Happy Pi Day, everybody!
- Thread starter Scribe1
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Heh... You beat me to it. I am two threads away from visiting my profile to remark on it being Pi Day.
A common misconception is that pi is random. It is not. It is deterministic and fixed. We credit Archimedes with doing the first rigorous study of pi.
It's also an 'irrational number'. The thinking at the time was that all things should be divisible by two rational numbers to a point where they've reached the lowest common denominator. That was rational. There should be a ratio between the numbers. Those irrational numbers can't be expressed as the ratio of two integers.
There being no ratio, it is not rational. Some of the old Greeks felt that the natural universe should be ordered, and things should be able to be expressed as a ratio.
These are some of the earliest steps into what is known as the philosophy of mathematics, or 'logicism' if you prefer.
Mathematics is both a philosophy and a language. I strongly disagree with those who say that mathematics is the language of the universe. No, the universe is indifferent. It's just the language we use to describe the universe, a very human thing. It's a tool for evaluating statements to ensure they/re logical. 1 + 1 = 2 because it's logical to conclude that.
Sometimes, it may not appear logical to some people. A good example is that 0.999... = 1. Well, it is if you're using the standard real number system.
Enjoy Pi Day!
A common misconception is that pi is random. It is not. It is deterministic and fixed. We credit Archimedes with doing the first rigorous study of pi.
It's also an 'irrational number'. The thinking at the time was that all things should be divisible by two rational numbers to a point where they've reached the lowest common denominator. That was rational. There should be a ratio between the numbers. Those irrational numbers can't be expressed as the ratio of two integers.
There being no ratio, it is not rational. Some of the old Greeks felt that the natural universe should be ordered, and things should be able to be expressed as a ratio.
These are some of the earliest steps into what is known as the philosophy of mathematics, or 'logicism' if you prefer.
Mathematics is both a philosophy and a language. I strongly disagree with those who say that mathematics is the language of the universe. No, the universe is indifferent. It's just the language we use to describe the universe, a very human thing. It's a tool for evaluating statements to ensure they/re logical. 1 + 1 = 2 because it's logical to conclude that.
Sometimes, it may not appear logical to some people. A good example is that 0.999... = 1. Well, it is if you're using the standard real number system.
Enjoy Pi Day!
Since pi is an infinitely long number, isn't it close enough? That is, if we use the first 100 or so digits to the right of the decimal point?
Pi to the 38th digit would be an accurate calculation for the entire universe, as memory serves. Some systems limit it to 15 digits. 3.14 is accurate for a circle the size of Mars' orbit (or something like that), it'll be something like a fraction of a centimeter off. I think I've heard it said that pi to the 15th digit is so accurate that if used to measure the Earth's circumference, it'd be like a molecule off.
We absolutely do not need to refine pi further. We have enough accuracy for the known universe.
Don't read that wrong. People are still going to continue calculate pi beyond what we have now.
Linus Media Group (yes, that Linus -- and, no, not that Linus) calculated pi out to 300,000,000,000,000 decimal points.
This is more or less pointless as far as uses in classical physics. I don't know if it'll help with quantum physics. While I do pay attention to some quantum physics information, it's outside of my wheelhouse, and our understanding has changed drastically since I was doing formal studies.
So, I'm not able to speculate on usefulness elsewhere.
From rockets to cancer research, here’s how the number pi is embedded in our lives
As people celebrate Pi Day this March 14, the mathematical constant continues to be part of everyday life for scientists and engineers.
www.nbcnews.com
This little history of pi just appeared in my news
Ah! Thank you for that. Most enlightening, Professor.
Well said! An engineer friend of mine said that 2+2=5 for sufficiently high values of 2, and we all laughed. The reason I'm math-phobic is that it seems too arbitrary to me. Yet, what a beautiful way to express things (I never thought I'd say that). And of course, one can do fun and beautiful things like fractals. Anyway, thanks, @KGIII !
LOL I typically avoid my honoraries, but thanks.
I do still give talks at a couple of schools. One of the schools is a local high school. One of them is a fairly local university. The last one would be a private school, specifically the one I attended long ago. As I keep ties, I might be considered a professor emeritus. To that end, I did a ton of TA work and taught some classes for a couple of years after I retired.
I have strong opinions regarding how we teach mathematics to our children. I firmly believe that we'd have greater numeracy if we taught mathematics like we teach languages, along with teaching a good understanding of the philosophy. Kids never really learn 'why' in mathematics. They're taught by rote.
I can pinpoint when my life changed. It wasn't a grand epiphany, nor did it suddenly make it easy to learn mathematics.
I was puzzling over finding the area of a triangle. The teacher was behind me and said that I could just turn it into a rectangle and divide it in half. It was at that moment that I realized it was a language and that you can express things in different ways.
Ah, well...
Remember, all squares are rectangles - but not all rectangles are squares!
Precisely my problem! My dad was a physicist and was enraged by my inability to "get" math. It was in a brief stint at a college, in a class commonly referred to as "math for dummies," that a young professor explained "rise over run" as applied to the roof of a house. BANG! I got it! that may be the only thing I really retained, because the rest was, well, arbitrary.
That it is.
But there's a difference between the following two statements:
"We do it this way because that is how it is done."
"We do it this way, and this is the reason why we do it this way."
I'm a fan of the latter.
But, yes, yes, it is very arbitrary.
In my above example, I shared the classic 0.999... = 1 statement. If you go to the head of the class and write that out on a chalkboard, you'll get to 0.999999999999 until you run out of space. At no time will you get to 1. You can write that out until the heat death of the universe (entropy is the final boss), and you'll never get to 1.
But, it does equal 1. Not only does it equal 1, I can prove it. (You can just Google this. It's a common way to describe some things and can be a bit of a puzzler for some.)
I do like the proof requirements.
Also, there's a series of work known as Principia Mathematica, by Alfred North Whitehead and Bertrand Russel. In that work, Russell took 162 pages to prove that 1 + 1 = 2. In the end, he pretty much just concluded that it was true because it's logical that it's true. Well, that and there's the very basic 'proof by counting' (combinatorial).
Also, now that I mentioned that, there are actually all sorts of ways to prove stuff in mathematics. Most folks will never go beyond algebraic proofs.
Rather than add 5 more paragraphs...
I just read a snippet about Einstein on the BBC. I'm no genius, but Einstein had one thing in common with me: in school, he only passed subjects that were interesting to him (philosophy and math).
I barely passed on math, even had to do the fourth grade over again.
Einstein passed math with flying colors, as everyone knows.
Still, I have an inner craving to grasp some advanced math subjects.
I barely passed on math, even had to do the fourth grade over again.
Einstein passed math with flying colors, as everyone knows.
Still, I have an inner craving to grasp some advanced math subjects.
Thank you! I can't thank you enough; Dad's been dead these many years, and math was a "thing" that got between us and kept us separated. Your remarks are quite powerful to me. Thank you so very much!That it is.
But there's a difference between the following two statements:
"We do it this way because that is how it is done."
"We do it this way, and this is the reason why we do it this way."
I'm a fan of the latter.
But, yes, yes, it is very arbitrary.
In my above example, I shared the classic 0.999... = 1 statement. If you go to the head of the class and write that out on a chalkboard, you'll get to 0.999999999999 until you run out of space. At no time will you get to 1. You can write that out until the heat death of the universe (entropy is the final boss), and you'll never get to 1.
But, it does equal 1. Not only does it equal 1, I can prove it. (You can just Google this. It's a common way to describe some things and can be a bit of a puzzler for some.)
I do like the proof requirements.
Also, there's a series of work known as Principia Mathematica, by Alfred North Whitehead and Bertrand Russel. In that work, Russell took 162 pages to prove that 1 + 1 = 2. In the end, he pretty much just concluded that it was true because it's logical that it's true. Well, that and there's the very basic 'proof by counting' (combinatorial).
Also, now that I mentioned that, there are actually all sorts of ways to prove stuff in mathematics. Most folks will never go beyond algebraic proofs.
Rather than add 5 more paragraphs...
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Isaac Newton?
Isaac Newton?
This will save me some time:
I strongly suggest not reading it. Also, some of it is out of date. Yes, mathematics changes. It's just slow to change.
As they say, "Science progresses one funeral at a time."
I agree with your point about π being deterministic. The “random” idea mostly comes from how its digits behave, not from how it’s defined. Same with irrational numbers in general, they push against the expectation that everything should reduce to clean ratios, which must have been a big philosophical shift at the time.
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You can thank (I suppose, if we must just pick one group) the Pythagoreans.
Now, it has been a while, and this is just a quick reply, but it seems to have 'started' (again, if we must point to one group of people, or to one person) when they worked out that different lengths of strings resonated with different notes and some of those notes complemented each other. These were simple whole-number ratios. So, they further reasoned that math made music, which meant that the whole universe could be described with such ratios.
I mean, sure, they had the general idea figured out -- but they were reaching pretty deep into the speculative arena.
And, yes, it was from that Pythagerous.
They also believed that some foods would arouse the eater and make them less spiritual...
They'd largely end up being subsumed by the Platonic movement later on, though the more accurate date of such eludes me at this moment in time.
The history of Western mathematics is fuzzy at times, but it is documented to some extent. Fortunately, much of the old Greek scholarly research was preserved and translated from the 8th to the 11th century. That period even has the 'title' of being the 'raeco-Arabic translation movement', which is nice and easy to remember.
Ah, well...
It does make me wonder where we'll be in a hundred years. Our understanding of the universe is still lacking in many ways, so it'll likely improve over that time frame.
Heck, imagine what will be considered 'normal' in the year 3026. I sure hope it doesn't include flying cars, because people can't drive worth a fart as it stands.
Things like pi are interesting to me (not that I'm smart or anything like that). Here's another of my favorites, the golden ratio:
https://en.wikipedia.org/wiki/Golden_ratio
https://en.wikipedia.org/wiki/Golden_ratio
The Golden Ratio is often used, especially in architecture. It's good stuff.
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