Symmetry

There is a strange, unsettling, engrossing symmetry
To us

Not a cold winter to warm spring to hot summer to
cool autumn to cold winter symmetry

Not a we grew up two blocks from each other but were
in different districts and didn't meet until we were
in college out-of-state symmetry

Not an opposites attract symmetry
Not a distance makes the heart grow fonder symmetry
Not an out of sight, out of mind symmetry

We do not dress alike
We do not root for the same teams
We do not like the same music
We do not read the same books
We do not order the same coffee at Starbucks

Ours is a symmetry of absence
Dissonance
Loneliness

You once asked me years ago
What it was
  I wanted, I needed
And my answer has remained
          unchanged
After all these years

The heart wants what it wants.

New Days

I would proffer my hand
  to you
But we really don't do that anymore
    (maybe someday)
And we talk about old friends and new

Did you hear what Billy said yesterday?
  Phil was pretty funny last night
    Frances really made me think this morning
      Mary Jo made me smile
        Katha and Galway were as irksome as always
Did you miss any of that? Did you see?

You blink and tell me of late
Your mind feels like
  blinds that haven't been dusted in months
  rugs that have pilled from overuse
  a layer of dust thicker than the table itself
Too many choices
None of them of interest
To an intellectual such as yourself

I feel much the same
A rabbit down the wrong hole
  in a warren not his own
Tired autumn leaves ready to
  give up and drop to the barren ground
Not enough choices
Too many interests
To a follower such as myself

I don't have time to miss the old days
I'm too busy wasting the new days

One Month Turns To Two

The space I inhabit
Colder than the rest of the house
(The windows leak, the fireplace is
not sealed)
More spacious than my last space
  But just as 
          confining and empty
The grey skies outside all eight windows
Mirror my heart

I shout into the void
But cannot be heard over
  the sounds of everyone else
Shouting
          into their own voids
  their own private
          hells-on-Earth
Everyone seems happy
  Everyone is lying
    This is fine

I mask my expression
Literal and figurative
I smile for a camera that
  No one sees
    Or can see
          or even wants to
I ask myself and everyone I see
    what the hell is wrong with you?
And their question of me
  mirrors my own of them

Long night into
  longer day into
    longer night into
A month of nameless days
  one month turns to two
    turns to three, turns to ten
And we are still nowhere
          or perhaps no-when
          or perhaps no-why
Hazy morning into wet afternoon

Alone.

Word Forward

Man, dating really sucks. At any age, but more so as you get older.

I’m not a really a go-to-a-bar-and-meet-someone kind of guy. I’m not terribly social. So, my only option, particularly in this time of quarantines and pandemics, is online dating. Which is the absolute worst.

Online dating is much like social media for single people. Between inaccurate (or totally fake) profile pictures and messages that never get returned, there is a decided cowardice and lack of respect in lying to and then ghosting someone. And from stories I’ve heard from female friends, it’s endemic to both sides of the aisle, as it were.

And when did “I love tacos!” become a way to demonstrate your date-ability?1 C’mon, now – aim higher. Everyone loves tacos. And hiking. And being on the water. And world peace.

The one thing that dating HAS done for me is to grow more comfortable talking about myself. In fact, dating off-and-on over the past nine months has helped me develop a bit of a spiel that I can jump into, at any point along the continuum, to keep an awkward first (or second) date moving along.

I’ve written about some of this before, in previous blog posts. Here, then, is the more detailed, expanded version – The Origin of 56-year-old Steve: The Special Edition:

I’ve spent the last four or five years jumping from one interest to the other. It started with music, and continued with a renewed interest in math, followed by dabbling in computer programming, and then delving deeper into philosophy…but wait, it goes back farther than that.

I was a decent enough student in high school. I could have been a straight-A student had I applied myself. However, that was not the case. I piddled around the entire four years. My final two years of high school were marked by my step-father dying, leaving my Mom as a single mother having to work to earn money to raise me, my four-year-younger sister, and my twelve-year-younger half-brother. Money as always tight, and she did the best she could.

But college was never really an option as I began my senior year. The focus was solely on me finishing high school so that I could get a job and start supporting myself. So a few weeks after graduation, I found myself delivering pizzas for Straw Hat Pizza for nine months, then it was off to Lackland Air Force Base in San Antonio, TX, for USAF basic training.

Two years into my military service, I was married to a fellow service member. We spent five years overseas together, then four more years in Denver, CO, before calling it quits. We split up shortly after I left the military (she had retired once we returned stateside), and I was on my own for a while.

Another whirlwind romance that ended in disaster brought me to Oklahoma, and this is where I’ve been for the past twenty-six-plus years.

Somewhere along the line, I began to regret not having concentrated more on my studies when I was in school. I went back to school, getting a business admin degree from the local community college. I dipped my foot in the pool of an actual B.A. program at OSU-Tulsa.

My kids are super smart – much smarter than I was at their age, and certainly much smarter than I am now. They began feeding me interesting YouTube videos. At first it was VSauce videos – Michael Stevens’ deadpan delivery of interesting science facts and thought experiments was extremely engaging. Then one of them started sending me Grant Sanderson videos, and we were off to the races.

I can still clearly recall one video where Sanderson was attempting to explain calculus in layman’s terms (relatively speaking), and I had a light bulb moment ten minutes into the video.

In this particular video, which was part one of a ten-part “Essence of Calculus” series2, less than ten minutes into the lesson, Sanderson clearly and concisely explained how, when solving for the area of a circle, you are actually solving for the area of a right triangle, and my mind, to say the least, was blown. I distinctly remember thinking to myself (and repeating to anyone who would listen), “If I’d had someone teach me this in high school, I’d be a rocket scientist by now!”

I immediately hit Kahn Academy and began relearning all of the algebra I’d forgotten over the years.

In the meantime, I’d become enamored with British mathematician Matt Parker and his YouTube channel, Stand Up Maths. He is also a frequent guest on another great channel, Numberphile. One of my favorite parts of Matt’s videos is, on occasion after working something out on butcher paper or a blackboard, will then reveal that he wrote a quick Python program to verify his results. That sealed it – I had to learn computer programming!

And so I did – I now know just enough JavaScript, HTML, CSS and (of course) Python to be extremely dangerous. My crowning achievement (so far) has been to write a Python script that will take a block of text and send it, word by work, to an unsuspecting cell phone user via SMS. Beauty!

. . . . .

My most recent obsession, philosophy, is another thing entirely. And yet, more of the same.

I’ve always been interested in philosophy – my first blog post dealt with stoicism, and that was almost two years ago. This time, I decided to approach it with the same academic rigor that I’ve explored math, science, and computer programming over the last few years. I started at the beginning with the father of pre-Socratic philosophy, Thales, and have been steadily moving forward through the different schools of thought over the last 2,500 years.

One of the things that fascinates me most about the study of philosophy is that the earlier philosophers had nothing else to go on but their five senses and their minds, and yet were able to develop such insightful, and often (overly) complicated explanations for everything.

While there existed schools of thought that invoked the four classical Greek elements (fire, water, earth, air) there was a school called the Atomists that supposed that everything was made up of smaller, unseen particles called atoms that actually made up everything we see – and they came to this conclusion 300 years B.C.E, nearly 2,000 years before the Janssen brothers in 1590 C.E.!

But more to the point, the early philosophers never gave up, and never stopped building upon the thoughts and ideas of their forebears, sometimes eloquently expanding on their ideas, sometimes developing totally new ideas and doctrines.

And that, more than anything, defines how I’ve overcome that lack of desire (and, to be honest, motivation) and am now attempting expand my horizons through the faux-academic study of things that interest me. I keep building upon what I’ve learned previously, always striving to expand my knowledge of the world around me. Or, in the words of philosopher Stephen West, to always “know more today than I did yesterday.”

The key (for me, anyway) is, word for word, to always keep moving forward.

. . . . .

Links

  1. Self-proclaimed “snarky dating poster” Sarah Kehoe tweeted something to this effect – @sarahkehoe. If you have a Twitter, follow her!
  2. “Essence of Calculus – Chapter 1” by Grant Sanderson (3blue1brown) – https://www.youtube.com/watch?v=WUvTyaaNkzM

. . . . .

If you haven’t had a chance to check out my first book, “What I’ve Learned: Random Thoughts on Various Subjects,” now it the perfect time! Bounce over to http://bit.ly/what_ive_learned and pick up a paperback copy, or download it to your Kindle!

A Deontologist, a Consequentialist, and a Virtue Ethicist Walk Into a Bar…

One of my philosophy professors asked an interesting question today, and after giving it some amount of thought, I think I have an answer.

His question was this: Who do you think would win in a bar fight between a deontologist, a consequentialist, and a virtue ethicist?

First, a quick recap:

  1. Deontology – simply put, these are the rules guys (and gals). Deontologists purport that there are certain universal moral truths that should guide our behavior and how we interact with others.
  2. Consequentialism – when considering whether an action is right or moral, we have to consider the outcome(s) of our actions, both intended and unintended. All actions have consequences.
  3. Virtue Ethics – good and bad, right and wrong, ethical and unethical – these are determined largely by one’s character. Moral people engage in moral thoughts and activities, so it is incumbent upon each of us to be the best person we can possibly be.

Each of these schools of thought have their advantages and disadvantages, and when taken in concert, create a bit of circular logic.

Aristotle believed that since each person’s thoughts and actions were under their own control, an individual could learn to be a good (or better) person. However, the issue with this line of thinking is two-fold.

First, if the individual is left to decide what is moral, then it is in fact his or her cultural and environment that is coloring their decision. In an often-used example, 200 years ago it was viewed as morally acceptable to own slaves. So, by extension, it was possible to own slaves and still be considered a moral, upstanding individual. Obviously this is wrong, but I can only say “obviously” because our culture has changed to the degree that we now understand (but don’t yet fully embrace, apparently) the inherent worth of each individual, regardless of race (or gender or sexual orientation).

Second, we are using the term to define itself, i.e. “A moral act is one that a moral person would engage in.” This is akin to saying, “The sky is blue because it is blue.” We are not defining anything here, really.

One of the ways around this line of thinking is to impose certain qualifications, such as “lying is always wrong, unless it lessens someone’s pain.” However, as we do this, we move further and further away from character in an infinite regress of “except for this” and “not counting that.” We begin to dilute the original meaning behind virtue ethics, reducing it to a series of best-case scenarios that become increasingly difficult to keep track of.

This is where deontology enters the picture. By having a hard-and-fast set of rules to guide human behavior, we make value judgements concerning morality much easier to deal with. Or do we?

We again run into the problem of having to qualify each moral judgement with some sort of disclaimer. “You should never break a promise” is a moral way to act, but what if you promised your buddy that you’d golf with him, and then one of your kids is in an automobile accident. Do you skip golf to care for your injured child? Deontologically speaking, you couldn’t – you’d made a promise to go golfing, and to break that promise would be morally unacceptable.

This leads to another issue – namely, where do you draw the line when devising your qualifications to all of these moral behaviors? Do we make exceptions for family only? Close friends? Co-workers? People who are less fortunate than yourself? And on top of that, who exactly is responsible for making these qualifications? Is it the individual? That won’t work – we’d have different standards for each individual, thus defeating the original purpose of having a set of set-in-stone rules in the first place.

And this brings us to utilitarianism, and the idea that moral acts can only be judged moral based on their outcomes. Consequentialism deals with this aspect specifically – what are the consequences of my actions, how do they affect not only me but those around me?

The issue with this line of reasoning is that you very quickly run into situations where immoral acts can lead to moral outcomes. The infamous Trolley Problem is the most famous example of this – is it okay to take the life of one individual to save five others? What if you personally know the one person? Is it then right to sacrifice the five individuals on the other track to save the one person you know?

Recent years have witnessed a return to a form of Aristotle’s original value ethics as the predominant method of determining good versus bad, but in all truthfulness this just leads us back to the top of the circle, ready to start the cycle anew.

So, back to the question at hand – who would win in a bar fight between a deontologist, a consequentialist, and a virtue ethicist?

I know which horse I’m putting MY money on – what are your thoughts?

Acta non Verba

One of the coolest things about immersing myself in some subject with which I already have some passing familiarity is being able to see how my views have shifted over the years. Streaming all four seasons of The Good Place recently has led to a reignited interest in philosophy in general. Since it’s been a hot minute since I’ve given philosophy any real thought or consideration, I decided to start from the beginning. And by that, I mean literally the beginning of philosophical thought, with Thales and the other pre-Socratic philosophers, of which I knew very little.

Another cool feature of this is that, as each new idea is presented and explained and demonstrated as a step forward in the evolution of philosophical thinking, I find myself going through the usual three stages of learning something new:

  1. Oh, that’s cool! I never thought of that!
  2. Oh, this is actually bullshit. Why did I think that made sense?
  3. Oh, this new bit of information is cool! I never thought of that! (see step #1)

With the study of philosophy, this constant cycle is significantly heightened. What makes sense one minute is revealed to be limited and not very insightful in light of subsequent thoughts, findings, and techniques. I’ve spent the last week reading books, listening to podcasts, and watching YouTube videos that (more or less) follow the development of philosophical thoughts and ideas from roughly 650 BCE up through today.

I’m quickly finding that the philosophers I relate to most closely are the ones who took action. Whether it is Thales laying the groundwork for future philosophical thought, or Pythagoras starting a new cult to prove that math is the language of the kosmos, or Plato utilizing the Socratic method of constant, insightful questioning to arrive at a conclusion, or Karl Popper questioning the scientific methods of Freud (pseudo-science) in comparison to Einstein (actual science), the philosophers that resonate with me are the ones who not only thought of something, but also did something about it.

Acta non verba – action, not words.

I mentioned in my previous post all of the changes I’ve attempted to make in 2020. While nullius in verba has become the defining principle of my life now, coming in close second is acta non verba. I have wasted so much time waiting for something to drop into my lap – financial success, new jobs with better pay, new passions – and I have largely been lucky in the sense that I’ve lived a bit of a charmed life compared to most.

How much more happier would I be, then, if I’d actually expended more than just the minimal effort required to reach my goals – if the fruits of my labor were a direct result of the effort I’d put into a task or activity? This is the true nature of the experiment I’m engaged in now. I seek to answer the question: what if I actually took control of my life and went after the things I desire, rather than just sit back and hope they will drop into my lap somehow?

It may very well be that I’m setting myself up for misery, or disappointment, or a fate worse than death – third marriage, anyone? But I don’t believe that to be the case. If I were a betting man, I’d wager that in the long run, I’ll meet with more success than failure. And isn’t that really what we all desire? To be successful more often than we fail?

If the ultimate goal of life is to be happy while minimizing (or eliminating) the sadness and dissatisfaction of others, then how much sweeter would that taste if it was by my own design rather than the luck of the draw, or fate? That may come across as a bit selfish, but that certainly is not my intent. I say it in this sense: How much more satisfying is it to be the master of one’s own fate, rather than leaving it to chance or the gods or God (or whatever your particular belief system happens to be)?

So, these are the questions I seek to answer, and I will be doing it via concrete action instead of mere rhetoric – acta non verba.

Nullius in Verba

I got my fourth tattoo today from Niah and the fine folks at Black Gold Tattoo here in Tulsa, OK. It had been a number of years since I have gotten any new ink, and today seemed like just as good a day as any. It is my brother’s birthday as well, and he is a tattoo nut, so this is in part for him as well.

The reactions have run the gamut from “Wow, cool!” to “But why?”. To those on the lower, disapproving end of the spectrum, I played it off as just something I wanted to do, or simply replied, “Why not?”

But the truth is that this phrase is the most important thing I have learned thus far in 2020, which is saying a lot. So far this year, I have had to learn to live on my own again, I have taught myself ukulele, I have tried to learn Python, I have begun studying philosophy again. Yet all of these things pale in comparison to the effect these three simple Latin words have had on my life in 2020.

Nullius in verba is Latin for “on the word of no-one.” More loosely translated, it is taken to mean “think (or do) for yourself.”

I have spent much of my life doing was I was told to do, believing what I was told to believe. From my religious upbringing, through my military service, through my varied jobs in the private sector, and through two failed marriages, I have always tried to do what I thought the other party felt was right.

Perhaps I paint with too-broad strokes here – it is not like I was a robot following orders. I have had my fun, and made my share of stupid mistakes that were 100% my idea alone. However, there were definitely times where I felt like an automaton, and this characterization is probably pretty accurate more, often than not.

This year, one of my (many) foci has been to attempt to figure out where I belong, where I fit in to the grand scheme of things. Everything else – ukulele, coding, philosophy, etc. – has been window dressing for the real search, the search for personal meaning and validation.

What these three simple words remind me of is this: there is no better judge of things than myself.

Does this mean I completely dismiss the words of subject matter experts and authority figures? Absolutely not.

What it DOES mean is that everything that is meaningful is also independently verifiable. Am I going to run my own lab tests to ensure the eventual COVID-19 vaccination works? Of course not. But will I pay more attention to who it is that is telling me that it works? Absolutely.

Am I going to vote for someone simply because they are a registered Democrat, or against someone because they are a registered Republican? Nope, not anymore. I have taken the time to actually delve into what each individual candidate stands for, what each individual ballot measure means and what the pass/fail ramifications are.

Closer to home: am I going to stop forcing my will on others because it is what I think is best for them? Can I accept that others know what is in their best interest, just like I have some idea of what is in my best interest? Hopefully.

And these are just a few of a million little things that bears closer scrutiny, starting with myself. It will be the ultimate introspective exercise. Socrates (via Plato) once indicated that “the unexamined life is not worth living,” and this is precisely what he meant. I’ve wasted so much of my life believing one thing and disbelieving another, simply because it was easier to follow the crowd instead of expending a little extra time and effort to do the research myself.

On the word of no-one; think for yourself.

12 Song Challenge

Among the new things I’ve been trying – podcasts, mathematics, computer programming, etc. – is trying to get a little bit better on the piano. I’ve set the guitar aside (for the most part) and focused more on piano-based songs.

For someone that doesn’t read music, this can be somewhat of a challenge. I play guitar chords on the piano. That is to say, when I play a “D” chord on the guitar, I know what those notes are, and I can then play the corresponding notes on the piano.

But to make a song sound like the actual song, it helps to be able to play at least a part of the actual song, be it a particular rhythmic part, or the introduction, or some distinctive part of a given song.

What follows are the intros to eleven songs and the “guts” of a twelfth. How many of these twelve songs do you recognize?

There’s a bit of an eclectic mix here:

-Three songs from the 70’s. Six from the 80’s. Three from the 2000’s.

-Three artists are represented twice – two are bands who had their hits in the 80’s, one band who is pretty big right now.

-Only one of the artists is not longer with us – the remainder are still making music, albeit some with different lineups.

I also mention the Parker Square at the beginning of the video. Matt Parker, British mathematician/comedian, discussed his aptly named Parker Square in a video for Numberphile a number of years ago, characterizing it as the “mascot of giving it a go” – in other words, don’t be afraid to try something new.

So, give this a go – how many of the songs can you recognize and name?

Good luck – and thanks for watching!

Links

Matt Parker’s Numberphile video is here.

The Almost Perfect Discovery

This is the story of my greatest mathematical “discovery” to-date.

In his entertaining video on leap years (link below), Numberphile contributor Matt Parker mentions towards the end of the video that the next “power year” will be in 2048.

That is to say, the next year that is a power of 2 will be 2048:  211 = 2048. The last one was the year 1024 (210) and the next one won’t be until way in the future – the year 4096 (212), to be exact.

A little quick math revealed that I will turn 84 in April of 2048 (assuming I’m still around to annoy people), and I thought it would be a mildly interesting blog post if I could associate 84 and 2048 somehow, preferably mathematically.

I spent about 15 minutes tackling this from different angles. They’re both divisible by 2, and their greatest common factor is (only) 4, which is surprising considering they both have 12 factors (including themselves):

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Factors of 2048: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048

No luck comparing prime factors, either:

Prime factors of 84: 2X2X3X7

Prime factors of 84: 211 (2x2x2x2x2x2x2x2x2x2x2)

Then I remembered Matt’s video on compatible pairs (also linked below), and I applied that system to these two numbers. Nothing special about 84, other than it is an abundant number, which is a number whose proper divisors (there’s another term for this, which I didn’t know about, and which we’ll get to shortly) sum to a number greater than itself. In this case, 84’s abundant (or excessive) number is 140 (1+2+3+4+6+7+12+14+21+28+42).

2048 was a different story. It immediately struck me that the sum of the proper divisors of 2048 came out to be 2047 – one less than the number itself. “How odd!”, I thought to myself, followed by “I wonder how prevalent this is?”.

I quickly threw together a spreadsheet listing the first 100 natural numbers along, with all of their factors. After the quick calculation of adding up all factors except the last one (the number itself), a pattern emerged. This appeared to be happening with every power of 2!

(At this point, I have to say that, had I been paying attention, what was happening would have been readily apparent. My number manipulation also revealed another pattern, one where the sum of the proper divisors of a number equaled the number itself. There are two numbers that this happens with in the set of natural numbers to 100…6 and 28. It honestly didn’t register with me what was happening…)

Here is the pattern that I discovered:

Sure enough, the powers of 2 less than 100, i.e. 20 through 26, followed this pattern, and they were the only numbers that were doing so! I did the same calculation for 27 through 212, and:

…the pattern held! I was super excited, thinking I’d found some hitherto unknown, or at least unexplored, feature of the powers of 2.

So excited, in fact, that I sent Matt Parker and email about it, detailing my “discovery” much as I’ve done above.

At this point, I closed everything down and went about my day. I went to listen to a friend play guitar at a consignment market, grabbed something to eat, washed my car. Went home and began surfing the interwebs a little bit.

That’s when I came across the Wikipedia page for Powers of 2 (yes, there is one, unsurprisingly – I’ve linked it below as well).

In short order, I found that:

-Summing the proper divisors has a technical name, the aliquot sum. I couldn’t find a proper etymology of the term, but this function has been known for thousands of years. Well, of course it has.

-What I “discovered” is known as the set of “almost perfect numbers” – almost perfect because their aliquot sum is one less than the number itself.

Which brings us to 6 and 28, who’s aliquot sum equals the number itself. As many videos I’ve watched regarding Fermat and Mersenne prime numbers, I should have recognized that these are the first two perfect numbers, so named because of the very quality I “discovered” in my spreadsheet above.

After sending Matt a follow up “never mind!” email, and texting everyone else that I was, in fact, not about to become math-famous, I shut down my computer for the day and enjoyed the unseasonable beautiful weather we were experiencing.

I still haven’t found a blog-worthy, interesting connection between 84 and 2048 – I’m sure they’re out there, I just need to dig a little deeper. Like I should have done initially when I made this amazing “discovery!”

Better luck next time, I guess!

Links:

Matt Parker’s Leap Years video on Numberfile

Matt’s Compatible Pairs video

Powers of 2 Wikipedia page

Irrational Behavior

Here’s some random stuff I learned this week:

File this one under “duh”: I’ve always assumed that the terms ‘rational’ and ‘irrational’ when relating to numbers was an indication of their behavior. 1/4, for instance, is a rational number, because it equates to 0.25 – it behaves in a rational way.

1/3, on the other hand, is 0.333333…, never ending – it behaves in an irrational manner.

Not only is the above NOT true mathematically, that is NOT where the terms come from.

Rational numbers are numbers that can be expressed as the ratio of two integers – i.e., a fraction:

1/4 = 0.25

1/3 = 0.33333…

98/100 = 0.98

On the other hand, numbers that CANNOT be expressed as a ratio between two integers – for example, π, e, or the square root of 2 – are all irrational numbers:

π = 3.1415926535…

e = 2.7182818…

√2 = 1.4142135…

The Golden Ratio (1.6180339…) is another example of an irrational number. The difference between the numbers in the Fibonacci Sequence, to use one example, follow the Golden Ratio as the sequence increases.

And speaking of cool numbers, if you divide 987654321 by 123456789 on your phone calculator, you may get an answer of 8. I’ve always enjoyed this, as I’ve always considered 8 to be my lucky number, based solely upon a fortune cookie I got when I was a kid. (This is also why I say my favorite color is red – the fortune cookie told me so – when it’s actually blue.)

However, you’re more likely to get an answer like 8.00000007, which seems much more accurate. But still close to 8, right?

But turn your phone calculator on its side to engage the scientific calculator functions, and the answer becomes 8.000000072900001. Much more accurate still!

But it doesn’t end there. The full remainder of this operation is another example of digits that repeat to infinity:

So, how many irrational numbers are there? That’s an interesting question, and involves countable versus uncountable infinity.

We already know that there are an infinite number of counting (or whole) numbers. “Countable” in this context doesn’t necessarily mean we can count them, just that it is possible to count them in theory. We can say n=1, for instance, and then apply the formula n=(n+1) to generate the series of natural numbers that increment by one.

Real numbers, on the other hand, are uncountable. In fact, there are more real numbers between 0 and 1 then there are natural numbers between 0 and infinity.

Think of it this way: where would you start counting the real numbers between 0 and 1? You can start with, say, 0.1. Well, there’s at least one number between 0 and 0.1 – it’s 0.01. And there’s at least one number between 0 and 0.01 – 0.001. And you can do this literally forever, and still not determine where to start counting the real numbers between 0 and 1.

Irrational numbers are similar – they are part of the set of uncountable infinite numbers. So in a sense, even though both the sets of rational numbers and irrational numbers are infinite, there are more irrational numbers than there are rational numbers, just as there are more real numbers than there are natural numbers, even though both sets are infinite.

My brain is now a smoldering heap, so that will be it for today. But tomorrow I have a nifty little thing to share regarding an interesting characteristic of numbers that are a power of 2, so until then – thanks for reading!

Further reading:

Math Is Fun Irrational Number Page: https://www.mathsisfun.com/irrational-numbers.html

Quora discussion on 987654321/123456789: https://www.quora.com/With-these-numbers-as-mirrors-why-does-987-654-321-divided-by-123-456-789-equal-8-and-is-this-a-coincidence-or-is-there-more-to-it

Maths Doctor UK rational vs. irrational discussion: https://mathsdoctor.co.uk/revision-help/gcse/number/rational-and-irrational-number/

Irrational Pi header courtesy of this Mathologer video.

Is your name in π?

Mmmm…pi pie.
Image courtesy of pillsbury.com

I preface this by saying zero original research went into this other than my interest in the subject and the God-given ability to Google stuff.

You probably already know that we use a base-10 number system to count stuff – 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. But there are other number bases – some of which you already know about, even if you didn’t realize it.

The language of computers, binary, is nothing more than a base-2 system, where every digit can be represented by zeroes and ones. Thus, when converting from base-2 to base-10, you’d count 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010…and so on. Here is what these base-2 numbers equate to:

0 = 0

1 = 1

10 = 3

11 = 4

100 = 5

101 = 6

110 = 7

and so on, so that (for example):

111001 = 57

(Links are at the bottom of the page if you want to explore this, as well as the other things discussed below)

But there are other bases – an infinite number of bases, actually. The Babylonians used a base-60(!) number system, and Mayans used a base-20 system. Hexadecimal (base-16) systems are relatively common, used in everything from computer programming (as a way to condense binary) to our everyday usage of pounds and ounces.

In fact, you use an octovigesimal system every day. Better known as base-28, this is what our current Gregorian calendar is (loosely) based upon.

So, what if I wanted to see if my name is represented in the digits of π? How would this even be possible? What would be the best way to convert the base-10 digits of π to letters?

Well, we’d want to use base-27 for this. Base-27 is represented thus:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P

To convert this to alpha, we’d assign 1=A, 2=B, 3=C and so on, letting zero represent a space, i.e. 0=””.

Doing this let’s us then convert π in base-27 from 3.3MQ53… to C.CVEZC. And we’re off and running!

(“Why not use base-26?” I hear you thinking. Well, we want to be able to represent each letter of the alphabet with a letter so that A=1, B=2, etc., and we need a 27th place to capture our zero value, so base-27 it is!)

There is a conversion tool online (also linked below) that allows you to enter your name (or any word, really) to see if it is represented in this converted base-27 π. My name, STEVE, appears starting with the 8,857,158th digit of π:

Image courtesy of Dr Mike’s Math Games

Some fun facts about the occurrence of words in (this version of) π-27:

The first spelled-out number appears at the 4,259th position: SIX

The first ordinal number appears at the 222,386th position, and of course it’s FIRST

JEDI appears at position 1,126,698, but the SITH overpowers them at the earlier position of 804,693.

Within the first 30,000,000 digits of our converted π-27, the word MATHEMATICS does not appear. Hopefully it will show up in the second group of thirty million characters…

And where does PI appear? Not surprisingly (given that it’s only two letters long), it pops up for the first time starting at position 18.

So, click on the link below (or right here if you’re impatient) and see where your name first appears!

One caveat, though – this database only includes the first ~30 million converted digits of π, and your chances of finding your name decreases substantially as the length of your name grows longer. If your name is three (or fewer) letters long, you have virtually a 100% chance of finding it. Five letter names (like mine) carry a 56% chance of success, but seven letter names have virtually zero chance of being represented.

Good luck, and let me know in the comments if you’ve found your name in the first thirty million digits of π!

Here are the links I promised:

Check π for your name here: http://www.dr-mikes-math-games-for-kids.com/your-name-in-pi.html

How π in base-27 works: https://www.atractor.pt/mat/fromPI/pib27-_en.html

How counting in different bases works: http://mae.engr.ucdavis.edu/dsouza/Classes/ECS15-W13/counting.pdf

Math.com’s π facts page: http://www.math.com/tables/constants/pi.htm

More stuff about π-27: http://www.dr-mikes-math-games-for-kids.com/your-name-in-pi-notes.html

More Than You Wanted To Know About The Sun

What color is the Sun? Be careful – it’s a trick question!

If you Google “What color is the Sun?” you’ll find that it is white. This probably runs counter to what you were taught in grade school – remember those bright, yellow smiley-face Suns you used to draw in the corner of you masterful work of art? Yeah – me, too.

The truth is that when it rises or sets, the Sun appears yellow (or orange, or red) because of the way light rays – or more specifically, photon packets – are dispersed through the atmosphere as they approach the Earth’s surface.

The core of the Sun, which is 25% of its area – but only 2%(!) of the entire Sun’s mass – is where the action starts. Like bumper cars, hydrogen atoms run into each other under the tremendous force of gravity at the center of our star. This hydrogen fusion creates helium 4, and this process generates photons which begin their journey from the core through the radiative zone, one micron at a time. This trip through the radiative and convection zones, then through the photosphere and chromosphere, and finally into the corona (the part of the Sun only visible on Earth during a solar eclipse), can take anywhere from 10,000 to 170,000 years. The final leg of this journey is the eight seconds it takes light (energy) to reach the surface of the Earth from the Sun. So, it’s entirely possible that if the Sun is shining on your face as you read this on your phone, you are viewing it in sunlight that was first generated 170,000 years and eight seconds ago.

So, if you’re viewing the Sun in the early morning or late evening, is it correct to say that the sun is yellow? Or orange? Or red?

And what about at midday, when the Sun is (more or less) directly overhead? In the morning, the shorter (blue & white) wavelengths of the electromagnetic spectrum are being diffused, causing the Sun to look orange. But when the Sun is high in the sky, the shorter wavelengths are dispersed, bouncing around the sky above your head and making the sky appear blue. The Sun itself takes on a blueish-white hue. Trust me on this – don’t look directly at it just to prove me wrong (or right).

So far, we’ve come up with a handful of colors – white, yellow, orange, red, blue-white…but which is it? What actual color is our actual Sun, minus all of the atmospheric interference and distractions?

Before we get there, let’s talk a little about the Kelvin scale.

In 1848, William Thomson (later made Lord Kelvin, from which the scale takes its name) devised a method of recording temperatures wherein absolute zero was denoted with a ‘0’ and subsequent degrees were incremented in a scale commensurate with the Celsius temperature scale. This “Kelvin scale” has since been widely used in the fields of engineering, astronomy, and science, particularly as a means of measuring noise temperature and color temperature.

A “blackbody” is a theoretical substance that absorbs (and transmits) the entire electromagnetic spectrum, and the gradations of the Kelvin scale are based on measurements of light refracted when a blackbody maintains a given temperature. (As an aside, a “whitebody” serves the opposite purpose, denoting a surface that reflects the entire electromagnetic spectrum. I know you were curious about that.)

Kelvin temperatures of 2000-3000K (we don’t use the term ‘degrees’ when discussing Kelvin temperatures) are characterized by reds-oranges-yellows (think sunrise and sunset); 3100-4500K are your yellow-whites (think late morning and afternoon); and 4600-6500K covers whites-blues (think midday). Anything less than 3000K is considered to be warm white; above that but less than 4500K is cool white; and anything over that is what we might consider daylight on a bright, sunny day.

So after all of that, have we come to a consensus on the color of the sun?

White? Red? Orange? Yellow? Warm white? Cool white?

I did mention that it was a trick question, didn’t I?

The truth of the matter is that the sun doesn’t have an inherent color. Electromagnetic waves themselves have no physical properties that would give them color. 

Color is purely a physical perception, a product of photons hitting the rods and cones of the retina in the eye. Humans have three cones, whereas dogs (for example) only have two cones. As a result, we humans are able to perceive a wider range of color than our canine friends. While dogs have better nocturnal vision than we do, their vision is usually in the 20/75 region, meaning we see much more detail than they do on the whole.

So, make no mistake – color is all in your (and my) head.

The Sun – and everything else in the universe, for that matter – is colorless. It is only through a fluke of evolution and natural selection that we are able to perceive it as a vibrant, beautiful rainbow of color.

Further reading:

More about the color of the sun: http://solar-center.stanford.edu/SID/activities/GreenSun.html

More about Kelvin color temps: http://www.westinghouselighting.com/color-temperature.aspx

Even more about Kelvin temps: https://www.larsonelectronics.com/blog/2017/12/12/led-lighting/lighting-101-color-temperature-what-is-the-kelvin-scale

Curious about blackbodies? https://en.wikipedia.org/wiki/Black_body

What is color, you ask? https://physics.info/color/

That’s cool, but how does it work? https://www.livescience.com/32559-why-do-we-see-in-color.html

Many thanks to JJ for inspiring this post. 

And as always, thanks for reading!