Exploring Logarithms, Exponentials, Hyperbolics and the Transcendental w

w is nothing special. But it teaches us a lot.

In a few past articles here on Popular Rationalism, we have seen the transcendental w = ln(10)/4.

In other settings, I noticed also that w = (ln(e)+ln(π))/(log10(e)+log10(π))/4. Not much ado about this if you ask some; surely, it’s just a parlor trick that, via prime factorization of 10 and some canceling out, we can see w = 1/4(ln(2) + ln(5)) as well. Mathematicians working in earnest call that “proof”, mathematicians criticizing pedestrians like myself will say “that’s not interesting, that’s just how math works”. I’d agree with the latter. And yes, I’m dropping teases.

Having found a function w = f(π) and thus implying there must be π = f(w), it’s either very impressive or obvious (that’s just how math works) that

π = 5cos^(−1)*((w(1+sqrt(5))/ln(10))

and

w = (ln(10)*cos(π/5))/(1+sqrt(5)).

Take your pick.

In a comment on an earlier article, I managed to explain the use of inscribed polygons within circles and the resulting hemicircles to understand their use in calculating (exactly) the area of a circle without using π as a function of w. It’s ugly, but the workup is too extensive to show here. But it works.

Of course I also share that sqrt(x) = e^(log10(x)*2w).

Like a caveman, scrawling in the sand, I also found that we can find the nth root of any via

f(x,n) = e^((4/n) * log10(x)*w = x^1/n

It’s an interesting fact and a complex way to find the nth root of any number, but x^1/n is far simpler. But still, there is w.

w and Hyperbolics

While first exploring w, Israel, a mathematician, generously pointed out to me that

w = arctanh(11/9-(2*sqrt(10))/9)

This led me to hyperbolics and a puzzle: Why such a peculiar, exact expression?

Polymaths will see some coordinate geometric potential, perhaps a relation to a distance formula, given the appearance of 2(sqrt(10)), another thread…

And I found

w = 1/2 ln((10 - sqrt(10))/(sqrt(10) - 1))

Which simplifies to

w = ln(10^(1/4))

Interesting, but not very exciting. Here’s something with more meat to chew on for the topic of today.

Calculation of the Area Under a Hyperbola Using w

The standard form of a hyperbola is

x^2/a^2 - y^2/b^2

where a and b are the hyperbola's semi-major and semi-minor axes, respectively.

Define w = ln(10)/4

The integration bounds are expressed in terms of hyperbolic functions. Define:

x1 = acosh(w) (Lower bound of integration)

x2 = acosh(w+k) (Upper bound of integration)

where k is an adjustable parameter representing the interval of integration.

The area A under the hyperbola between x1 and x2 is given by

A = ab/4[sinh(2(w+k)) - sinh(2w)] - (ab/2)k

The fact that

w = arctanh(11/9-(2*sqrt(10))/9) (Israel)

and the requisite

tanh(w) = (11/9 - 2(sqrt(10))/9

exist, and that we have now defined a function to calculate the area of a hyperbola in terms of w shows that an intrinsic relationship exists between hyperbolic functions and logarithms; logarithms are the bridge between exponentials and hyperbolics.

That relationship seems more than descriptive to me. After all, we do find w in hyperbolics, exponentials, logarithms, and geometrics (and trigonometrics, not shown yet).

In fact, the constant w seems to encapsulate properties useful to transform problems… and allows us to move between and among complex expressions, whether in logarithmic, exponential, or hyperbolic forms.

The hyperbolic arctangent function serves as a tool for expressing these transformations in another, and more unified way.

The value of w connects geometric interpretations (like angles and areas) with algebraic transformations (like roots and logarithms).

This interconnectedness underscores the versatility and importance of w in both theoretical and practical contexts.

Admittedly, it is not the most straightforward way, but the relationships exist. Nevertheless, one less from w appears to be:

(1) Accurately executed, simplification in math leads to information compression without loss. But sometimes there is loss - loss in the structure of the equation, identity, function, or expression that conveys more meaning.

It also implies that

(2) There exists a map between domains of mathematics that can be traversed directly using functions (known) but also using constants like w (prior to my work, known by few, appreciated by fewer, and, I expect, never understood fully by any).

The mathematical constant w = ln(10)/4 provides an example of a numerical constant that bridges hyperbolic, logarithmic, and geometric domains (and trigonometric, not shown yet), providing a unified framework for understanding and transforming various mathematical expressions. These relationships implicitly suggest the utilitarian value of w, highlighting its significance as a transformative constant.

I’m having fun, of course, by showing these relationships, which will further amuse some mathematicians to no end and call me out for having a sense of humor:

w = (ln(e)+ln(π))/(log10(e)+log10(π))/4

w = (ln(cosh(π/x))+ln(sinh(π/x)))/(log10(cosh(π/x))+log10(sinh(π/x)))/4 for all x > 0

Mathematicians can play with these a while; if you have comments, drop them below. In fact, your reactions overall to all about w are welcome.

Just so we know where we stand, as a reminder for the numberists who like to see values…

3.1622776601684… is the square root of 10 and is an irrational number;

ln(sqrt(10)) = 1.151292546497…

w = 1/2 of 1.151292546497… or w = 0.5756462732485…

Numbers That Act Like Functions and the Infinity of Counting Systems

Next week, we will explore some more elegant places w comes into view involving relationships among other constants.

We will revisit the idea that numbers can act like constants (I am calling them “bridging scalars” or “universal modulators”) philosophically later in an article on the special significance sometimes attributed to constants like π, i, e, and ϕ, and perhaps w. It will be a blast.

Errors are attributed to me, of course. To let them persist would be just cruel to humanity. Let me know!

PS At this point, I want to speculate that there are 2∞ modulators like w of the form ln(x)/y, any one of which might reveal functional maps among mathematical domains. So, for those who might think w is not special, I would tend to agree.