A ≡ ⟨ A_q, A_r, A_s, A_t ⟩

Another identity defines the meaning of a quintright in terms of real numbers:

A ≡ 2 A_q cos (18°) + 2 A_r cos (36°) + 2 A_s cos (54°) + 2 A_t cos (72°)

It is convenient to give names to certain quintrights:

Q ≡ ⟨ 1, 0, 0, 0 ⟩
R ≡ ⟨ 0, 1, 0, 0 ⟩
S ≡ ⟨ 0, 0, 1, 0 ⟩
T ≡ ⟨ 0, 0, 0, 1 ⟩
Z ≡ ⟨ 0, 0, 0, 0 ⟩

With these, one can write:

A ≡ A_qQ + A_rR + A_sS + A_tT

The quintrights are patently a subset of the real numbers. Note that the integers are denumerable, and ordered quadruplets of them are also denumerable. Since the real numbers are not denumerable, there must be some reals that cannot be written as quintrights, therefore the quintrights are a proper subset of the real numbers.

Quintright comes from quint, meaning one-fifth, and right, meaning right angle. The angles in this formula are all multiples of eighteen degrees, which is one-fifth of a right angle. The first subscript, q, happens to be the first letter of quintright, and the last subscript, t, is also the last letter of quintright.

A + B = ⟨ A_q + B_q, A_r + B_r, A_s + B_s, A_t + B_t ⟩
A − B = ⟨ A_q − B_q, A_r − B_r, A_s − B_s, A_t − B_t ⟩

Multiplication by an integer scalar n is not a problem:

nA = An = ⟨ A_qn, A_rn, A_sn, A_tn ⟩

Here is the rather complicated rule for general multiplication (derivation):

(AB)q	= (AqBr + ArBq)	+ (ArBs + AsBr)	+ (AsBt + AtBs)
(AB)r	= (AqBs + AsBq)	+ (ArBt + AtBr)	+ (3AqBq + 2ArBr + 2AsBs + AtBt)
(AB)s	= (AqBr + ArBq)	+ (AqBt + AtBq)	− (AsBt + AtBs)
(AB)t	= (AqBs + AsBq)	− (ArBt + AtBr)	− (2AqBq + ArBr + 3AsBs + 2AtBt)

This table may help to clarify:

Multipli- cation	Q 1.902113…	R 1.618034…	S 1.175571…	T 0.618034…
Q 1.902113…	3R − 2T 3.618034…	Q + S 3.077684…	R + T 2.236068…	S 1.175571…
R 1.618034…	Q + S 3.077684…	2R − T 2.618034…	Q 1.902113…	R − T exactly 1
S 1.175571…	R + T 2.236068…	Q 1.902113…	2R − 3T 1.381966…	Q − S 0.726543…
T 0.618034…	S 1.175571…	R − T exactly 1	Q − S 0.726543…	R − 2T 0.381966…

In general, division does not work, because it often leads to non-integers as elements of the quotient; and division by zero always fails. However, there is a scheme that will find the quotient of A and B whenever it exists. Let C = A ÷ B; then CB = A, and the multiplication rule can be written this way:

Aq	= Cq(Br)	+ Cr(Bq + Bs)	+ Cs(Br + Bt)	+ Ct(Bs)
Ar	= Cq(3Bq + Bs)	+ Cr(2Br + Bt)	+ Cs(Bq + 2Bs)	+ Ct(Br + Bt)
As	= Cq(Br + Bt)	+ Cr(Bq)	+ Cs(− Bt)	+ Ct(Bq − Bs)
At	= Cq(Bs − 2Bq)	+ Cr(− Br − Bt)	+ Cs(Bq − 3Bs)	+ Ct(− Br − 2Bt)

Regard this as a system of four linear equations in the four unknowns C_q, C_r, C_s, C_t and solve. If the four results are integers, division was successful.

The Bs can be regarded as elements of a four-by-four matrix:

(Br)	(Bq + Bs)	(Br + Bt)	(Bs)
(3Bq + Bs)	(2Br + Bt)	(Bq + 2Bs)	(Br + Bt)
(Br + Bt)	(Bq)	(− Bt)	(Bq − Bs)
(Bs − 2Bq)	(− Br − Bt)	(Bq − 3Bs)	(− Br − 2Bt)

a lengthy expression for the determinant of which is:

5B_q⁴ − 10B_q³B_s − 5B_q²B_r² − 10B_q²B_rB_t − 5B_q²B_s² − 10B_q²B_t²
+ 10B_qB_r²B_s + 10B_qB_s³ − 10B_qB_sB_t² + 1B_r⁴ + 6B_r³B_t − 10B_r²B_s²
+ 11B_r²B_t² − 10B_rB_s²B_t + 6B_rB_t³ + 5B_s⁴ − 5B_s²B_t² + 1B_t⁴

A + B = B + A
(A + B) + C = A + (B + C)
A + Z = A

Similarly, multiplication is commutative and associative, and R − T is the multiplicative identity:

AB = BA
(AB)C = A(BC)
A⟨ 0, +1, 0, −1 ⟩ = A

Further, multiplication distributes over addition, and when anything is multiplied by the additive identity Z, the result is Z itself:

A(B + C) = AB + AC
AZ = Z

Although these quintrights do form an integral domain, they do not form a division ring because many of them fail to have multiplicative inverses. For instance, the reciprocal of ⟨ 1, 0, 0, 0 ⟩ would be ⟨ +0.4, 0.0, −0.2, 0.0 ⟩.

Quintrights of the form ⟨ 0, n, 0, m ⟩ constitute an integral subdomain.

The cosines can be written in radical form:

Q	= 2 cos(18°)	= ½ √(10 + √20)	= 1.902113…
R	= 2 cos(36°)	= ½ (√5 + 1)	= 1.618034…
S	= 2 cos(54°)	= ½ √(10 − √20)	= 1.175571…
T	= 2 cos(72°)	= ½ (√5 − 1)	= 0.618034…

Observe from these that ⟨ 0, +n, 0, −n ⟩ = n, thus the integers are a subset of the quintrights. Because cos(18°) is irrational, so is ⟨ 1, 0, 0, 0 ⟩, hence the integers are a proper subset (more specifically a subdomain) of the quintrights.

The golden ratio is R = 1.618034…, while T = 0.618034… is the golden mean. Powers of the golden mean exemplify a sequence of quintrights where the terms approach zero, but where the components of the terms are unbounded:

T 1	= ⟨ 0,	0,	0,	+1 ⟩	= 0.618034…
T 2	= ⟨ 0,	+1,	0,	−2 ⟩	= 0.381966…
T 3	= ⟨ 0,	−1,	0,	+3 ⟩	= 0.236068…
T 4	= ⟨ 0,	+2,	0,	−5 ⟩	= 0.145898…
T 5	= ⟨ 0,	−3,	0,	+8 ⟩	= 0.090170…
T 6	= ⟨ 0,	+5,	0,	−13 ⟩	= 0.055728…
T 7	= ⟨ 0,	−8,	0,	+21 ⟩	= 0.034442…
T 8	= ⟨ 0,	+13,	0,	−34 ⟩	= 0.021286…
T 9	= ⟨ 0,	−21,	0,	+55 ⟩	= 0.013156…
T10	= ⟨ 0,	+34,	0,	−89 ⟩	= 0.008131…
T11	= ⟨ 0,	−55,	0,	+144 ⟩	= 0.005025…
T12	= ⟨ 0,	+89,	0,	−233 ⟩	= 0.003106…
T13	= ⟨ 0,	−144,	0,	+377 ⟩	= 0.001919…
T14	= ⟨ 0,	+233,	0,	−610 ⟩	= 0.001186…
T15	= ⟨ 0,	−377,	0,	+987 ⟩	= 0.000733…
et cetera

This shows that is not generally possible to approximate a quintright by approximating its components.

Section 4. Given any two quintrights B and C, if these four conditions apply:

• B_q = C_q
• B_r = C_r
• B_s = C_s
• B_t = C_t

To prove the converse, however, is more complicated. First, it can be established that if A = 0, then A_q = A_r = A_s = A_t = 0. Second, to investigate B = C, write B − C = 0. Then

(B − C)_q = (B − C)_r = (B − C)_s = (B − C)_t = 0
B_q − C_q = B_r − C_r = B_s − C_s = B_t − C_t = 0
B_q = C_q, B_r = C_r, B_s = C_s, B_t = C_t

Component unicity is the name of this property that, when two quintrights are equal, their respective components must also be equal. It does not necessarily exist when the right angle is divided into other than five parts. For instance, it fails when the right angle is divided into nine parts because one of the angles created is sixty degrees, the cosine of which is rational. Here is an example of two "nonarights" that have different components but which are nonetheless equal:

2 (0) cos (10°) + 2 (+1) cos (20°) + 2 (0) cos (30°) + 2 (−1) cos (40°) +
2 (0) cos (50°) + 2 (+1) cos (60°) + 2 (0) cos (70°) + 2 (−1) cos (80°) = 1

2 (0) cos (10°) + 2 (0) cos (20°) + 2 (0) cos (30°) + 2 (0) cos (40°) +
2 (0) cos (50°) + 2 (+1) cos (60°) + 2 (0) cos (70°) + 2 (0) cos (80°) = 1

More concisely,

⟨ 0, +1, 0, −1, 0, +1, 0, −1 ⟩ = 1
⟨ 0, 0, 0, 0, 0, +1, 0, 0 ⟩ = 1