# A new proof of the Pythagorean Theorem, generalization of the usual algebra
norm, and novel algebra isomorphism invariants

Fred Greensite

###### Abstract.

A new proof of the Pythagorean Theorem is presented, utilizing George
Birkhoff’s version of the postulates of Euclidean geometry as incorporating
$\mathbb{R}$. Extrapolation of the mechanism of proof leads to a
generalization of the usual algebra norm of a real finite-dimensional unital
associative algebra, and resulting novel computationally accessible algebra
isomorphism invariants.

Phrased in modern terms, Decartes’ introduction of the Cartesian plane as
emerging from the classical Euclidean plane entails construction of an
isometric isomorphism between the intrinsic vector space resulting from the
Euclidean plane with a distinguished origin and the direct sum
$\mathbb{R}\oplus\mathbb{R}$ augmented by the inherited Euclidean notion of
length (the four postulate formulation of Euclidean geometry due to Birkhoff
[1], which fundamentally incorporates $\mathbb{R}$, provides a rigorous
environment for effecting that program). The isomorphism relies on the
parallel postulate, and an important feature is that such an isomorphism can
be constructed from any pair of perpendicular lines through a chosen origin in
the Euclidean plane. We will present a proof of the Pythagorean Theorem based
on that feature. The proof is believed to be new - at least the idea of this
proof is not found in the large compendium [2], the more recent survey [3], or
the present version of the Wikipedia article “Pythagorean theorem”. More
interesting than the proof itself is extrapolation of the basic rationale to
the setting of real finite-dimensional unital associative algebras, which
leads to a generalization of the usual notion of an algebra norm, and some
novel computationally accessible algebra isomorphism invariants [4]. In fact,
the proof was originally inspired by the algebraic application rather than the
other way around [5].

Section 1 presents a synopsis of the proof, and Section 2 generalizes the
proof’s argument to real finite-dimensional unital associative algebras. Full
details of the proof, which are straightforward given the synopsis, are
relegated to Appendix A.

##  1\. The new proof of the Pythagorean Theorem

With respect to Birkhoff’s geometry postulates, we assume an understanding of
$\mathbb{R}$ as in Real Analysis, along with Differential Calculus in one
dimension, and usual notions of limits and continuity.

The original motivation for the Cartesian plane was that it can provide a
convenient arena for computations relating to features of the classical
Euclidean plane. An example relevant to our current endeavor concerns the
“Euclidean length function” $\ell$ which specifies the length of a line
segment from a distinguished origin $O$ to any other point $P$ of the
Euclidean plane. Define a “direction” to be any point $C$ of the unit circle
around the origin. The directional derivative of $\ell$ at $P$ in reference to
a given direction $C$ can be defined in terms of the labeling of $O,P,C$ by
$(0,0),(x,y),(u,v)\in\mathbb{R}\oplus\mathbb{R}$ as determined by the above
isomorphism resulting from a selected pair of perpendicular lines intersecting
at $O$ (“a pair of orthogonal coordinate axes through the origin”), via the
function $D\ell(x,y;u,v)$ supplying the directional derivative value according
to the usual definition based on vector space operations taking place in a
one-dimensional subset of $\mathbb{R}\oplus\mathbb{R}$. This value is
invariant under a change to an alternative pair of orthogonal coordinate axes
through the origin. This is because the spaces defined by these alternative
sets of coordinate axes, each being isometrically isomorphic to the above
Euclidean plane with distinguished origin vector space (“the Euclidean plane
vector space”), are thus isometrically isomorphic to each other. $D\ell(P;C)$,
as the directional derivative at $P$ in direction $C$ with respect to the
Euclidean plane vector space, is thereby unambiguously defined.

###### Synopsis of the argument.

It is first observed that for $(u,v)$ taken to be a fixed direction,
$D\ell(x,y;u,v)$ is a continuous function of $(x,y)$, for $(x,y)$ varying in a
domain excluding the origin (this continuity can ultimately be established
with a “one-dimensional” argument due to the Euclidean property that $\ell$ is
a degree-1 positive homogeneous function). Combined with the Mean Value
Theorem of one-dimensional Calculus, it is then easy to show that
$D\ell(x,y;u,v)=\frac{\partial\ell(x,y)}{\partial
x}u+\frac{\partial\ell(x,y)}{\partial y}v$, where
$\frac{\partial\ell(x,y)}{\partial x}$ and $\frac{\partial\ell(x,y)}{\partial
y}$ are the directional derivatives of $\ell$ at the point labeled by $(x,y)$
with respect to the directions associated with the first and second coordinate
axes (these directions being determined by the direction of increasing values
of $\mathbb{R}$ on the respective coordinate axes). Thus, for fixed
$(x,y)\neq(0,0)$, $D\ell(x,y;u,v)$ is a linear functional for $(u,v)$ now
varying over all of $\mathbb{R}\oplus\mathbb{R}$. Evidently, this linear
functional is the dual vector of $\left(\frac{\partial\ell(x,y)}{\partial
x},\frac{\partial\ell(x,y)}{\partial y}\right)$. It follows that $D\ell(P;C)$
is a linear functional for fixed $P$ with $C$ now varying throughout the
Euclidean plane vector space. In view of the isomorphism, the dual of this
linear functional is the point $G$ of the Euclidean plane vector space labeled
by $\left(\frac{\partial\ell(x,y)}{\partial
x},\frac{\partial\ell(x,y)}{\partial y}\right)$ (i.e., the isomorphism maps
vectors to vectors and dual vectors to dual vectors). The point $G$ as
obtained from $D\ell(x,y;u,v)$ via the isomorphism is thereby invariant under
change to an alternative pair of orthogonal coordinate axes through the origin
since, as observed above, the linear functional $D\ell(P;C)$ as determined by
$D\ell(x,y;u,v)$ is invariant under this change. Thus, we can choose an
alternative pair of orthogonal coordinate axes through the origin (with
coordinate axes and components of points distinguished by the symbol ′) such
that $P$ is on the positive portion of the first coordinate axis of the new
coordinate system, so that the Eucldean plane vector space point $P$
originally labeled by $(x,y)$ is now labeled by $(x^{\prime},0)$, and $G$ is
now labeled by $\left(\frac{\partial\ell(x^{\prime},0)}{\partial
x^{\prime}},\frac{\partial\ell(x^{\prime},0)}{\partial
y^{\prime}}\right)=(1,0)$, where the right-hand-side is an easy calculation
since determination of the two directional derivatives on the left-hand-side
only involves evaluation of $\ell$ on the coordinate axes. Thus, according to
the (isometric) isomorphism, we have $\ell(G)=1$ (i.e., $G$ is a point on the
unit circle) and $G$ is labeled by a positive multiple of $(x^{\prime},0)$
(i.e, $G$ is labeled by a point on the ray from the origin through
$(x^{\prime},0)$). From the Euclidean geometry postulates, $\ell$ is a
degree-1 positive homogeneous function. This means that with respect to the
original pair of orthogonal coordinate axes through the origin, we have
$\left(\frac{\partial\ell(x,y)}{\partial x},\frac{\partial\ell(x,y)}{\partial
y}\right)=\left(\frac{x}{\ell(x,y)},\frac{y}{\ell(x,y)}\right)$, since the
right-hand-side (uniquely) is both a point on the unit circle while also being
a positive multiple of $(x,y)$ \- as are now seen to be required for a
labeling of $G$ via the isomorphism. With respect to this last equation, one-
dimensional indefinite integration (anti-differentiation) can be applied
separately to the equality of the first coordinates, and then to the equality
of the second coordinates. Reconciliation of the two resulting expressions for
$\ell$ in the context of the nonnegativity of $\ell$ and its degree-1 positive
homogeneity, quickly leads to $\ell(x,y)=\sqrt{x^{2}+y^{2}}$, from which the
Pythagorean Theorem easily follows via an argument involving parallelograms to
measure lengths of arbitrary line segments using $\ell$. ∎

##  2\. The proof’s structure leads to a generalization of the usual norm
associated with a unital associative algebra, and novel isomorphism invariants

Surpisingly, the form of the argument presented in the proof synopsis is quite
robust as regards its potential for generalization. The climax of the synopsis
centers on two equations that can be recast as,

(2.1) $s=\ell(s)\nabla\ell(s),$

and,

(2.2) $\ell(\nabla\ell(s))=1,$

for $s\in\mathbb{R}^{n}$ not at the origin (with $n=2$, though we will not use
that restriction going forward), where “$\nabla\ell(s)$” is the ordered
$n$-tuple of coordinate-wise one-dimensional derivatives as in the proof
synopsis. Indeed, solution of the above two displayed equations by themselves
yields the Euclidean norm. But (2.2) is itself just a consequence of (2.1) and
degree-1 positive homogeneity of $\ell(s)$ (simply apply $\ell$ to both sides
of (2.1) and invoke the homogeneity condition). And (2.1) is appealing on its
own, since given the degree-1 positive homogeneity constraint it seems to be a
“simplest” expression for a relationship between $s$ and $\ell(s)$. That is,
the Euler Homogeneous Function Theorem states that
$s\cdot\nabla\ell(s)=\ell(s)$. For $\alpha>0$, replacing $s$ by $\alpha s$
then leads to $\alpha s\cdot\nabla\ell(\alpha s)=\ell(\alpha
s)=\alpha\ell(s)=\alpha s\cdot\nabla\ell(s)$, implying $\nabla\ell(\alpha
s)=\nabla\ell(s)$ (where $\nabla\ell(\alpha s)$ means that $\nabla\ell$ is
evaluated at $\alpha s$). So, although $s$ and $\nabla\ell(s)$ are both
members of $\mathbb{R}^{n}$, they behave very differently when $s$ is replaced
by $\alpha s$. On the other hand, for degree-1 positive homogeneous $\ell$,
the expressions $s$ and $\ell(s)\nabla\ell(s)$ do behave the same way when $s$
is replaced by $\alpha s$ (each expression simply being multiplied by
$\alpha$). So what could be more natural than linking them via (2.1) (and thus
forwarding the Euclidean norm)?

But without unduly compromising the simplicity of the rationale, we could
alternatively propose that (2.1) be replaced by

(2.3) $Ls=\ell(s)\nabla\ell(s),$

for a linear transformation $L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, since
we would still have $Ls$ and $\ell(s)\nabla\ell(s)$ behaving the same when $s$
is replaced by $\alpha s$, $\alpha>0$. Applying the dot product with $s$ to
both sides of (2.3), the Euler Homogeneous Function Theorem then implies

(2.4) $s^{\prime}Ls=s\cdot Ls=\ell^{2}(s),$

where $L$ in this equation is understood to be the matrix associated with the
above linear transformation $L$ with respect to the standard basis of
$\mathbb{R}^{n}$. According to (2.3), $L$ must be a real symmetric matrix
since $Ls$ is a gradient (i.e., the right-hand-side of (2.3) is equal to
$\frac{1}{2}\nabla\ell^{2}(s)$). According to (2.4), $L$ must be positive
semi-definite. From the Polarization Identity, it is seen that the above leads
to arbitrary inner product spaces on $\mathbb{R}^{n}$ \- a nice, if simple,
generalization.

However, the above argument can easily be further generalized to the task of
attaching a “norm” to $\mathbb{R}^{n}$ when the latter is the vector space of
elements of a unital associative algebra. That is, the basic rationale of the
argument is easily modified to allow the new feature of the inverse of an
element to influence selection of a norm - something not explicitly addressed
by the well-recognized “usual norm” of a finite-dimensional associative
algebra as the determinant of an element’s image under the left regular
representation (e.g., the algebra’s norm as defined by Bourbaki [7], and whose
norm format is also fundamental more generally, such as in Algebraic Number
Theory).

Thus, we wish to exploit the existence of an element’s inverse to aid in
construction of a “norm-like” real function $\mathfrak{u}(s)$ on the units of
the algebra. By analogy with the vector space argument in the first two
paragraphs of this section, one recognizes that $s^{-1}$ and
$\nabla\mathfrak{u}(s)$ are members of $\mathbb{R}^{n}$ if $s$ is a unit of
the algebra, and $\nabla\mathfrak{u}(\alpha s)=\nabla\mathfrak{u}(s)$ for
$\alpha>0$ if degree-1 positive homogeneity is again mandated. While the
expressions $s^{-1}$ and $\nabla\mathfrak{u}(s)$ thereby behave differently
when $\alpha s$ replaces $s$, this time it is $s^{-1}$ and
$\frac{\nabla\mathfrak{u}(s)}{\mathfrak{u}(s)}$ that behave the same under
that replacement, in that both expressions are simply multiplied by
$\frac{1}{\alpha}$ \- mirroring the situation that pertains to the first
paragraph of this section. So now, based on that latter treatment (and its
“minimalist” approach for attaching a norm to a vector space), we might be
tempted to equate $s^{-1}$ and $\frac{\nabla\mathfrak{u}(s)}{\mathfrak{u}(s)}$
\- except that the latter is the gradient of $\log\mathfrak{u}(s)$, while
$s^{-1}$ is in general not a gradient. That problem resolves if we can find a
linear transformation $L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that
$Ls^{-1}$ is a gradient, i.e., satisfies the exterior derivative condition,

(2.5) $d\left([Ls^{-1}]\cdot ds\right)=d\left((ds)^{\prime}Ls^{-1}\right)=0.$

Here and in the sequel, the standard basis and standard topology on
$\mathbb{R}^{n}$ are assumed. Thus, $L$ in (2.5) is the matrix associated with
the above linear transformation with respect to the standard basis.

But there is also the Euler Homogeneous Function Theorem to deal with, wherein
$\frac{s\cdot\nabla\mathfrak{u}(s)}{\mathfrak{u}(s)}=1$. Denoting the
multiplicative identity of the algebra as ${\bf 1}$, we can define $\|{\bf
1}\|^{2}$ to be the number of independent entries on the main diagonal of the
matrices comprising the left regular representation of the algebra. We are
free to constrain $L$ by the requirement that ${\bf 1}\cdot(L{\bf 1})={\bf
1}^{\prime}L{\bf 1}=\|{\bf 1}\|^{2}$. It follows that we can now propose,

(2.6) $Ls^{-1}=\|{\bf
1}\|^{2}\frac{\nabla\mathfrak{u}(s)}{\mathfrak{u}(s)}=\left(\frac{\|{\bf
1}\|^{2}}{\mathfrak{u}(s)}\right)\nabla\mathfrak{u}(s),$

if $L$ satisfies both (2.5) and

(2.7) $s^{\prime}Ls^{-1}=s\cdot Ls^{-1}=\|{\bf 1}\|^{2},$

since constraint (2.7) is a necessary condition for a solution of (2.6) to be
degree-1 positive homogeneous (e.g., take the dot product of both sides of
(2.6) with $s$ and apply the Euler Homogeneous Function Theorem). Note that
(2.7) must hold if $s$ is replaced by $s^{-1}$ as both are units, so we expect
$L$ to be a real symmetric matrix - and we will impose that condition going
forward. This makes $L$ akin to an inner product matrix. More precisely, $L$
is then a scalar product matrix (i.e., associated with a symmetric bilinear
form) with the important feature is that $L$ need not be positive semi-
definite (exploitable in a relativistic physics context [6]). Thus, (2.7) can
be interpreted to mean that multiplicative inverses $s,s^{-1}$ also behave
inversely as vector space members with respect to scalar product $L$. If $L$
satisfying (2.5) and (2.7) can be found, which we refer to as a “normalized
uncurling metric”, then one can provide $\mathfrak{u}(s)$ by integrating
(2.6), at least in a simply connected neighborhood of $\mathbf{1}$ that
excludes non-units.

Equation (2.6) is analogous to (2.3) (which we emphasize with the format of
the second equality in (2.6)), and (2.7) is analogous to (2.4). But the marked
difference is that for the “naked” vector space application (where (2.3) and
(2.4) pertain), $L$ can be any real symmetric $(n\times n)$-matrix (allowing
for possibly complex-valued $\ell(s)$), while in the unital algebra
application (where (2.6) and (2.7) pertain) the additionally required (2.5)
very significantly constrains real symmetric $L$. It is this constraint (which
also allows for the further constraint (2.7)) that makes things interesting,
because different algebras have very different collections of admissible
choices for $L$, and these differences distinguish the algebras in novel ways.

Furthermore, a straightforward argument shows that $\mathfrak{u}(s)$ has the
following remarkable attributes.

  1. (1)

$\mathfrak{u}(s)$ is a nonnegative degree-1 positive homogeneous function.
That is, $\mathfrak{u}(\alpha s)=\alpha\mathfrak{u}(s)$ for $\alpha>0$.

  2. (2)

$\mathfrak{u}(s)$ behaves inversely with respect to the algebra’s
multiplicative inverse operation. That is,
$\mathfrak{u}(s^{-1})=\big{(}\mathfrak{u}(s)\big{)}^{-1}$.

  3. (3)

$\mathfrak{u}(s)$ and $s^{-1}$ generate a scalar product such that
multiplicative inverses behave inversely with respect to this scalar product.
The scalar product matrix $L$ is determined by
$Ls=\|\mathbf{1}\|^{2}\mathfrak{u}(s)\nabla\mathfrak{u}(s^{-1})$, where
$\nabla\mathfrak{u}(s^{-1})$ is defined as the evaluation of
$\nabla\mathfrak{u}$ at $s^{-1}$.

Based on the above, $\mathfrak{u}(s)$ would seem worthy of being called a
“unital norm” associated with a unital algebra - though that appellation will
be even more compellingly justified shortly. Of course, in this
generalization, “norm” is understood as something satisfying a subset of the
kind of attributes that the usual algebra norm has (as the determinant of an
element’s image under the left regular representation [7]), rather than the
attributes of a vector space norm.

Equation (2.6) indicates that each pair $(L,\mathfrak{u}(s))$ satisfying the
equation characterizes the algebra’s multiplicative inversion operation
$s\rightarrow s^{-1}$, on the domain orthogonal to the kernel of $L$ in a
simply connected neighborhood of $\mathbf{1}$ devoid of non-units (i.e.,
$s^{-1}$ on that domain results from application of the pseudoinverse
$L^{\dagger}$ to both sides of (2.6)). However, it is the aggregate of
possible $(L,\mathfrak{u}(s))$ that will be most useful in identifying novel
algebra isomorphism invariants.

Nevertheless, an individual $\mathfrak{u}(s)$ does have its own interest.
Compared to the vector space entity $\ell(s)$, the algebra entity
$\mathfrak{u}(s)$ can lead to more elaborate possibilities for the geometry of
$\mathbb{R}^{n}$, following from the fact that $\mathfrak{u}(s)$ is not itself
descriptive of a quadratic space - nor is the usual algebra norm. So the
viewpoint here is that the usual algebra norm is a geometrical structure given
its interpretation in terms of the volume of an implied parallelpiped, and an
algebra’s unital norms (plural!) can each be also given a geometrical
interpretation based on how they alter (and in aggregate subsume) the usual
algebra norm. The context is that the solitary vector space $\mathbb{R}^{n}$
supports many quadratic space norms stemming from choices of symmetric $L$.
Similarly, a solitary algebra may support many unital norms stemming from
choices of a normalized uncurling metric $L$ \- and (it turns out) one of
these unital norms is always essentially equivalent to the usual algebra norm,
while the other unital norms modify it in a transparent manner. Furthermore,
the space of all uncurling metrics associated with an algebra forwards novel
computationally accessible algebra isomorphism invariants (such as the
dimension of that space). The latter is significant since the algebra
isomorphism problem is recursively undecidable, e.g., [8, 9, 10] (greatly
contrasting with the isomorphism problem for quadratic spaces, which is
completely solved by computing matrix signatures).

So, the following summarize some of the reasons for bothering with all of
this.

  * •

Characterizations of an algebra’s multiplicative inverse operation via the set
of pairings of normalized uncurling metrics and unital norms lead to a
geometrically-significant generalization of the algebra norm.

  * •

Novel isomorphism invariants can be derived from the space of an algebra’s
uncurling metrics, and some of these invariants are always computationally
straightforward.

  * •

The above invariants impact not only the recursively undecidable algebra
isomorphism problem, but also the more general epimorphism problem - via a
particular functor.

  * •

For some purposes, a parametrized symmetric matrix depiction of the space of
an algebra’s uncurling metrics leads to an alternative to the explicitly
ideal-based expression of algebra structure.

The above are developed in detail in [4], which also presents an alternative
motivation to the one we have used here.

##  Appendix A Details of proof of the Pythagorean Theorem

To flesh out the proof synopsis in Section 1, in this appendix we will

  * •

Present the four postulates of Birkhoff’s formulation of Euclidean geometry,
along with the implied vector space and degree-1 positive homogeneous length
function $\ell$,

  * •

Derive the resulting isometric isomorphism with $\mathbb{R}\oplus\mathbb{R}$
based on a pair of orthogonal coordinate axes through a distinguished origin
in Euclidean space,

  * •

Prove that the length function $\ell$ is continuously differentiable on
domains excluding the origin,

  * •

Show that, for fixed $(x,y)\neq(0,0)$ with $(u,v)$ varying over
$\mathbb{R}\oplus\mathbb{R}$, the expression $D\ell(x,y;u,v)$ is a linear
functional having the form prescribed in the synopsis.

###  A.1. Birkhoff’s postulates for plane Euclidean geometry, and the
resulting “Euclidean plane with distinguished origin” as a vector space, with
a homogeneous length function

Slightly reworded, Birkhoff’s postulates entail the following [1]:

  1. (1)

“Postulate of Line Measure”: There is a bijection between $\mathbb{R}$ and the
set of points on any line, so that the length of any line segment is the
absolute value of the difference of the real numbers corresponding to the
segment’s endpoints.

  2. (2)

“Point-Line Postulate”: There is one and only one line that contains any two
given distinct points of the plane.

  3. (3)

“Postulate of Angle Measure”: There is a bijection between the set of rays
from a point $O$ and the real numbers $(\mbox{mod}\,2\pi)$ such that, for a
line not containing $O$, continuously varying a point on the line implies
continuous variation of the bijection-assigned real number value of the ray it
lies on. The angle formed by a first ray from $O$ assigned real value
$\alpha_{1}$ and a second ray from $O$ assigned real value $\alpha_{2}$, is
$\alpha_{1}-\alpha_{2}$ (mod $2\pi$).

  4. (4)

“Postulate of Similarity”: If a triangle has two sides having lengths related
to the lengths of two sides of another triangle by the same positive
proportionality constant, and the absolute values of the angles formed by
these two sides of each triangle are equal, then these are similar triangles,
i.e., they have the same three angles up to a sign, and the lengths of the
third side of the two triangles are related by the same proportionality
constant.

Regarding the identification of points of the plane with points of
$\mathbb{R}\times\mathbb{R}$, we’ll let [1] speak for itself: “We are now
prepared to define a rectangular coordinate system $(x,y)$ with axes the
directed perpendicular lines $Ox,Oy$ intersecting in the origin $O$. Choose a
system of numeration along $Ox,Oy$ so that $O$ is marked $0$ on both lines,
and the numbers increase algebraically in the positive direction. Drop the
unique perpendiculars from $P$ to $Ox$ and $Oy$ respectively meeting these
lines at $Px$ and $Py$. The rectangular coordinates of $P$ are then the
numbers $x_{P}$ and $y_{P}$ attached to $Px$ and $Py$ respectively. These are
evidently uniquely determined numbers.” This bijection, explicitly derived in
[1], is a straightforward consequence of Playfair’s “parallel axiom” (Theorem
IX in [1]) - which follows from the Postulate of Similarity without use of the
Pythagorean Theorem.

Birkhoff’s formulation of Euclidean geometry makes intrinsic use of
$\mathbb{R}$. Accordingly, we take $\mathbb{R}$ to be as understood in Real
Analysis, and we thus assume the accompanying notions of limits and
continuity, as well as Differential Calculus in one dimension.

We can now fashion a vector space with a length function out of the Euclidean
plane with origin $O$.

######  Definition A.1.

The Euclidean plane vector space is constituted as follows:

  1. (1)

There is a chosen point $O$ of the Euclidean plane termed “the origin”.

  2. (2)

There is a length function $\ell$ such that for $P$ in the Euclidean plane we
have $\ell(P)$ as the length of the line segment from the origin to $P$.

  3. (3)

For $P,P^{\prime}$ in the Euclidean plane vector space that are not collinear
with the origin $O$, we use the parallel postulate (a consequence of the
Postulate of Similarity) to define $P+P^{\prime}$ to be a third point
$P^{\prime\prime}$ in the Euclidean plane vector space as the point of
intersection of two unique lines, the first line being the line through $P$
parallel to the line segment from the origin to $P^{\prime}$, and the second
line being the line through $P^{\prime}$ parallel to the line segment from the
origin to $P$. The case of points $O,P,P^{\prime}$ being collinear is
addressed using the Postulate of Line Measure: If $P$ and $P^{\prime}$ do not
lie on the same ray from $O$, then $P^{\prime\prime}$ has length given by the
absolute value of the difference of the lengths of the two points, and resides
on the ray from the origin through the point $P$ or $P^{\prime}$ as determined
by which point has the greater length. If $P$ and $P^{\prime}$ do lie on the
same ray from $O$, then $P^{\prime\prime}$ has length equal to the sum of the
lengths of $P$ and $P^{\prime}$, and lies on the ray from the origin on which
$P$ and $P^{\prime}$ lie.

  4. (4)

Multiplication of a point $P$ by $\alpha\in\mathbb{R}$ is defined so that, for
$\alpha\neq 0$ and $P$ not at the origin, $\alpha P$ is one of two points on
the line containing the origin and $P$ that have length $|\alpha|\ell(P)$. If
$\alpha>0$, the point $\alpha P$ is such that $P$ and $\alpha P$ lie on the
same ray from the origin (and the new point is termed “parallel” to $P$). For
$\alpha<0$, the point $\alpha P$ is such that $P$ and $\alpha P$ do not lie on
the same ray from the origin (and the new point is termed “antiparallel” to
$P$). For $\alpha=0$, or if $P$ is the origin, the point $\alpha P$ is the
origin.

It follows that $\ell$ is a degree-1 positive homogeneous function.

###  A.2. The isometric isomorphism between the Euclidean plane vector space
and $\mathbb{R}\oplus\mathbb{R}$

######  Lemma A.1.

Consider the bijection, $B:\text{\rm Euclidean plane vector
space}\longrightarrow\mathbb{R}\oplus\mathbb{R}$, that results from a pair of
orthogonal coordinate axes through the distinguished origin in the Euclidean
plane. Suppose $\mathbb{R}\oplus\mathbb{R}$ is endowed with a length function
similarly notated to that of the Euclidean plane vector space, i.e.,
$\ell:\mathbb{R}\oplus\mathbb{R}\rightarrow\mathbb{R}$, such that
$\ell(x,y)\equiv\ell(P_{x,y})$ where $P_{x,y}$ is the member of the Euclidean
plane vector space corresponding to $(x,y)$ under the bijection. Then the
bijection is an isometric isomorphism.

###### Proof.

Using the bijection $B$, we can denote points of the Euclidean plane vector
space as $P_{x,y}$, with $(x,y)\in\mathbb{R}\oplus\mathbb{R}$. From the
structure of the Euclidean plane vector space (Definition A.1), we can write
$P_{x,y}=P_{x,0}+P_{0,y}$. Consequently,
$P_{x_{1},y_{1}}+P_{x_{2},y_{2}}=P_{x_{1},0}+P_{0,y_{1}}+P_{x_{2},0}+P_{0,y_{2}}=P_{x_{1}+x_{2},0}+P_{0,y_{1}+y_{2}}=P_{x_{1}+x_{2},y_{1}+y_{2}}$.
We then have,

(A.1) $\displaystyle B(P_{x_{1},y_{1}}+P_{x_{2},y_{2}})$ $\displaystyle=$
$\displaystyle
B(P_{x_{1}+x_{2},y_{1}+y_{2}})=(x_{1}+x_{2},y_{1}+y_{2})=(x_{1},y_{1})+(x_{2},y_{2})$
$\displaystyle=$ $\displaystyle B(P_{x_{1},y_{1}})+B(P_{x_{2},y_{2}}).$

Furthermore, for $\alpha\in\mathbb{R}$, we have

(A.2) $B(\alpha P_{x,y})=B(P_{\alpha x,\alpha y})=(\alpha x,\alpha
y)=\alpha(x,y)=\alpha B(P_{x,y}),$

where the first equality follows from the Postulate of Similarity. Equations
(A.1) and (A.2) indicate that $B$ is a linear isomorphism from the Euclidean
plane vector space to $\mathbb{R}\oplus\mathbb{R}$. By definition,
$\ell(B(P_{x,y}))=\ell(x,y)=\ell(P_{x,y})$. Consequently, $B$ is an isometric
isomorphism. ∎

###  A.3. The length function $\ell(s)$ is continuously differentiable on
domains excluding $s=0$

######  Definition A.2.

The value of the directional derivative of the length function
$\ell:\mathbb{R}\oplus\mathbb{R}\rightarrow\mathbb{R}$ at a point
$(x,y)\neq(0,0)$ with respect to direction $(u,v)$ is given by

(A.3) $D\ell(x,y;u,v)\equiv\lim_{\epsilon\rightarrow
0}\frac{\ell\left((x,y)+\epsilon(u,v)\right)-\ell(x,y)}{\epsilon}.$

######  Lemma A.2.

Directional derivatives of $\ell$ in all directions exist at each point
exclusive of the origin.

###### Proof.

From Definition A.1, $\ell(\alpha P)=|\alpha|\ell(P)$, i.e., $\ell$ is a
degree-1 positve homogenous function. Also note that, using a parallelogram
construction, the length of any line segment in Euclidean plane vector space
can be expressed using $\ell$ and vector subtraction. That is, for
$P,P^{\prime}\in\text{Euclidean plane vector space}$, the length of the line
segment with endpoints $P,P^{\prime}$ is $\ell(P-P^{\prime})$. Using the
triangle inequality, the continuity of $\ell$ immediately follows (i.e.,
$|\ell(P^{\prime})-\ell(P)|\leq\ell(P^{\prime}-P)$, the implied “reverse
triangle inequality”).

The Postulate of Line Measure assures that every non-origin point is a
positive multiple of a direction. Thereby, evaluation of (A.3) immediately
indicates that the directional derivative exists in the directions parallel
and antiparallel to $(x,y)\neq(0,0)$, and these values are respectively $+1$
and $-1$. Thus, consider the directional derivative in some other direction.
We assert that the expression inside the limit in (A.3) is monotonic as
$\epsilon\rightarrow 0^{+}$ and is monotonic as $\epsilon\rightarrow 0^{-}$.
If that were not the case, since $\ell$ is continuous, there would have to be
two choices of $\epsilon$ having the same sign such that the expressions
inside that limit are equal. We can call these choices $\epsilon^{\prime}$ and
$\frac{1}{k}\epsilon^{\prime}$, with $0<k<1$. Using the degree-1 positive
homogeneity of $\ell$, simple manipulation of the equality of the two
expressions leads to

(A.4)
$\ell\big{(}(x,y)+\epsilon^{\prime}(u,v)\big{)}-\ell\big{(}k(x,y)+\epsilon^{\prime}(u,v)\big{)}=\ell(x,y)-\ell\big{(}k(x,y)\big{)}=(1-k)\ell(x,y).$

But according to the (reverse) triangle inequality, the absolute value of the
left-hand-side above must be strictly less than the absolute value of the
right-hand-side, since $k\neq 1$, and $(u,v)$ and $(x,y)$ are not collinear.
Thus, (A.4) is not possible, so the above monotonicity assertion is verified.
Furthermore, the triangle inequality indicates that the expression inside the
limit in (A.3) is bounded by the values of the the directional derivatives of
$\ell$ in the directions parallel and antiparallel to $(x,y)$. A monotone
convergence argument then establishes the existence of the directional
derivative. ∎

######  Lemma A.3.

The directional derivative of $\ell$ in a given fixed direction as a function
over the points of the plane is continuous at each point exclusive of the
origin.

###### Proof.

From the existence of directional derivatives in all directions at
$(x,y)\neq(0,0)$ (Lemma A.2), and the degree-1 positive homogeneity of $\ell$,
it easily follows from the directional derivative expression (A.3) that a
fixed-direction directional derivative of $\ell$ at any point distinct from
the origin depends only on the ray on which the point lies (i.e., it is
constant on a ray). This means that to demonstrate continuity of the
directional derivative at $(x,y)\neq(0,0)$, the argument need only evaluate
the limit of the difference of the directional derivatives at $(x,y)$ and
$(x^{\prime},y^{\prime})$ as the ray on which the latter point lies approaches
the ray on which the former point lies - so it is sufficient to examine the
case where $(x^{\prime},y^{\prime})$ approaches $(x,y)$ along the circle with
center $(0,0)$ containing $(x,y)$. In fact, the derivative is continuous at
$(x,y)$, if and only if it is continuous at the point of the unit circle
$\frac{(x,y)}{\ell(x,y)}$.

Thus, without loss we can assume that $(x,y)$ and $(x^{\prime},y^{\prime})$
are both points on the unit circle centered at the origin, so that it is then
sufficient to show that the difference of their directional derivatives in a
fixed direction $(u,v)$ tends to zero as $(x^{\prime},y^{\prime})$ approaches
$(x,y)$ along the unit circle. That is, we only need to show that the
directional derivative in the fixed $(u,v)$-direction is a continuous function
over the points of the unit circle. But, the directional derivative in a
direction $(u,v)$ for a point $(x,y)$ on the unit circle depends only on the
angle formed by $(x,y)$ and $(u,v)$, since the directional derivative is
coordinate system-independent, and the latter vectors both have unit length.
Hence, the following symmetry pertains: demonstrating continuity of the
directional derivative in a fixed direction as a function over all points of
the unit circle, is equivalent to demonstrating continuity of the directional
derivative taken at a fixed point of the unit circle as a function over all
directions.

To demonstrate continuity of the latter function, we invoke Proposition 24 of
book 1 of Euclid’s Elements, which states that if two sides of one triangle
are congruent to two sides of a second triangle, and the [absolute value of
the] included angle of the first triangle is larger than the [absolute value
of the] included angle of the second triangle, then the third side of the
first triangle is longer than the third side of the second triangle. Now label
the rays from the origin via $\theta\in(-\pi,\pi]$, with $\theta=0$ labeling
the ray containing $(x,y)$. From the Postulate of Angle Measure, it follows
that $\theta$ continuously labels the set of directions. Applying the above
Proposition 24 to the directional derivative expression (A.3), it is easily
seen that the directional derivative at $(x,y)$ as a function of the direction
labeled by $\theta$ is monotonic for $\theta\in(-\pi,0]$ and also monotonic
for $\theta\in[0,\pi]$ \- and is thus a continuous function from a well known
result that a derivative which exists at each point of a closed interval, and
is monotonic on that interval, is continuous on the interval. In view of the
statement at the end of the last paragraph, this establishes the continuity of
the directional derivative taken in a fixed direction over the non-origin
points of the plane. ∎

###  A.4. The directional derivative $D\ell(x,y;u,v)$ has the prescribed form

Suppose non-origin point $P$ in the Euclidean plane vector space is labeled by
$(x,y)$ via the isomorphism. Lemma A.2 states that all directional derivatives
exist at $(x,y)$. Let $\frac{\partial\ell(x,y)}{\partial x}$ and
$\frac{\partial\ell(x,y)}{\partial y}$ be the respective directional
derivative values of $\ell$ at $(x,y)$ in the $(1,0)$ and $(0,1)$ directions.
From the mean value theorem of one-dimensional Calculus,

(A.5) $\displaystyle\ell\left((x,y)+\epsilon(u,v)\right)-\ell(x,y)$
$\displaystyle=$
$\displaystyle\big{[}\ell\left((x,y)+\epsilon(u,0)\right)-\ell(x,y)\big{]}$
$\displaystyle+\big{[}\ell\left((x,y)+\epsilon(u,v)\right)-\ell\left((x,y)+\epsilon(u,0)\right)\big{]}$
$\displaystyle=$ $\displaystyle\frac{\partial\ell({\bf p}_{1})}{\partial
x}u\epsilon+\frac{\partial\ell({\bf p}_{2})}{\partial y}v\epsilon,$

where ${\bf p}_{1}$ is in the line segment with endpoints $(x,y)$ and
$(x,y)+\epsilon(u,0)$, and ${\bf p}_{2}$ is in the line segment with endpoints
$(x,y)+\epsilon(u,0)$ and $(x,y)+\epsilon(u,v)$, and by constructing an
appropriate parallelogram we already know that the respective lengths of these
line segments are just $u\epsilon$ and $v\epsilon$ since the line segments are
parallel to the $x$-axis and $y$-axis and the values of $\ell$ are known on
these axes (i.e., $\ell(a,0)=|a|$ and $\ell(0,b)=|b|$ from the Postulate of
Line Measure).

We know from Lemma A.3 that the directional derivative of $\ell$ in a fixed
direction is continuous at $(x,y)\neq(0,0)$. Thus, dividing both sides of
(A.5) by $\epsilon$ and taking the limit as $\epsilon\rightarrow 0$, we find

$D\ell(x,y;u,v)=\lim_{\epsilon\rightarrow
0}\frac{\ell\left((x,y)+\epsilon(u,v)\right)-\ell(x,y)}{\epsilon}=\frac{\partial\ell(x,y)}{\partial
x}u+\frac{\partial\ell(x,y)}{\partial y}v.$

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