated infinity
```
In the ultrareal layer there is only positive unbounded value. The negative sign belongs to orientation.
## Euler's Rotation
Euler's identity gives the standard notation for this rotation:
```text
e^{i theta} = cos(theta) + i sin(theta)
```
This expression represents a point on the unit circle in the complex plane. Changing `theta` rotates the point.
At a quarter-turn:
```text
theta = pi/2
e^{i pi/2} = i
```
So:
```text
i = e^{i pi/2}
```
Squaring `i` doubles the rotation:
```text
i^2 = e^{i pi} = -1
```
Therefore:
```text
i = sqrt(-1)
```
more precisely:
```text
sqrt(-1) = +/- i
```
This is the arithmetic reason negative values can be understood as rotated positive square-values. The negative sign is a half-turn in value-space, produced by a quarter-turn in the inner square-root layer.
## General Rotation
Let the inner value be rotated:
```text
a = u e^{i theta}
```
Then:
```text
a^2 = u^2 e^{i 2theta}
```
The outer orientation is doubled. A quarter-turn of the inner value becomes a half-turn of the squared value:
```text
theta = pi/2
(u e^{i pi/2})^2 = -u^2
```
This is why `u^2` can be negative if `u` is not real. The negative square is not an ultrareal value; it is the result of rotating the inner value before squaring.
## Rotation-Aware Joining
If two inner values carry orientations,
```text
a = u e^{i alpha}
b = v e^{i beta}
```
then their positive joined value is:
```text
|a + b|^2
```
Expanded:
```text
|a + b|^2 = u^2 + v^2 + 2uv cos(alpha - beta)
```
The relation term depends on relative orientation.
Aligned joining:
```text
alpha - beta = 0
|a + b|^2 = (u + v)^2
```
Opposed joining:
```text
alpha - beta = pi
|a + b|^2 = (u - v)^2
```
Relation-erased joining:
```text
alpha - beta = pi/2
|a + b|^2 = u^2 + v^2
```
Ordinary addition is therefore not the whole operation. It is the case where the relation term is absent, canceled, or ignored.