# Pythagorean Addition and Angles
Pythagoras appears naturally in ultrareal arithmetic.
The relation-aware addition rule is:
```text
u^2 +_{theta} v^2 = u^2 + v^2 + 2uv cos(theta)
```
The angle `theta` records how the inner values meet.
## Aligned Addition
When:
```text
theta = 0
```
we have:
```text
cos(0) = 1
```
so:
```text
u^2 +_{theta=0} v^2 = u^2 + v^2 + 2uv = (u + v)^2
```
This is aligned ultrareal addition.
## Orthogonal Addition
When:
```text
theta = pi/2
```
we have:
```text
cos(pi/2) = 0
```
so:
```text
u^2 +_{theta=pi/2} v^2 = u^2 + v^2
```
This is the Pythagorean case.
If:
```text
c^2 = u^2 + v^2
```
then:
```text
c = sqrt(u^2 + v^2)
```
So the Pythagorean theorem is standard arithmetic recovered as orthogonal ultrareal addition.
Standard addition is not the absence of structure. It is the case where the relation term vanishes.
## Opposed Addition
When:
```text
theta = pi
```
we have:
```text
cos(pi) = -1
```
so:
```text
u^2 +_{theta=pi} v^2 = u^2 + v^2 - 2uv = (u - v)^2
```
Opposition still produces a positive square-form.
If `u = v`, the result is:
```text
(u - u)^2 = 0
```
That is cancellation, not negative magnitude.
## The Law Of Cosines
The same formula recovers the law of cosines.
For joined inner values:
```text
|u + v e^{i theta}|^2
= u^2 + v^2 + 2uv cos(theta)
```
For the opposite side of a triangle, the conventional sign is:
```text
c^2 = u^2 + v^2 - 2uv cos(C)
```
The difference is orientation convention. The content is the same: the square-value depends on the relation between the inner values.
## Angle Addition
Euler's rotation rule also recovers the standard angle-addition formulas.
Rotations multiply:
```text
e^{i alpha} e^{i beta} = e^{i(alpha + beta)}
```
Using:
```text
e^{i theta} = cos(theta) + i sin(theta)
```
the left side becomes:
```text
(cos alpha + i sin alpha)(cos beta + i sin beta)
```
Expanding and collecting real and imaginary parts gives:
```text
cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta
```
and:
```text
sin(alpha + beta) = sin alpha cos beta + cos alpha sin beta
```
So trigonometry is also relation arithmetic. It is the arithmetic of how inner values are oriented before the square-value is evaluated.
## Why This Matters
The Pythagorean theorem is often taught as a special geometric fact.
In ultrareal arithmetic, it becomes one case of a larger addition rule:
```text
relation term present: u^2 + v^2 + 2uv cos(theta)
relation term zero: u^2 + v^2
relation term negative: u^2 + v^2 - 2uv
```
So Pythagoras is not outside the new arithmetic.
It is what standard arithmetic looks like when the inner values meet at right angle.