Worked Examples in the Real Algebra

Introduction

This article is the computed companion to the algebraic articles of the real algebra. Where Real Algebra states the definitions and proves the general properties, the present article exhibits them on explicit elements, so that the general statements can be read against a concrete computation: the field operations, the sole involution and its fixed-point subspace, the associated direct-sum decompositions, and the order with its Archimedean and completeness consequences. It is the first rung of the ladder that the worked-example articles of the complex, quaternion and biquaternion algebras continue.

The elements used throughout are

$$ c = -\tfrac{3}{4}, \qquad d = \tfrac{7}{2}, $$

chosen because their arithmetic is exact, because their squares are the small squares $c^2 = \tfrac{9}{16}$ and $d^2 = \tfrac{49}{4}$, and because they have opposite signs, so that the order of $\mathbb{R}$ is visible on the worked pair. Where a statement needs a degenerate example, the elements $0$, $1$ and $-1$ are used. The conventions are those of Real Algebra: the basis is $e_0 = 1$, the sole involution is the identity $\operatorname{id}$, and $\lvert a\rvert$ is the absolute value defined by the order.

Every numerical value below is exact and rational; no decimal is used.

Everything computed below is algebraic. The metrical reading of the algebra — the norm and the modulus of Real Norm and Invertibility, the sign decomposition of Real Polar Element Representation, and the Cayley matrix of Real Regular Element Representation — belongs to the later slots of the system and is not used here.

The Algebra on Concrete Elements

Sum, difference and product. With $c = -\tfrac{3}{4}$ and $d = \tfrac{7}{2}$,

$$ c + d = -\tfrac{3}{4} + \tfrac{14}{4} = \tfrac{11}{4}, \qquad d - c = \tfrac{14}{4} + \tfrac{3}{4} = \tfrac{17}{4}, $$

and, applying the field multiplication,

$$ c d = -\tfrac{3}{4}\cdot\tfrac{7}{2} = -\tfrac{21}{8}, \qquad d c = \tfrac{7}{2}\cdot\bigl(-\tfrac{3}{4}\bigr) = -\tfrac{21}{8}, $$

the two orders agreeing, the concrete form of commutativity.

Powers and the absence of zero divisors. The squares are

$$ c^2 = \bigl(-\tfrac{3}{4}\bigr)^2 = \tfrac{9}{16}, \qquad d^2 = \bigl(\tfrac{7}{2}\bigr)^2 = \tfrac{49}{4}, $$

and $c^2 = (-c)^2$ although $c \neq -c$: the square forgets the sign while the element does not. The product of the two nonzero worked elements is again nonzero, $cd = -\tfrac{21}{8} \neq 0$, the concrete form of the absence of zero divisors in a field.

Associativity and distributivity are illustrated by the two products above agreeing and by $(c+d)^2 = c^2 + 2cd + d^2$, a computation of the same rules which needs no separate display.

The Involution on Concrete Elements

The real algebra carries one involution, the identity.

The identity. $\operatorname{id}(c) = c = -\tfrac{3}{4}$ and $\operatorname{id}(d) = d = \tfrac{7}{2}$. The identity fixes every element, and it is the only involution of $\mathbb{R}$: an $\mathbb{R}$-algebra involution is a field automorphism of $\mathbb{R}$, and there is no nontrivial one, so there is no analogue here of the conjugation of $\mathbb{C}$ or of the quaternion conjugation of $\mathbb{H}$ and $\mathbb{B}$. The absence is the mathematical content of the unitary case.

It is an involution, $\operatorname{id}(\operatorname{id}(a)) = a$ for every $a$, and the relation is immediate because $\operatorname{id}$ is the identity map.

element	$\operatorname{id}$	$\operatorname{id}$ applied twice
$c = -\tfrac{3}{4}$	$-\tfrac{3}{4}$	$-\tfrac{3}{4}$
$d = \tfrac{7}{2}$	$\tfrac{7}{2}$	$\tfrac{7}{2}$
$cd = -\tfrac{21}{8}$	$-\tfrac{21}{8}$	$-\tfrac{21}{8}$

The Fixed-Point Subspace on Concrete Elements

The fixed-point subspace of the identity involution is the whole algebra, and there is no anti-fixed subspace, because the only candidate involution is trivial.

The fixed subspace. An element is fixed exactly when it equals itself, which is always true. For the worked pair,

$$ \operatorname{id}(c) = c, \qquad \operatorname{id}(d) = d, $$

so both $c$ and $d$ lie in the fixed subspace $\mathbb{R}_{\mathbb{R}}$; indeed every real number does. The fixed subspace is the whole algebra, $\mathbb{R}_{\mathbb{R}} = \mathbb{R}$, a real vector space of dimension $1$.

The anti-fixed subspace is empty. An element is anti-fixed exactly when $\operatorname{id}(a) = -a$, i.e. $a = -a$, i.e. $2a = 0$, i.e. $a = 0$. The anti-fixed subspace is therefore the zero subspace $\{0\}$, and it is not a subspace of positive dimension:

$$ \{a \in \mathbb{R} : \operatorname{id}(a) = -a\} = \{0\}. $$

This is the concrete form of the absence of the imaginary direction: $\mathbb{C}$ has the imaginary axis $i\mathbb{R}_{\mathbb{C}}$ as its anti-fixed subspace, and $\mathbb{R}$ has only $0$.

The Decompositions, Worked

The eigencomponents of the identity. Because the involution is trivial, the two eigenprojections coincide on the fixed subspace and the anti-fixed component vanishes:

$$ c_+ = \tfrac{1}{2}\bigl(c + \operatorname{id}(c)\bigr) = \tfrac{1}{2}(c+c) = c = -\tfrac{3}{4} \in \mathbb{R}_{\mathbb{R}}, $$

$$ c_- = \tfrac{1}{2}\bigl(c - \operatorname{id}(c)\bigr) = \tfrac{1}{2}(c-c) = 0 . $$

The result is the trivial decomposition $\mathbb{R} = \mathbb{R}_{\mathbb{R}} \oplus \{0\}$ in which the second summand has dimension $0$. This is the real decomposition of Real Algebra, and its triviality is the statement that the general involution decomposition has one summand here.

The Cartesian decomposition. The decomposition of an element into a fixed part plus an anti-fixed part, $\tfrac{1}{2}(a+\operatorname{id}a) + \tfrac{1}{2}(a-\operatorname{id}a)$, reduces on the worked elements to

$$ c = -\tfrac{3}{4} + 0, \qquad d = \tfrac{7}{2} + 0, $$

with both anti-fixed parts zero.

The Order, the Archimedean Property and Completeness, Worked

The order on the worked pair. The two worked elements have opposite signs,

$$ c = -\tfrac{3}{4} < 0 < \tfrac{7}{2} = d, $$

and the order separates the two elements while the field structure alone does not: $c < 0 < d$, yet the two have positive squares.

The Archimedean property. For $a > 0$ and any $b$ there is $n \in \mathbb{N}$ with $na > b$. With $a = \lvert c\rvert = \tfrac{3}{4}$ and $b = \lvert d\rvert = \tfrac{7}{2}$, the inequality $n\cdot\tfrac{3}{4} > \tfrac{7}{2}$ holds exactly when $n > \tfrac{14}{3}$, so the smallest witness is

$$ n = 5, \qquad 5\cdot\tfrac{3}{4} = \tfrac{15}{4} > \tfrac{7}{2}. $$

No infinitesimal occurs: the multiples of any positive element eventually exceed any bound, which is the Archimedean property checked on the worked numbers.

Completeness. Let $S = \{c, 0, d\} = \{-\tfrac{3}{4}, 0, \tfrac{7}{2}\}$. It is non-empty and bounded, and

$$ \sup S = \tfrac{7}{2} = d, \qquad \inf S = -\tfrac{3}{4} = c, $$

both attained. A bounded set whose supremum is not attained is $T = \{\tfrac{n}{n+1} : n \in \mathbb{N}\}$, for which $\sup T = 1$ and $1 \notin T$; the existence of the supremum is the completeness axiom, and it is the property that distinguishes $\mathbb{R}$ from $\mathbb{Q}$.

Existence of roots. The positive elements of the worked pair have the rational square roots

$$ \sqrt{\tfrac{9}{16}} = \tfrac{3}{4} = \lvert c\rvert, \qquad \sqrt{\tfrac{49}{4}} = \tfrac{7}{2} = \lvert d\rvert, $$

and the non-negative root of the first is the absolute value of $c$ although $c$ is negative; the existence and uniqueness of the non-negative root is a consequence of completeness.

Density and the irrational. Between $c$ and $d$ there lies the rational $\tfrac{1}{2}$, and also the irrational $\sqrt{2}$; the set $\{a \in \mathbb{Q} : a^2 < 2\}$ is bounded above in $\mathbb{R}$ with supremum $\sqrt{2} \notin \mathbb{Q}$, which is the completeness of $\mathbb{R}$ used to manufacture an irrational from a rational set.

The Place in the Ladder

The real algebra is the first rung of the tensor ladder $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{B}$ of the corpus, and every structure computed above is the one-dimensional case of a structure that the higher rungs enrich.

structure	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{H}$	$\mathbb{B}$
involution	identity only	complex conjugation	quaternion conjugation	quaternion and complex
fixed subspace	the whole algebra	the real line	the real line	Hermitian subspace
anti-fixed subspace	$\{0\}$	the imaginary line	the pure quaternions	anti-Hermitian subspace
nonzero elements invertible	yes, a field	yes, a field	yes, a division algebra	no, there are zero divisors

The real column is the base case that the other worked-example articles continue: Worked Examples in the Complex Algebra adds the nontrivial conjugation and the circle, Worked Examples in the Split-Quaternion Algebra adds the indefinite form and its two regimes, and Worked Examples in the Biquaternion Algebra adds the complex norm and its zero divisors. Each of them reuses the algebraic identities computed here — commutativity, the triviality of the involution decomposition, and the invertibility of every nonzero element — in the case where the involution is no longer trivial and the algebra is no longer a field.

Summary

On the worked pair $c = -\tfrac{3}{4}$ and $d = \tfrac{7}{2}$ the real algebra is exhibited concretely: the sum $\tfrac{11}{4}$, the difference $\tfrac{17}{4}$, and the product $cd = -\tfrac{21}{8}$ with $dc = cd$, the concrete form of commutativity.

The only involution is the identity, so its fixed-point subspace is the whole algebra and its anti-fixed subspace is $\{0\}$; the involution decomposition is the trivial $\mathbb{R} = \mathbb{R}_{\mathbb{R}} \oplus \{0\}$. Both worked elements are units — their inverses are the field inverses $c^{-1} = -\tfrac{4}{3}$ and $d^{-1} = \tfrac{2}{7}$ — and $0$ is the only non-unit, the concrete form of the field property.

On the order, the worked pair straddles the origin, the Archimedean witness for $\lvert c\rvert$ against $\lvert d\rvert$ is $n = 5$, and the completeness axiom is checked on the bounded sets $\{c,0,d\}$ and $\{\tfrac{n}{n+1}\}$, whose suprema are $\tfrac{7}{2}$ and $1$. The article is the first rung of the ladder continued by the complex, quaternion and biquaternion worked-example articles.

Summary of Notation

symbol	meaning
$\mathbb{R}$	the real algebra, the complete ordered field, basis $e_0 = 1$
$c = -\tfrac{3}{4}$, $d = \tfrac{7}{2}$	the worked elements
$\operatorname{id}$	the identity involution, the sole involution
$\mathbb{R}_{\mathbb{R}}$	the fixed subspace of $\operatorname{id}$, the whole algebra
$\{0\}$	the anti-fixed subspace of $\operatorname{id}$
$\lvert a\rvert$	the absolute value, defined by the order
$a^{-1} = 1/a$	the inverse of a nonzero element
$\sup S$, $\inf S$	the supremum and infimum of a bounded set