Limits

 

(Some random math humor found without attribution on the internet ...)

π Day!

 

Although I haven’t posted about it on the blog since 2020, π Day is celebrated every March 14 here at the Hopper household with unbridled gusto! Champagne, party hats, a few roman candles, plus dancing until the wee hours of the morning! Since we’ve entered Lent, and it’s a meatless Friday, we’re skipping the filet mignon and charcuterie board.

 

Now, with the assistance of AI, I offer you the Weirdest Fact About π –

 

The sequence 123456 will not be found in the first million digits of π. And π has been calculated out to over 62.8 trillion digits, so we’ll have to wait a bit before the location of that sequence is found.

 

And a bonus fact –

 

 A sequence of six nines (999999) can be found in π at the 762nd position in the digit expansion. This block of nines is known as the Feynman point, after physicist Richard Feynman (whose biography I read earlier this year), joked that he could recite all the digits of π up to this point.

 

Finally, about 17.3 billion digits in, you can spot the sequence 0123456789. (Is this the first appearance of 123456? Let me get a pencil and check …)

 

And really finally, there’s a website out there that will find the location of your birthday in the digit expansion of π. Haven’t checked it out yet, but it’s some fun to save for the weekend.

 

Happy π Day!

 

Synchronicity or Syzygy?

 

“God teaches the soul by pains and obstacles, not by ideas.” – Fr. Jean-Pierre de Caussade, Abandonment to Divine Providence

 

“What stands in the way becomes the way.” – Marcus Aurelius, Meditations

 

S = ∫ (t1 to t2) L dt

 

Measured in joules / second, or accomplishments per unit of life.

 

What is the Midway Point?

 

Here’s a neat mathematical riddle to use on your friends to prove your genius bona fides. It sounds unsolvable until, well, you hear the solution.

 

Question:

 

What is the exact middle point between zero and infinity?

 

In other words, on this number line from negative infinity to positive infinity, what is the halfway point between zero and positive infinity on the right?

 

Any guesses?

 

Hmm?

 

Seems kinda impossible to figure out, right? At first I thought so, because infinity, that sideways-number-eight, is not really a number, like 3, 17/50, or π^cubed is a number. Yeah, 3 and 17/50 have exact locations on the number line, and even though π^cubed, like pi itself, is not an exactly defined number (it is an irrational number whose decimal expression goes on, it has been proven, forever), it pretty much has an exact location on the number line. But infinity is not a specific number but an idea. A mathematical concept. So it really doesn’t have a location on the number line, except a vague neighborhood that lives ever, ever, ever rightward as you heading that way down the number line.

 

Hint #1 (minor):

 

So the trick is not to think of the question spatially. Not as in the case of 18 inches being the midway point of a yard, or 500 meters the halfway point of a kilometer.

 

Think of numbers themselves, as in types of numbers.

 

Any guesses?

 

Hmm?

 

Hint #2 (major):

 

Every number on the number line can be expressed as a reciprocal. A reciprocal of a number is one-over-that-number. The reciprocal of x is 1/x. The reciprocal of 3 is 1/3. The reciprocal of 17/50 is 50/17. The reciprocal of π^cubed is 1/π^cubed.

 

So what’s the halfway point between zero and infinity?

 

Answer: 1

 

The reciprocal of 1 is 1/1, or 1. 1 is its own reciprocal. But for every single number greater than 1, from 1.0000000000000001 to a googolplex (10 raised to the power of 1 with 100 zeros following it), there is a corresponding reciprocal. Every single one. And that reciprocal is LESS than 1. Every number greater than 1 has a reciprocal less than 1. Therefore, 1 is the midway point between zero and infinity. Not physically, as in a spatial distance sense, but in the number of actual numbers that occupy the intellectual space between 0 and 1 and 1 and infinity.

 

Q.E.D., as they say.

 

Now go and riddle your most intelligent friend.

Mathematical Jerk Redux

 

I was scrolling through Twitter over the weekend and saw this pic: 

 with the phrase, “DON’T BE A …” right in front of it.

 

Yes! It took me a while to decrypt this (then I had to resort to google) but this is the mathematical expression of a Jerk.

 

No, not that kind of a jerk, not the kind the witty Twitter user was referencing. This kind of a jerk is what you’d experience if you were speeding up the highway and suddenly a force, say a huge gust of wind, pushes your vehicle quickly and unexpectedly to one side.

 

Now, “speed” here is a relative term. In physics, it’s called “velocity” because direction is generally though not necessarily indicated. Velocity is distance per time. It can be expressed in an equation relating these two variable. Throw some Calculus 101 in the mix, and you can obtain what’s called the second derivate of this equation. Since velocity is the change in distance over time, the second derivative represents the change in velocity over time. It’s called acceleration. Now, the third derivative (if you apply the derivative-obtaining technique to the second derivative) represents the change in acceleration over time. This is called “jerk.”

 

Like the beard-second, like the jiffy, math and physics has some interesting and humorous * terms. I had known about jerk from my calculus classes back in the early 90s, but had forgotten. However, I have never heard the technical terms “snap,” “crackle”, and “pop” in mathematics. Now I have and now you, if you have followed me up to this point, have also.

 

For the layman,

Acceleration is the change in velocity over time

Jerk is the change in acceleration over time

Snap is the change in jerk over time

Crackle is the change in snap over time

and

Pop is the change in crackle over time

 

And this is the Euler’s-honest truth!

 

Edit: After writing and publishing this, I see that I had done a similar blog post on it, here, on January 14, 2011, over thirteen years ago! It’s a great exhibit about the fickleness of memory. If you have a mathematical bent, I’d recommend reading that short post, ’cuz I particularly like the analogy used way back then.

Fibonacci Kilometers

 

I just learned a nifty little item a few days ago that only stuck out to me since I posted a bad math joke on Fibonacci numbers about a month ago.

 

First, a refresher for those mathematically challenged. Don’t worry; it’s pretty easy.

 

The Fibonacci sequence is a sequence of numbers obtained by adding the two prior to numbers together to get the next number in sequence. It starts with a 0 and 1, then you get the following:

 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

 

It’s got about a billion uses in math and computer science and shows up in such various areas as nature, architecture, and the subject of beauty. I never got too deep into it in my college days a few decades back. Might look into it as an anti-Alzheimer’s medicine in a few years, though.

 

Anyway, when you take the ratio of a Fibonacci number with the one prior to it and go out through the sequence to infinity, that ratio closes in on 1.618…, or what’s called the Golden Ratio.

 

What’s special about this ratio is that it’s very close to the ratio between a kilometer and a mile. Since a mile is longer, for every mile you travel, you travel 1.609 kilometers. Very close to that 1.618.

 

So, to know how many kilometers you’ve traveled when you know how many miles you’ve traveled, simply go to the Fibonacci sequence above and move one number to the right.

 

For example, traveling 5 miles is equivalent to traveling 8 kilometers.

 

It goes in the opposite direction to convert kilometers to miles. If you’re in Europe and a city is 55 kilometers away, that translates to 34 miles in distance.

 

How neat is that?!

 

This Brought a Slight Smile to My Face

 

“This joke about the Fibonacci sequence is just as bad as the two that preceded it.”

 

Answer Key:

Bee Nerds

 

I’m pleasantly meandering my way through a science fiction anthology I picked up from the local library, The 1976 Annual World’s Best SF, and each tale wows me in some little way. True, most have a doomsday vibe, but others often have little nuggets of weird awesomeness that blow me away and give me interesting bits of ephemera to tease out.

Like this one, from “The Bees of Knowledge,” written by Barrington J. Bayley, an SF author whom I have never read:

 

… the Bees are much interested in mathematics, but theirs is of a type that not even he would be able to understand (any more than I could, except intuitively when I was in the grip of the trance). What would he have made, with his obsession with numbers, of the Bees’ theorem that there is a highest positive integer! To human mathematicians this would make no sense. The Bees accomplish it by arranging all numbers radially on six spokes, centered about the number One. They then place on the spokes of this great wheel certain number series which are claimed to contain the essence of numbers and which go spiraling through it, diverging and converging in a winding dance. All these series meet at last in a single immense number. This, according to the theorem, is the opposite pole of the system of positive integers, of which One is the other pole, and is referred to as Hyper-One. This is the end of numbers as we know them. Hyper-One then serves as One for a number system of a higher order.

 

Hyper-One! I love that. This will be forever filed away in my memory as the Theorem of Hyper-One.

“The Bees of Knowledge” is a gentle, weird tale with more than a bit of existential horror tucked in. The “he” mentioned at the beginning of the above excerpt is a man-sized Fly who understands mathematical processes at least up to exponentiation.

And the Bees are ten-foot sized insects that inhabit the planet Handrea, upon which our narrator crash lands, the sole survivor in a malfunctioning life pod from an interstellar passenger ship which unexpectedly explodes. He’s seized and taken by these curious Bees to their hive, which must be something of the size of the Great Pyramid hunched atop Grand Central Station, and spends the rest of his life there. Where does he stand? What he can do to survive, and how can he communicate to these oddly intelligent Bees? We wind up very metaphysical and surreal by story’s end.

Like I stated earlier, I have never read Bayley before (nor had I heard of him). But a quick web search reveals a body of work consisting of at least 16 novels and 87 short stories stretching over a half century (1954 to 2008). His name goes on the Acquisitions List and I will definitely pick up more of his writings should I come across them in my used book store travels.

Hyper-One!

The Great Pyramid and Tau

 

Okay, I read about this a month or so ago and wanted to post something but haven’t had the time, energy or inclination. Now, lucky reader, I do.

Did you know that there is a relationship between our favorite mathematical concept, π, the irrational and transcendental constant, and the Great Pyramid of Giza, seen here:

 

 Yep. There is.

But first, let’s review a simple formula. The circumference of a circle:

 

C =2πr

 

C stands for the circle’s circumference, r for the radius. This 2π thing is also known, to those in the know, as “tau.” It has been trendy in recent years, from what (little) I understand, to push tau over π, arguing that it makes mathematical formulae easier. I don’t know if that’s worth all the effort to overthrow centuries of mathematical foundation, but let’s consider tau for this discussion.

 

Tau = 2π

 

Now, π = 3.14159…, so 2π, or tau, = 6.28318…

 

Roughly 6.283.

 

All well and good – but where does the Great Pyramid come in?

[This is really cool!]

The height of the Great Pyramid is 481.4 feet. Its base length, the length of one of the four bases along the ground, where the pyramid meets the desert sand, is 756.4 feet.

Got that?

Since there are four base lines at the, er, base of the pyramid, the total base length is 3,025.6 feet. 4 x 756.4 = 3,024.6.

So let’s take this total base length and divide it by the pyramid’s height:

 

3,024.6 / 481.4 = …

 

Ready?

 

3,024.6 / 481.4 = 6.28292548

 

Or rounded to the thousandths decimal place:

 

6.283

 

And tau, from above, equals, roughly, 6.283.

Tau = the base length of the Great Pyramid divided by its height!

 

Wow! Are you honestly not blown away by that? More than a coincidence, no? Has to be, right?

 

Indeed …

 

Dementia-proof

 

Don’t know if I’ve mentioned it here in these electronic pages before, but I’ve been thinking about some personal inevitabilities lately. Now that my two daughters attained significant milestones – oldest entering college, youngest entering high school – I’ve been musing about my later years. Specifically, my health.

Now, I’m a bit overweight, but I know how to lose poundage: Keto, walking, and weightlifting. It’s just a question of motivation. Which comes and goes in bursts. I’ve had some other minor dings and dents to the frame, but a trip to the doctors office should take care of those. Nothing major, and nothing to worry about. Since my heart issues a dozen years ago, I’ve been lucky to be fairly healthy.

It’s my mind I’m concerned with. Specifically, keeping it intact. The brutal reality is that I can probably expect 20, maybe 25 more years of lucid thinking before the dueling dance with dementia begins. How to gain a proactive advantage, how to start strengthening the mind, how to prevent senility from gaining a toehold, early or not?

“They” say you need to keep the mind active in your older years. Engaged. Curious. That shouldn’t be a problem for me. I’m curious by nature. Engaged somewhat, depending on my fascination du jour. But what would be the best course of action for me specifically to take?

It should be no surprise to anyone who’s read the Hopper to know that Hopper likes to read. Maybe a little too much, if it can be argued that too much reading is a thing (I’m not sure). So I thought back, meta-like, upon my reading habits.

Read a lot as a kid, but the quantity went down significantly as a teen. In my twenties I was focused on music, friends, partying, that sort of thing, and didn’t read much. Maybe a half-dozen books each year, if that. Then, in my thirties, I started reading again. Re-reading great stuff from my youth, exploring other works of great fiction first-time, and a lot of science, some religion.

My reading took off in my late 30s / early 40s. Broadly – very broadly – speaking, my nonfiction focus was primarily religion and philosophy. Why am I here blah blah blah. Then, curiously, my reading habits morphed into history over the past ten years or so, heavily into history since I turned 50. Still enjoy it, and still read it. Have three WW2 books on deck, a book on the JFK assassination, a book on the Crusades, one on Christopher Columbus, and am looking for definitive works on the Holy Roman Empire and the Great Schism of 1054.

All well and good.

Now, and I know I’ve written about this before, every Fall when that crisp chill gets in the air (happens late September in New Jersey, late November in Texas), I get the itch to investigate some math. Yes, I know I’m weird. But I did go to school for this stuff 30 years ago and continually kick myself for not finishing with it. So the brain conflates the September nip with the first days of school and exciting new classes. I absolutely love chipping away at higher math, whether it’s calc, number theory, transcendental numbers, infinite series, you name it. Truth be told I’m horrible at it and forget half the stuff I learn a few days after I learn it, but it excites me in a “thrill of discovery” sort of way.

I’ve joked about this with the wife, but the idea has somehow crept from the absurd to the practical. When I turn 60 I decided to buy either

A)    The best all-around college math textbook I can find

B)    My calculus textbook from my Seton Hall physics days

C)    Both

and truly, deeply, delve into the mysteries of math and try to completely understand what I learn before I move on to the next concept. This experiment could last a week or it could be my new obsession, like military history has been to me since 2012. I’ll only know when I cross the threshold into my seventh decade.

Good Lord, “seventh decade”! Where does the time go!

 

Pilish

“Yes, I want a small breakfast of banana bread and apple fritters, expensive alcohol, improving all my old features”

That is something I just wrote in Pilish.

What is Pilish? you may be asking.

Well, Pilish is writing in a way that the number of letters in each word corresponds to a digit of π, in proper order.

To recap, the fist 20 digits of π are:

3.14159 26535 89793 23846….

If you double check the line I wrote above, you’ll see that, yes, it is written in authentic Pilish.

Pilish has been around since the early 20th century. One early example, one much much more witty than my somewhat average example, was written by Physicist Sir James Jean:

“How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics.”

Now that’s a great Pilish line!

I encourage you today, Pi Day 2020, to try your hand at the Pilishtic dialect, especially now that we’ve all been confined to small groups within our homes to save mankind from the superflu. One man, I’ve read, has even written an entire book in Pilish, and there is a sub- sub- subgenre of poetry called Pi-ku, which uses the standard 5-7-5 syllabic pattern of the traditional haiku, but word length must follow the digits in Pi.

Go ahead and try it! It’s less restrictive than a crossword puzzle and more creative than sudoku.

Happy π Day everyone!

Mathematician Discipline

Q: How did the mathematician scold his child?

A: “If I told you n times, I’ve told you n + 1 times …”

Joke courtesy of Patch, trying to cheer up her clumsy dad recovering from an ankle sprain.

Loose Lips

During the Civil War, how did a general know the size of the opposing force?

McClellan had on hand Alan Pinkerton, of the well-known Pinkerton Detective Agency, to do his snooping for him. Unfortunately, whether through incompetence, indifference or ill-will, Pinkerton frequently overestimated Confederate numbers, to which McClellan would add a hefty ten, fifty, or a hundred percent, usually to obtain more men or supplies, but more often than not, to provide a cover to justify his own procrastination.

In May of 1863, facing the Army of the Potomac under a new commander, Joseph Hooker, Lee found an ingenious way to gauge to size of his re-tooled foes. Both sides had spies everywhere, and Lee was sent a Union newspaper article containing an interview with Hooker’s medical director. The director is quoted as stating that the Army of the Potomac currently had 10,777 men on the sick roster. A few sentences later, this same man in this same journal goes on to extrapolate that this number amounted to 67.64 men per 1,000.

The rest is simple math, which Lee did:

67.64 / 1,000 = 10,777 / x

Cross-multiplication yields

67.64 x = 10,777,000

Divide both sides by 67.64 and one gets

10,777,000 / 67.64 = x = 159,329

So Lee concluded from this newspaper report that he was facing 159,329 federal soldiers. (Which was pretty close to the truth. Officially, Hooker was marching with 163,000 men, but this number included teamsters, cooks, horse wranglers, etc.) This bit of intel helped Lee develop a strategy so that his meager force of 58,800 would completely route Hooker in the Battle of Chancellorsville a few days later.

Too bad the phrase “loose lips sink ships,” and the idea behind it, did not become popular until World War II seventy-five years later …

Furlongs a Fortnight

I was reading an article on why the metric system never took off in the US the other day (it was slow at work). In the comments, someone suggested that we use the units that make the most sense or are most familiar to us, i.e., we use “miles per hour” instead of “furlongs per fortnight.”

Furlongs per Fortnight! I absolutely love it!

Because it got me thinking …

What is a furlong? Aside from the two minutes a year the wife watches the Kentucky Derby and I happen to be in the same room, I don’t think I’ve never heard it enough to bother about a definition. It turns out a furlong is one-eighth of a mile – 220 yards, or 660 feet. Twice around the big oval track two towns over I used to walk daily during my bout of unemployment a few years’ back.

How about a fortnight? This I knew. A fortnight is a two-week period. How a two-week period was deemed important enough to earn its nomen I know not, though I believe it has something to do with either wages being paid out biweekly decades or centuries ago, or something to do with a common length of time enjoyed by various sports tournaments. Or maybe something else entirely, I’m not sure.

So –

A furlong is 660 feet.

A fortnight is 14 days.

Now, a furlong is distance and a fortnight is time, so the variable measured by distance over time is speed. A furlong per fortnight is a measurement of speed.

Here’s where the fun begins.

Let’s take a common, familiar speed, say 60 miles per hour. Side note: I remember being absolutely fascinated when, as a small child, I was driving in the car and my father told me that going 60 miles an hour means you’re traveling exactly one mile a minute.

How would you transpose 60 miles per hour into furlongs per fortnight?

Let’s reason this out.

A mile is 5,280 feet which is 8 furlongs.

So 60 miles per hour would be (60 x 8) 480 furlongs per hour.

But an hour is a fraction of a fortnight. A fortnight is 14 days x 24 hours, or 336 hours. Thus, an hour would be 1 / 336th of a fortnight, or 0.00297619 of a fortnight.

Thus, 60 miles per hour would be 60 miles per .00297619 fortnight. To determine the miles traversed over a fortnight at 60 miles per hour, you would divide 60 by .00297619 and obtain 20,160.

So 60 miles per hour would be 20,160 miles per fortnight.

And now for the best part. What would be the furlong / fortnight equivalent of miles / hour?

Multiply 60 by 8 and divide by 1 divided by 336.

(60 x 8) / (1 / 336)

(480) / (.00297619)

161,280.

60 miles an hour is as fast as 161,280 furlongs a fortnight.

There!

And that’s the principle behind the why that we in the United States don’t use the metric system!

This Made Me Laugh

Musings on 196

Take any number greater than two digits and add its palindrome. For example,

26 + 62

The answer will also be a palindrome.

26 + 62 = 88

Sometimes, you have to do the operation more than once:

48 + 84 = 132; 132 + 231 = 363

Sometimes, many times more than once.

59 + 95 = 154; 154 + 451 = 605; 605 + 506 = 1,111

The operation of adding palindromes has to be done twenty-four times for the number 89 to reach a palindrome.

But, consider this.

Do all numbers undergoing addition with their palindromes eventually wind up as palindromes?

Possibly not.

Numbers that may not eventually end up palindromic are known as Lychrel numbers. But Lychrel numbers have not yet been proven to exist; numbers that don’t quickly revert to palindromes when added to their palindromes are really candidates for the designation Lychrel number.

A clarification: Lychrel numbers have not been proven to exist in base 10; they have been proven in other bases, such as binary and hexadecimal.

The lowest candidate for Lychrel numberhood in base 10 is 196. The next two, going up, are 879 and 1,997.

Now, here’s a thought experiment. Or rather, the set-up to one:

Imagine yourself up one level of being. Not God, but something more powerful than a man. What a man is to an animal, you in your new self are now to man.

Imagine what new powers of reasoning you would have. Much greater than anything imaginable now, so I guess my command is a little self-defeatingly paradoxical. But, anyway, try to guess at what new intellectual capabilities you might now have.

For instance, I think I might have the capacity to move into another “higher” dimension. From this vantage point, I can look down upon that thing we call MATHEMATICS. I can see the whole gridwork our reality lies upon. Or within. Or analogous to. However it’s described, I can see it all as I “look” down upon it.

I bring my focus like a scalpel on the subject of Lychrel numbers. I can see how the pattern evolves, where it leads, how it twists and turns. I can see it because I can somehow transform this gridwork into more than just three spatial and one temporal dimensions. As many as I need until the pattern coagulates out of the mist. Then I have the ability to predict which numbers go Lychrel – if any indeed do.

Once envisioned, I have the ability to retain this information as I sink down back to the level of man. And I can teach man what I’ve learned, because though a man can’t teach an animal anything beyond the simplest of reward-punishment behaviors, can’t teach it to reason, I as a superior being can teach man to be over-rational. Supra-rational.

I wonder – can man alone bring about such a state of over-being? Meditation? Mind-altering drugs? A random knock on the head from a non-fatal accident? Or is this simply a case where an over-being needs to reach its “hand” down to “lift us up”? Did Einstein and Newton enter such realms without even knowing it?

Or will it simply and simplistically devolve down to developing a computer with enough chip power and space to grind out all the permutations 196 must go through to determine if its Lychrel or not?

Will we reach supra-being-hood before 196 is declared Lychrel?

My money is on the former.

Three Penny Madness

Let’s say you take three pennies and toss them up in the air simultaneously. What are the odds they all come down on the same side, either heads or tails?

Well, obviously, there’s a 50/50 chance a flipped coin will result in either a head or tail. A one in two chance, or 1/2.

For three coins, the probability of all landing all the same would be

(1/2) x (1/2) x (1/2)

Which is 1/8. One out of eight times, or 12.5% of the time.

But …

When you toss three coins up in the air, at least two will come down the same, right? If they don’t all land either all heads or all tails, at least two will be heads or tails.

So that leaves the third coin in question.

It has a 50/50 chance of agreeing with the other two coins. A one in two chance, or 1/2.

Therefore, the odds of flipping three coins simultaneously and getting three matches would be

(1/2)

Which is 1/2. One out of two times, or 50% of the time.

So ….

Which is the correct probability? One-eighth of the time or one-half?

Why?

Why Can't Daylight Savings

Happen every weekend?

I am so refreshed – really needed that extra hour of sleep. I am sure you are, too. So, why can’t we do this every weekend?

Unless I’m not quite awake yet, I think it can be done. In order to do it, though, we’d have to shorten the amount of time a second measures. Here, let me explain.

There are 168 hours in a week. For a weekly daylight savings to occur, we’d need to have 169 hours, so we can get an extra hour of sleep every Saturday night / Sunday morning.

Every second would then need to be 168/169th in duration, so we could squeeze in that extra hour over the week that we could take back in precious, delicious sleep over the weekend.

168/169 = 99.408 percent as long.

So we’d have to shorten every second by 0.592 percent. Hardly noticeable.

I say let’s do it! Parents, are you with me? Someone start a petition!

Or am I still asleep, sleep posting ….

Quest

“A mathematician is a blind man in a dark room looking for a black cat which isn’t there.”

- (in all probability erroneously) attributed to Charles Darwin

Can also describe Hopper trying to figure out something worthwhile to do …

Math Humor

If Twitter existed back then