Some Details on the Rational Unit ℧₁ for Logarithmic and Information‑theoretic Quantities
One of the formulas most often cited as “the most beautiful equation in mathematics” is
eiπ ＋ １ ＝ ０.
It contains the additive and multiplicative identities 0 and 1, the imaginary unit 𝑖,
and the two most famous transcendental numbers, 𝑒 and 𝜋.
The symbols are linked through multiplication (𝑖 and 𝜋), exponentiation (𝑒 and
𝑖𝜋),and addition (𝑒𝑖𝜋 and 1), and finally the entire left-hand side collapses
to 0 through the equality sign.
Tracing this back, the equation is simply the special case 𝜃= 𝜋 of
eiθ ＝ cos θ ＋ i sin θ
In other words,
trigonometric functions can be interpreted as exponential functions along the imaginary axis.
Everything above is today’s prelude.
The real topic begins below: how to choose the unit symbol.
Why I wrote “(2/π) arc sin(1)” in tables.pdf
In the explanation of the radian in tables.pdf,
I intentionally wrote the expression
(2/π) arc sin(1),
which looks like a trivial identity.
The point was this:
If arc sin(1) is taken as one quarter of the rational full‑turn plane angle Ω1,
then the unrational unit radian is simply (2/𝜋) times that.
This brings us to a long‑standing question.
What symbol should represent the rational unit corresponding to the irrational unit neper?
Until now, I had used “f” (for “figure”) as the unit symbol:
fk ＝ log(2k), k = 1(bit), d(figure), 4(nibble), 8(byte),…,
But “f” is too generic, and many readers may not find it intuitive.
This issue remained unresolved for years—until recently,
when I suddenly realized that the prelude above connects directly to this problem.
If trigonometric functions are exponentials along the imaginary axis, and if
the rational and irrational units of plane angle are related by inverse trigonometric functions,
then the rational unit of logarithmic quantities—defined by inverse exponentials along
the real axis—should naturally be written as ℧.[1]
In other words:
For plane angle, the rational unit is Ω.
For logarithmic quantities, the rational unit should play the same structural role.
That role is perfectly matched by ℧.
The more I think about it, the more it feels as though there was never any other choice.[2]
℧ was created long ago for a completely different purpose, and its original usage faded away
at just the right time—not conflicting with the Universal Unit System.
It is as if the structure itself had been “saving the symbol” for this moment.
Therefore, in the next revision,
the rational unit of logarithmic and information‑theoretic quantities will be written as
℧k ＝ log(2k), k = 1(bit), z(figure), 4(nibble), 8(byte),…[3]
Notes
[1] For the duality between plane angle and logarithmic quantities as units,
see Appendix A.2 of univunit-e.pdf.
[2] When defining the solid angle Ω2 so that its role serves as the dual quantity to
the impedance ♮Ω, the full plane angle naturally became Ω1 as a consequence.
What surprises me is that this earlier decision, made for an entirely different reason,
turned out to be completely unrelated to the present line of reasoning.
[3] As a side note, the 12‑based digit z (formerly d) satisfies log2(12.) = z.
Thus, the main value of b1.58 (2024‑07‑16) can be expressed as ℧z-2 (trit?).
Note: ℧1 ≒ 3 dB as a logarithmic quantity.
Related Article 2026-01-12 Some Details on the Unit mol for Amount of Substance
-> Japanese
eiπ ＋ １ ＝ ０.
It contains the additive and multiplicative identities 0 and 1, the imaginary unit 𝑖,
and the two most famous transcendental numbers, 𝑒 and 𝜋.
The symbols are linked through multiplication (𝑖 and 𝜋), exponentiation (𝑒 and
𝑖𝜋),and addition (𝑒𝑖𝜋 and 1), and finally the entire left-hand side collapses
to 0 through the equality sign.
Tracing this back, the equation is simply the special case 𝜃= 𝜋 of
eiθ ＝ cos θ ＋ i sin θ
In other words,
trigonometric functions can be interpreted as exponential functions along the imaginary axis.
Everything above is today’s prelude.
The real topic begins below: how to choose the unit symbol.
Why I wrote “(2/π) arc sin(1)” in tables.pdf
In the explanation of the radian in tables.pdf,
I intentionally wrote the expression
(2/π) arc sin(1),
which looks like a trivial identity.
The point was this:
If arc sin(1) is taken as one quarter of the rational full‑turn plane angle Ω1,
then the unrational unit radian is simply (2/𝜋) times that.
This brings us to a long‑standing question.
What symbol should represent the rational unit corresponding to the irrational unit neper?
Until now, I had used “f” (for “figure”) as the unit symbol:
fk ＝ log(2k), k = 1(bit), d(figure), 4(nibble), 8(byte),…,
But “f” is too generic, and many readers may not find it intuitive.
This issue remained unresolved for years—until recently,
when I suddenly realized that the prelude above connects directly to this problem.
If trigonometric functions are exponentials along the imaginary axis, and if
the rational and irrational units of plane angle are related by inverse trigonometric functions,
then the rational unit of logarithmic quantities—defined by inverse exponentials along
the real axis—should naturally be written as ℧.[1]
In other words:
For plane angle, the rational unit is Ω.
For logarithmic quantities, the rational unit should play the same structural role.
That role is perfectly matched by ℧.
The more I think about it, the more it feels as though there was never any other choice.[2]
℧ was created long ago for a completely different purpose, and its original usage faded away
at just the right time—not conflicting with the Universal Unit System.
It is as if the structure itself had been “saving the symbol” for this moment.
Therefore, in the next revision,
the rational unit of logarithmic and information‑theoretic quantities will be written as
℧k ＝ log(2k), k = 1(bit), z(figure), 4(nibble), 8(byte),…[3]
Notes
[1] For the duality between plane angle and logarithmic quantities as units,
see Appendix A.2 of univunit-e.pdf.
[2] When defining the solid angle Ω2 so that its role serves as the dual quantity to
the impedance ♮Ω, the full plane angle naturally became Ω1 as a consequence.
What surprises me is that this earlier decision, made for an entirely different reason,
turned out to be completely unrelated to the present line of reasoning.
[3] As a side note, the 12‑based digit z (formerly d) satisfies log2(12.) = z.
Thus, the main value of b1.58 (2024‑07‑16) can be expressed as ℧z-2 (trit?).
Note: ℧1 ≒ 3 dB as a logarithmic quantity.
Related Article 2026-01-12 Some Details on the Unit mol for Amount of Substance
-> Japanese
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