I have some things I need to work on so it may be a minute before I get this finished to a finish so this will remain as it is for a spell.
One, two, three, four…
These are the numbers we all first learn. They are often called the natural numbers or counting numbers.
Our present journey is on a number of its own system but one borne from counting and definitive of the natural order.
e
It is difficult to start in the middle when discussing most of maths or its history but we are on a subject singular in its own ways.
The number was first located by John Napier (or maybe William Oughthred), and then made more of a thing by Euler, who called it, though was not the first to do so, e. It was a convenient letter. Leonard Euler is one of the most brilliant minds of documented history so e became commonly called Euler’s number. In maths, where it is powerful number, it is e or the base of the natural log.
Our two great universal languages are math and music with all other arts being music of a different medium. One can carry a dialogue across land and time within the discourse allowable by maths. My first year teaching high school was rough. That’s how it is for everyone.
We had a lot of fun and there was some learning done but it could be a hot mess at times. I had one calculus class and that was nice respite from the other sections. There were 2 cousins in the class, one who spoke little English and one who was not as good at math but fluent in Arabic. Because the dialogue was centered on the math, we found little trouble in getting things sorted out. I had Cantonese student in an Algebra 2 class that same year. Her English was limited but her work demonstrated understanding and mastery. Perhaps, one reason international students can often do well in math classes is because the core language is shared.
Students first come across e studying compound interest or logarithms. Logarithms sprung from a most relatable problem. They are a way to multiply and divide huge numbers when you do not have a calculator. John Napier prefaces his Mirifici logarithmorum canonis descriptio,
“Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”
No one likes to do long division and mistakes will happen, particularly when numbers get longer. Oceanic navigators and pilots can ill afford the results of errors from tedious calculations. A degree off at sea can put one hundreds or more miles off track. Navigating seasons and distances has piloted mathematical development since earliest recorded history. This continues as we push the very limits of our horizons.
Until the late 1900s, math students learned to use logarithms via slide rules and log tables as way to do calculations. William Oughtred made the first slide rule in 1622, based on a variation of Napier’s Bones, created by Napier in 1617.
Napier’s goal was to turn the numbers into a different type of number so that multiplying and dividing would be like adding and subtracting. He
When you multiply like bases, you add the exponents.
x2*x5=x7
It’s easy to see when you extend it out.
x*x * x*x*x*x*x = x7
In a more generic sense, we can use a and b to mean any two numbers and state that:
xa*xb = x(a+b)
Logs are exponents of a specified base. log(2)(8) is the number we raise 2 to in order to get 8, 3. 2
Traditionally, they have been written with a subscript for the base value.
log464 = the exponent of 4 that yields 64.
Computers were trained on regular syntax and so we write it like so log(4)(64) for coding. Some people can understand it more clearly that way.
43 = 64 so log464 = 3.
Remember, when multiplying common bases, we add the exponents.
log2(4*2) = log24+log22
This seems like very little but what we have done is turn multiplying into adding and dividing into subtracting.
Base 10 logs are most common. These have 10 as the base value and the log is the exponent we raise the 10 to in order to get the desired value.
Logs written without a specific base value have an understood base of 10.
log100 = the power of 10 that yields 100, 2. 102=100
log1000 = the power of 10 that yields 1000, 3. 103 = 1000
100*1000= 102*103=105
log100000=5
log(100*1000)=log100 + log1000 = 2+3=5.
More numbers have messy logs than neat ones. log(3) is a wacky decimal. So is log(3.1) and every other number but rational powers of 10.
For logarithmic calculations to work, you need to have way to translate a number to its log and back. Folks of my age and older learned how to use log tables at some point in school.
These rows and columns are like the secret decoder that so anyone with the know-how could take advantage of the logarithms to do more.
Back in the sixteen-teens, they did not have such tables. John Napier devoted much of his life to calculating tens of thousands of values for his log tables. This was something he felt of real value for the world. Considering he turned division into subtraction, he did.
In the appenix to the second printing of his work, there are tables that use 2.718 as a constant. This section is believed to be written by William Oughtred.
The logarithm with a base of e gets its own name, the natural log or ln.
Bernoulli and infinite limits of definition.
What happens when you divide by a very large number?
What happens when you divide by a very small number?
The nature of these two questions lead us to the extremes limited by 0 and infinity.
1/100 is much smaller than 1/10.
1/1 is way bigger than 1/1000.
What of numbers less than 1?
Diving by ½ is the same as multiplying by 2. One divided by ½ is 2.
Dividing by a 1/1000000 is the same as multiplying by a million.
The one number we can not divide by is 0. We can get ever closer but can only reach it when we can count to infinity. The limit allows us to provide a formal “answer”. Dividing by zero turns it into infinity and dividing by infinity yields 0. Neither is possible but that does not matter because we are in the land of maths so saying that’s what it would be if we could get there is enough.
Limits are a way to solve problems that we can’t solve by imagining how it would look if we could.
Lim x->oo 1/x = 0.
Lim x->0 1/x = oo.
Calculus came from exploring these places. Both Isaac Newton and Wilhelm Leibniz are guilty. There was a controversy going back to the 1680s. Bridges were burned and blood bittered over who is the creator of calculus.
“The rules for calculus were first laid out in Gottfried Wilhelm Leibniz’s 1684 paper Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales, quantitates moratur, et singulare pro illi calculi genus (A New Method for Maxima and Minima as Well as Tangents, Which is Impeded Neither by Fractional Nor by Irrational Quantities, and a Remarkable Type of Calculus for This).” Newton had come up with his theory of fluxions prior to this but it was not published yet. The reality is that each contributed so much to the formation of the field that there is no controversy. Leibniz thought of things more as a mathematician while Newton was focused on objects in motion.
Lebniz also taught Jacob Bernoulli a bit of calculus and it was Jacob who, upon messing around with a question on compounding interest, identified the existence of the limit that is one definition of e. He found the number between 2.5 and 3. Calculating it further was not crucial to the actual question he was exploring,“Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?” (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?”
Compound interest is a trick by creditors to make more by adding interest accured to the original amount borrowed so the interest is more. Remember, interest is a fee for borrowing money and one way those that with capital can profit off of those without.
The basic interest formula looks like this, A=p(1+r)t .
The (A)mount is what you end up with. The (p)rincipal is what you start with. r is the rate. It’s the decimal form of a percent so 7% is .07. t is time. When they compound it, they take the interest accrued over a small period of time and put it in the pot and they do that repeatedly over the time and so the formula for compounding interest looks like so.
A=p(1+r/n)nt
In maths, you can divide any span of time into an infinite number of shorter spans and Bernoulli was a first to do so on a unique construct.
(1+1/n)n is the term of interest to find out about interest.
1
What happens when n gets super big? We already vistied some of the key components to solving answering that. The 1/n part will look more and more like 0.
One raised to any power is 1. No matter how you multiply one by itself, it is still itself. If the 1/n part could ever become 0, then the inside part would be 1+0, which is just 1 and so the whole thing would be 1 but it can’t so it won’t.
Any number bigger than one will drift towards infinity if you raise it to a high enough power. The inside part is larger than one so the whole thing would usually run toward infinity but in this situation, the inside part is making it 1 at the same rate so it all balances out to a number bewteen 2.5 and 3, e. (1+1/n)n becomes e and so the continuously compounding equation is just A=pert. The maths to make it work is one of the identifying features of e.
The calculus student comes to appreciate e almost immediately because it is the answer to its own analysis. The curve formed by the function f(x)=ex is self describing. The derivative of ex is ex and its integral is ex+c (can’t forgot that +c). Calculus is the study of change over time. Newton was focused on an object’s position related to time and how that changed, or its velocity. To better understand velocity, one needs to understand how much it changes, acceleration. Differentials and integrals are ways to look at these changes at an instant by dividing things into an infinite number of parts and putting them back together again. Newton was able to use this math to support his laws of motion, which are amazing because they so accurately describe 99% of the observable universe in a few statements. I had a physics professor in college who marveled at the beauty of these laws. It was in the main lecture hall of a building devoted to the study and its applications at the highest level. They designed and ran telescopes for NASA. Physics for engineers lectures can be dry and so I slept through much of it. The gentleman’s name has slipped my mind but he left an imprint in a few ways. On his first day, he explained how he wore a shirt and tie out of respect for his students. I never wore a tie to work until my first day as a teacher and I wore one almost everyday I went to work in the classroom. It was usually a little loose and under a sweater vest but it was there.
He would marvel at the beauty of F=ma and how it described so much of our physical universe, not our spiritual one, so accurately. My professor made a point to separate the physical from the spiritual and divine. The man was busy researching the very edges of what we can know and had no struggle with the existence of both science and god. He was, in fact, moved to appreciate both the physical and spiritual more for it.
Exponential functions, those of the form f(x)=ex, don’t show up much in classical physics but they are prominent in life sciences because they model natural growth.
One of my favorite math terms is asymptote. Maybe it’s the way it sounds. I also like the idea. The further you go to the left on the graph, the closer the curve will get to touching the x-axis. It will never make it. The x-axis is a horizontal asymptote.
Going from left to right, ex remains almost a horizontal line until the ”hockey-stick” curve and then exponential growth really kicks in and the graph goes increasingly higher and faster at a rate of ex.
Organic growth follows an exponential model. As does decay. It’s the way the physical universes composes and decomposes. The time may vary but the curve is the same. One of the mantras of 2020 was to “flatten the curve”, that is, to prevent COVID from spreading at exponential levels by vaccines, masking, etc. Redefining what is the natural course is just not easy. Many would say that it is not possible. Radioactive waste can take longer than the span of human history to decompose. Good thing we keep making more of it.
The exponential curve is far more fun to look at when you see it’s variations on a polar plane. That gives us pretty spirals. The self-similarity is key.
Jacob also studied spirals, logarithmic spirals, in particular. He called them spirific mirabilis.
Bernoulli wanted a logarithmic spiral and the motto Eadem mutata resurgo (‘Although changed, I rise again the same’) engraved on his tombstone. He wrote that the self-similar spiral “may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self.” Bernoulli died in 1705, but an Archimedean spiral was engraved rather than a logarithmic one.[11]“
Growth in the downward spiral.
Logarithmic spirals abound in the world around. And to here I will follow the turn.
I had been doing much of the reading for this piece I was supporting myself doing a lot of weeding. Other things, as well, but weeding most of all. Most of that work was getting beds cleaned up after the winter. Greens start sprouting from the browns. The skeletal branches grow brighter every day, before long flowers begin to form.
Spirals abound and each stage is exponential more than the previous. There is good reason we have been drawn to these shapes but seeing them as clearly as they are provides plenty as it is.
The gardenerers persepective is mighty fine one to watch it all come together from. I carry a camera with me so you can see some of it without tearing up your hands.
Phi, the famous golden mean, is just one from an infinity of infinity bases for the spiral defined by e.
This one uses the fun Fibonacci sequence to make a good enough approximation. Therein lies the magic. I can’t draw and can still make something that looks cool because I can add 1 and 1, as can you.
You find the next number in the sequence by adding the second to last to the last number. It can start with any two numbers but generally starts with 1. 1,1, 1+1, 2+1, 3+2, or 1,1,2,3,5,8,…The further along you get in the sequence, the closer the ratio of the last two terms, n/n-1, gets to phi (approximately 1.618).
Start with a square if side 1. Draw an adjacent square of side 1. The long side of the rectangle is the side of your next square, this one has a side of 2. The next one has 3, and so on. If you have a compass, you can use it to draw quarter circles where I drew a line. The result will be a surprisingly elegant figure. There are as many ways as you can imagine to decorate. I did the above one on some notebook paper and used another sheet of paper to get things measured and straight. I was sitting with a friend, helping them to get people to register to vote, outside a Stop the Violence held by a local dance company, and was a bit lost in this study and needed something to keep my hands busy.
More than some find divine in numbers. Much of early maths was developed around sacred geometries and holy days. The only way to accurately keep track of the days for most of human history has been the night sky. The position of the stars and the phases of the moon are the basis for calendars across lands and time. We have forgotten this knowledge and let technology take it’s place but it is at the very heart of where all that came from. Machines run our lives in the world we built. They can do extraordinary things but we must be wary of of the programming. If selfish means are the motivation, the ends can be no better. Our destruction of the planet and each other so the few can profit has reached catastrophic levels because the machines we built have enabled us to make it so.
Stars not only help locate one in time, but space. Astronomy and trigonometry. The latter starts as nothing more than the study of the most basic triangle and leads to the mechanics of frequencies and all the things that can come from harmony. Music is an expression of time. I can pinpoint myself to exact moments of my life hearing the chords to a chorus of a song. Maybe it was playing on the radio in van coming back from a field trip or riding with pretty lady with the top down or when you saw the band perform it. I am not even a musician. It’s not something I take to with ease. I will need to study at it some to get to where it comes naturally enough for me to say that I am. Music will speak and reflect all that is around me in so many more levels when I get to that point so it will be well worth the effort.
Here, on some rudimentary level, one can see how time and space are not the same but operate as one. The science says we can not know both a thing’s time and space. Expressions from the creative form challenge this concept. A good painting can capture it strikingly well. A well told story will put you in the place. Science is bound by the physical universe. It can take us to infinity and beyond within that space but it is limited. What exists outside of that existence depends on the individual and their life force or god.
This started as a writing about my favorite number, e. It decomposed and evolved until it was something of a different timbre.
“It is what we were put here to do.” The painter said. The return of the second line interrupted and we came toward the street to watch the party stroll by. The man for which the party was being held was shot and killed by a stranger a few weeks prior.
I would have been a witness to the wickedness if I had gone to work like I had planned. I was painting the ceiling and trim of a large therapy room at the end of the tiny block where it happened. It was a Friday evening and I ended up sleeping instead. He was family of good friends, his brother was killed by a home intruder some months prior. The brothers loved each other dearly and were wonderful fathers.
This is our reality and I hope my hope is not in vain that we can make something that nourishes the decent in the world.
….It was in watching the day turn to night and in my never-ending studying of the ways of trees and things that I could better see the transcendent power in natural growth. We and our ability to create new through expression leads to another level at the most dynamic and brightest spots, blossoming, filling the empty space in tones inspiring our dreams of color.
The plant grows in the path of energy from sky and ground. Ones that have been nourished and can gather enough of the sun’s infinty build into the ancients of the forest or the most flourising of flowers, offereing life to the world.
Once again, I was painting at the house I keep finding myself working on. It was a very nice January day with severe winter storms on the way. I was up on the ladder, high enough to be able to be eye level with the trees and my wandering mind turned to science and religion and the nature of god. Science is studying the world around us to understand it better. It’s a formal and technical study so you build new knowledge and the asking of questions to ask better questions. A botanist studies the way and nature that plants grow. These are the workings of the gods. Through that study and tending to all the things to understand, the botanist finds imprints of the divine, however they understand it to be. The Demiurge works an artist of all creation so one can study spirals, stones, stars, sounds, cells, colors, cultures, confections or whatever they fancy and see the handiwork of the gods. It is not easy because being wrong is a best way to learn how to be right.
This study, so far, has failed to even to get to Euler. There is only so much I can manage at once and so much one can learn at a go. Who knows, maybe I succeeded in helping someone else get lost in learning maths. That’s good enough for me. The rest is gravy.