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!
Thank you - I'm going slow due to the importance of the full set of discoveries. w provides a pathway to understanding theoretical mathematics across the board, with exceptional potential in the applied realms.
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.
Why use the decimal logarithm and mix it with natural exponential? Not everyone in the universe has 10 fingers. Generally, a^b = exp (b ln a), which means that when you have the complete theory for exp and ln, then you also have the complete theory for f(x)^g(x) (i.e., limits, continuity, differentiability, derivatives). Then other base logs can be written as log_b (x) = (ln x)/(ln b), so knowing how the natural logarithm works means you know how all other logarithms with any other base works. The point of base e is that the derivative of exp(x) is equal to itself. This is the main reason why it is interesting. Another approach is to start by defining ln x first by integrating 1/x; can't do that for any other base, because starting with no log functions defined, you need log functions to define log functions via rational integral for every base except for the natural one.
I quite agree with you on the foundational importance of natural logarithms and exponentials in mathematics. While base 10 logarithms are convenient in certain contexts, the natural logarithm and exponential functions provide a universal framework that simplifies and unifies many mathematical concepts. The blending of bases in this earlier exploration showcases the flexibility of logarithmic and exponential transformations, but the underlying principles always return to the elegance and utility of the natural base 𝑒 , as we will see. I gave a hint of that in the article.
The deeper theoretical question is defining the meaning a^b, which is easy enough when b is a natural number, then integer, then rational number. The piano falls on our head when b is a real irrational number. One way out is to define a^b = exp (b ln a), prove this is consistent with the previous definitions for b rational, and then prove that this extended definition for a^b still respects all the properties that you want powers to respect. A schizoid thing about a^b is that as you broaden the definition in terms of b, you have to intensify the restrictions imposed on a. Finally, the next big headache is when you want b to a complex number and even worse, when you want a to be a complex number too.
I can tell you have a glimpse of where this is all going. We'll be going through Euler's formula (where complex exponentiation can be expressed in terms of trigonometric functions) and its use in this context for complex numbers. Likely in 3-4 weeks. You're also right about the complex numbers forms, and operations, of course. These are tools, however, for studying w, which has quite a few interesting relationships worth considering by itself. So much so that it may in fact have implications for simplifying the standard exploration of complex logarithm and power functions, Riemann surfaces... there may be additional information or much, much simpler ways of seeing through that multivalued solution space. We'll generalize late, but I think ln(x)/y has great potential for moving around in maths. I'm taking it slow to help people check things out on their own if they would like. It should be interesting to see what your thoughts are as layers are revealed each Friday. PS While not everyone in the universe has ten fingers, most will be able to count ours. I'm hoping they will teach us a few things about counting.
A lot of these extensions to the a^b definition may seem to be arbitrary and wacked out, but for the most part they can be justified by a uniqueness argument. Meaning that, if you want powers to satisfy certain basic algebraic properties, then only one definition extension is available. I think this goes a bit haywire when you move into complex numbers, but the uniqueness argument holds up throughout real numbers. By the way, one lovely controversy is the definition of 0^0 (zero to power zero). I am of the opinion that 0^0 = 1, even though the corresponding limit form is indeterminate; in the exact sense, I don't think there is a problem. The problem arises when one or both of the zeroes in 0^0 are not exactly zero, then the expression misbehaves a bit, but is still properly defined.
The big question is why you’d want a more difficult and expensive calculation than what you started with. Calculating roots is much “cheaper” than logarithms.
Thank you yes, correct all across and you have notice the equation for the nth root of any number x. Well done. In the first article (published last Friday) I explained the origins of this area of inquiry. Philosophically, however, simplifying things in math sometimes stops us from appreciating relationships among functions and among areas of inquiry. Some like classical music, I like baroque kind of thing. I'm sure my random walk will amuse you when it's all told.
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!
Thank you - I'm going slow due to the importance of the full set of discoveries. w provides a pathway to understanding theoretical mathematics across the board, with exceptional potential in the applied realms.
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.
Why use the decimal logarithm and mix it with natural exponential? Not everyone in the universe has 10 fingers. Generally, a^b = exp (b ln a), which means that when you have the complete theory for exp and ln, then you also have the complete theory for f(x)^g(x) (i.e., limits, continuity, differentiability, derivatives). Then other base logs can be written as log_b (x) = (ln x)/(ln b), so knowing how the natural logarithm works means you know how all other logarithms with any other base works. The point of base e is that the derivative of exp(x) is equal to itself. This is the main reason why it is interesting. Another approach is to start by defining ln x first by integrating 1/x; can't do that for any other base, because starting with no log functions defined, you need log functions to define log functions via rational integral for every base except for the natural one.
I quite agree with you on the foundational importance of natural logarithms and exponentials in mathematics. While base 10 logarithms are convenient in certain contexts, the natural logarithm and exponential functions provide a universal framework that simplifies and unifies many mathematical concepts. The blending of bases in this earlier exploration showcases the flexibility of logarithmic and exponential transformations, but the underlying principles always return to the elegance and utility of the natural base 𝑒 , as we will see. I gave a hint of that in the article.
The deeper theoretical question is defining the meaning a^b, which is easy enough when b is a natural number, then integer, then rational number. The piano falls on our head when b is a real irrational number. One way out is to define a^b = exp (b ln a), prove this is consistent with the previous definitions for b rational, and then prove that this extended definition for a^b still respects all the properties that you want powers to respect. A schizoid thing about a^b is that as you broaden the definition in terms of b, you have to intensify the restrictions imposed on a. Finally, the next big headache is when you want b to a complex number and even worse, when you want a to be a complex number too.
I can tell you have a glimpse of where this is all going. We'll be going through Euler's formula (where complex exponentiation can be expressed in terms of trigonometric functions) and its use in this context for complex numbers. Likely in 3-4 weeks. You're also right about the complex numbers forms, and operations, of course. These are tools, however, for studying w, which has quite a few interesting relationships worth considering by itself. So much so that it may in fact have implications for simplifying the standard exploration of complex logarithm and power functions, Riemann surfaces... there may be additional information or much, much simpler ways of seeing through that multivalued solution space. We'll generalize late, but I think ln(x)/y has great potential for moving around in maths. I'm taking it slow to help people check things out on their own if they would like. It should be interesting to see what your thoughts are as layers are revealed each Friday. PS While not everyone in the universe has ten fingers, most will be able to count ours. I'm hoping they will teach us a few things about counting.
A lot of these extensions to the a^b definition may seem to be arbitrary and wacked out, but for the most part they can be justified by a uniqueness argument. Meaning that, if you want powers to satisfy certain basic algebraic properties, then only one definition extension is available. I think this goes a bit haywire when you move into complex numbers, but the uniqueness argument holds up throughout real numbers. By the way, one lovely controversy is the definition of 0^0 (zero to power zero). I am of the opinion that 0^0 = 1, even though the corresponding limit form is indeterminate; in the exact sense, I don't think there is a problem. The problem arises when one or both of the zeroes in 0^0 are not exactly zero, then the expression misbehaves a bit, but is still properly defined.
Ah, 0. My favorite number given it's status thereof be decree, and post-hoc at that. I can see we are going to have much to compare notes on...
I pondered for ten minutes what your numbers say
I wasted 10 whole minutes of my equationless math day.
LMAO thanks, Maurine. Give it time. There will be something for you appreciate, I'm sure.
I do appreciate this. You intelligence in this area stuns me. I have to rely on the right side of my brain. Thanks
I’m interested to see what comes of your constant w. But here’s a quick summary of the algebra:
sqrt(x) = x^(1/2) = exp(ln(x^(1/2)) = exp(ln(x)/2) = exp(log10(x)/(2log10(e))) = exp(log10(x)*y) where y = 1/2log10(e) = 1/(2ln(e)/ln(10)) = ln(10)/2ln(e) = ln(10)/2*1 = ln(10)/2 = 2w, where w = ln(10)/4.
And
f(x,n) = exp((4/n)*log10(x)*w) = (exp(log10(x)*2w))^(2/n) = (sqrt(x))^(2/n) = (x^(1/2))^(2/n) = x^((1/2)*(2/n)) = x^(2/2n) = x^(1/n) = nth_root(x)
The big question is why you’d want a more difficult and expensive calculation than what you started with. Calculating roots is much “cheaper” than logarithms.
Thank you yes, correct all across and you have notice the equation for the nth root of any number x. Well done. In the first article (published last Friday) I explained the origins of this area of inquiry. Philosophically, however, simplifying things in math sometimes stops us from appreciating relationships among functions and among areas of inquiry. Some like classical music, I like baroque kind of thing. I'm sure my random walk will amuse you when it's all told.