Another Mathematical Function Using the Transcendental Constant w

We are taking our time to let people with rusty math skills catch up if they would like to...

In mathematics, numbers do not only serve to represent quantitative measures such as counts, lengths, and volumes. Some serve as constants that help us transform information. This is generally true for any dimensioned number (such as 1 second, or 2 hours, the dimension being time) in that they carry information. But it is also true that some dimensionless numbers, like 𝜋 = 3.141592653, have the ability to help us transform information in profound and interesting ways.

The transcendental number 𝜋 represents the ratio of the circumference of a circle to its diameter, both of which are lengths, so their units (e.g., inches) cancel out. This characteristic means 𝜋 can be used in various mathematical and physical formulas without affecting the units of measurement in those equations. For example, in the formula for the circumference of a circle, 𝐶=2𝜋𝑟, the radius 𝑟 has units of length, and the 2𝜋 factor does not introduce any additional dimensions. The same principle applies to formulas involving area (𝐴=𝜋𝑟2) and volume where 𝜋 appears.

Constants like 𝜋 can act as bridges, linking diverse concepts and simplifying complex calculations. At first glance, our number w = ln(10)/4, first introduced as a transformative number that connects natural logarithms (ln) and common logarithms (log10), has no special place in mathematics. In fact, it looks like it’s just a part of the bigger picture in logarithms and an incomplete part at that.

So far, we have explored how w was empirically discovered and utilized in deriving the square root of numbers. Today, we extend this exploration to uncover how w can be used in calculating the nth root of any number, showing a new and somewhat unexpected mathematical computation.

First, a little background on logarithms.

The Role of Logarithms

Logarithms, by their nature, transform multiplication into addition, an elegant simplification that has powered many technological and scientific advances. The natural logarithm, denoted as ln(x), uses the transcendental number e as its base. On the other hand, the common logarithm, or log10, operates with 10 as its base, often used for its convenience in calculations involving powers of 10. As one reader hinted, log10 exists because we count with our fingers.

It’s true that, by far, most of our mathematical framework is grounded in the decimal system, a base-10 structure that naturally aligns with how we understand and interact with numbers daily. Given this fundamental reliance, it is unsurprising that a logarithmic function based on ln(x) could reveal connections across various mathematical domains. Do things look different if we count differently?

Bridging ln and log10 with w

The constant w can be seen as a mediator between these two logarithmic scales. In previous articles, we hinted at its ability to transcend traditional roles, acting as a number and a functional tool in mathematical transformations. This quality of w positions, perhaps somewhat uniquely, as a bridge, making it less about the specific values and more about the relationships it fosters between different logarithmic operations and, as we will see, various domains of mathematical inquiry.

The connection between w and the common logarithm function, log10(x), arises from the properties of logarithms and their relationships across different bases. Here’s a detailed, basic breakdown of how w links to log10(x):

Basic Logarithmic Relations

To understand the connection, recall the change of base formula for logarithms:

logb(a) = logc(a)/ logc(b)

where

logb(a) is the logarithm of a with base b

logc(a)/ logc(b) are the logarithms of a and b with another common base c

Using b = 10 and c = e (natural logarithm), the formula to convert from natural logarithm to common logarithm is:

log10(x) = ln(x)/ln(10)

This is known as the change of base formula.

A distracting illustration that brings some of it all back…

Recalling that w = ln(10)/4, rearranging gives:

ln(10) = 4w

Substituting back into the change of base formula, we get:

log10(x) = ln(x)/4w

This shows that log10(x) can be expressed as a function of w. In fact, w can be seen as a scaling factor that can transform the natural logarithm of any number x into its common logarithm log10 through multiplication by the reciprocal of 4w. This relationship highlights w’s potential role as a bridging scalar that directly connects the natural and common logarithmic scales, leveraging the fundamental logarithmic properties across different bases, particularly focusing on the interplay between e and 10.

This link underscores w's potential utility in mathematical calculations involving logarithms, especially where conversions between natural and common logarithms are frequent or necessary.

Towards the Nth Root

The beauty of logarithms lies in their ability to demystify exponentiation. By extending the logarithmic properties through w, we find a novel approach to nth roots.

The equation of interest is

nth root of x = f(x,n) = e^((4/n)*log10(x)*w

Eww, right?

(Reader challenge: derive f(x,n) from first principles. Showing all of your steps will demonstrate your agility in the area of logarithms and exponentiation.)

First, for fun, here are the first 519 digits of w, courtesy Wolfram alpha…

w = ln(10)/4 = 0.5756462732485114210044978636710910519002753721571932440083319752418931524193381200589993012723995745854919460105715621583523813662707016891665718422746954223707268020813886702109499737065582996320983763272413444331572115408415555719245549716866359168686010608185912887622335787348478699048511000555262754285437000922021003161770171391935804057088805028701165928914780343362686964236920865904198025451611267662000069375671229186637646714233918355167645284107306138601439731431052060328673922254189735064194077839229823...

w is transcendental. A transcendental number is a real or complex number that is not algebraic. In other words, it is not the root of a non-zero polynomial with rational coefficients. The most well-known transcendental numbers are π (pi) and e. These numbers cannot be expressed as solutions to any finite-degree polynomial equation with rational coefficients. While only a few classes of transcendental numbers are known, they are not rare: almost all real and complex numbers are transcendental. It’s important to note that all transcendental real numbers are irrational, but not all irrational numbers are transcendental.

But wait… That may be all fine and well, but… I’m sure you’re wondering… why would anyone want such an ugly equation like

nth root of x = f(x,n) = e^((4/n)*log10(x)*w

when the simple expression 𝑥^1/𝑛 represents the nth root of 𝑥 - and shows how it can be calculated much more efficiently?!?

• Understanding Transforms: Using a more complex function like 𝑓(𝑥,𝑛) can be educational, helping students or researchers understand how different mathematical concepts like exponentials, logarithms, and roots interrelate. It can also provide a deeper insight into the nature of exponential and logarithmic functions. (The fact that x^1/n = e^((4/n)*log10(x)\*w itself is interesting. However, proving it is not trivial and involves prime factorization, but you can check it empirically.

• Conceptual Bridging: Experimenting with alternative forms of familiar functions can lead to new insights or methods for those exploring mathematical theory or developing new mathematical tools. For instance, expressing roots as logarithms and exponentials might lead to new generalizations or specialized tools useful in specific contexts.

• Simplicity for its Own Sake Can Lead to Information Loss. A general principle in mathematics is that equations should be as simple as possible when presented in their final form. As a result, some physicists marvel at the uncanny goodness of fit between the empirical world and mathematics. (They are, of course, ignoring the trashbin bias). However, it is valid and sometimes important to leave functions and equations in an undignified state to show the functional relationships among the moving parts if known. This is more than a philosophical position. This is more than a philosophical position; it ensures that we do not overlook essential aspects of the relationships involved. If a generation of mathematicians, for example, canceled out an important factor because they assumed all of the dimensions of those variables were the same when, in fact, they had fundamental differences, the correct equations themselves might be more complex than the results in which, say, time was canceled out across the board.

Without revealing too much, we will dive into specific relationships in the forthcoming discussions, harnessing the properties and relationships of w to explore what has traditionally been seen as completely understood calculations. We will show the peculiar ubiquitousness of w as a constant, as it pops up in many places across mathematics if we bother to look for it. This approach can broaden our mathematical toolkit, and can also enhance our understanding of the fundamental properties of numbers and operations. We might learn something or two about abstract mathematics, number theory and realms of mathematics not often tread.

As we explore w, each step unveils more about its potential to connect and transform. Stay tuned as we continue to uncover the capabilities of w, promising to enlighten and inspire innovation in mathematical thinking.

w Puzzle of the week (Watch out, it may be a trick question!) and a teaser for next week’s article

I’m a big fan of mathematical constants and important numbers like sqrt(2), sqrt(5), and 1/137. I enjoy studying them, their relationships with each other, and their notable roles in certain equations. 𝜋, for example, is found in hundreds of equations someone, somewhere, sometime found - and found useful. (See Wikipedia page). They highlight relationships in mathematics by making things more readily computable and sometimes understandable; sometimes, they are just a mystery.

So, here’s the puzzle:

In noodling around with w, I happened to notice this equation also yields w:

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

Whoa! Does this mean that w is somehow related in a fundamental way to e and 𝜋?

The answer is easier than you might expect….

If you know the answer, write “Yes” or “No” in the comments, but don’t explain why until 5/24/2025.

Hint for the w Article for Next Week: w does indeed have special, perhaps surprising relationships with some key transcendental constants that reveal that it may in fact be… the missing constant… How much I will reveal I have not yet decided… Or will we see how it’s useful in different domains of math? Hmmm… nice to have options…