w, The Missing Constant, Revealed. Article 2. An Equation for Finding the Square Root of Any Number
More is revealed about this unusual number.
A few years back, twiddling at 3 am on an expression I thought might lead to an equation to find the square root of any number, I was delighted to see that I could achieve values very close to sqrt(x) using
sqrt(x) = e^(log10(x)*y)
I recall zooming in on my spreadsheet and toggling values of y, so I could visualize how far from sqrt(x) these values were. Of course, I could have solved for y - had I evidence that the equation was reliable, that would have been justified. However, remember, this was a challenge to find an equation unknown to find sqrt(x), from intuition.
As I zeroed in a value of y using various values of x, I found I could, by adding more digits, achieve higher accuracy. This was a good sign. The improved accuracy of the estimate persisted over values of x. Successive approximation at its best.
The value I was zeroing in on was
y ≈ 1.151292546497…
I kept going…
y ≈ 1.15129254649702284…
Clearly, for this equation to work across all values of x, 1.15129254649702284…, or y, had to have some clear meaning.
Satisfied, I went to sleep.
As I lay there, I turned the equation over in my mind
sqrt(x) = e^(log10(x)*1.15129254649702284…)
sqrt(x) = e^(log10(x)*y)
I considered the equation and wondered if y itself was a function of other numbers, scaled via log10, the natural log, ln.
Upon this possibility, I once again returned to my spreadsheet to toggle values. Eventually, I found that y was also exactly found via 2ln(10)/4, or one-quarter of twice the natural log of 10. Clearly, this is an odd way to express a number, since y is also exactly ln(10)/2. We’ll see why the former was preferred in due course.
If this was important, ln(10)/4 it might need a symbol. The first one that came to mind was w.
So, w = ln(10)/4.
And my equation became
sqrt(x) = e^(log10(x)*2w) where w = ln(10)/4.
The mix of exponents and two forms of logarithms seemed unusual. It can be observed in the equation for finding the square root of a number, this number w bridges base 10 and the natural log.
Therefore, I suspected that y, or w might have other features.
Next time, you’ll see I was right… w indeed has interesting features. Many.
One of them is
f(x,n) = e^((4/n)*log10(x)*w
As a challenge, I’ll leave that there for the interested to ponder, and we’ll discuss this function, most likely next Friday.
If you’re with me this far in recounting my journey, I am glad. If you’re lost, that’s on me.
Feel free to leave any comments or questions. As we work through this series, you will come to know and love w = ln(10)/4 = 0.57564… as well as you know (or should know) pi, i, e, and φ. Stay with the series, you’re in for some surprises.
pi - Wikipedia
e - Wikipedia
i - Wikipedia
φ - Wikipedia
Related
I applaud your genuine dedication to solving these math questions of which I have never pondered. God bless your expansive brain power. I cheer for your success and maybe someday I will understand one tenth of a scintilla of your words!
Where were you when I needed you?
I only took a few college classes at Fairleigh Dickinson University and one of them was Calculus. The professor, if you can call him that, spent more time discussing his past lives than teaching! Somehow, all his past lives were heroic characters going back to ancient Rome.
I barely passed the exams but was graded B+ because he hated failing anyone and thus graded on a curve.
The only thing I really liked about him is that he would set up chairs and hold classes out on the lawn in nice weather.