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| numb3rs |
 Posted: Sat Mar 15, 2008 1:12 pm Post subject: pi 3.14... |
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calculate pi as far back as you can go 
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
72458 70066 06315 58817 48815 209
u can use a calculater but no online cheating. 
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| Faron |
| Posted: Wed Mar 19, 2008 10:20 pm Post subject: Re: pi 3.14... |
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Joined: 27 Dec 2007
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| numb3rs wrote: |
calculate pi as far back as you can go
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
72458 70066 06315 58817 48815 209
u can use a calculater but no online cheating. |
that's a full PI? I thought it was infinite. |
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| serpicojr |
| Posted: Wed Mar 19, 2008 10:35 pm Post subject: |
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| The decimal expansion of π does go on forever without repeating. |
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| DivideByZero |
| Posted: Thu Mar 20, 2008 9:32 am Post subject: |
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if a decimals 0.abcdef... goes of forever will it be rational or irrational?
(assuming a, b, c, d, etc are completly random numbers from 0 to 9) |
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| Chemboy |
| Posted: Thu Mar 20, 2008 10:11 am Post subject: |
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| DivideByZero wrote: |
if a decimals 0.abcdef... goes of forever will it be rational or irrational?
(assuming a, b, c, d, etc are completly random numbers from 0 to 9) |
If the numbers are completely random, it will be considered irrational.
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| serpicojr |
| Posted: Thu Mar 20, 2008 12:19 pm Post subject: |
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To be a little more precise, if the digits eventually repeat, then the number is rational. By eventually repeat, I mean something along the lines of:
1/6 = 0.1666...
where the 6 repeats forever. More generally, I'm talking about expansions like:
0.a1...anb1...bmb1...bmb1...bm...
where the string b1...bm repeats forever. This includes the case when the decimal expansion terminates, as you can think of a terminating expansion as ending with an infinite number of 0's. You can also show that any terminating decimal expansion is equal to a decimal expansion with 9's repeating forever.
If the above does not happen, then you have an irrational number. So we can construct irrational numbers using this fact, for example:
0.101001000100001000001000000100000001...
where I keep sticking one extra 0 between successive 1's.
(This whole discussion is independent of the base B we're using, except replace 9 by B-1 in the statement about terminating decimal expansions and, of course, 1/6 is not equal to that expansion when B ≠ 10.) |
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| William McCormick |
| Posted: Thu Apr 03, 2008 9:12 pm Post subject: Re: pi 3.14... |
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Joined: 03 Apr 2008
Posts: 388
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| numb3rs wrote: |
calculate pi as far back as you can go
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
72458 70066 06315 58817 48815 209
u can use a calculater but no online cheating. |
One thing that bothers me, is what they are doing with a number, and calling it pi. Years ago in my school they took pi very seriously, in a good way.
Pi was the lifetime work of a man called Archimedes. He was stated to have created a very large marble wheel, and then a very long marble path for it to roll upon. He spent years polishing and honing his wheel and path.
When he was done, he had a ratio that was just larger then 22/7. He wanted perfection, however at the time without calculators, he could not imagine anyone being interested in his ratio, if it was a complex fraction. So he decided after years and years of working on the problem to call pi 22/7.
But today I hear individuals claim that pi is 3.1459......, the truth is that pi is actually larger then 22/7, in real life.
By actual building and testing of a precision wheel, I found many phenomena that are overlooked, by people claiming to be experts, or claiming to be the authority on pi.
This all started because my family worked for an Aero Space contractor, that also knew pi was larger then 3.14159, but felt that at the time when we were just getting over a war with Vietnam, that this point would not be exciting enough to start a debate.
Years later on the Internet I mentioned it, and someone dared me to make a wheel and measure it. Even though I had pointed out, that the methods they had used to measure a wheel were obviously flawed.
They had used a tape measure, but a band contracts and stretches as it is bent around a round object. So that the actual point the band measures, is at the diameter of the wheel being measured, plus one thickness of the metal tape used to measure it. Or half a thickness on either side.
I do this kind of work so it is common knowledge to me.
What I found by making the wheel and rolling it, was truly incredible.
I had just made a very nice pass with the cobalt cutter, and decided to take the wheel off the lathe and do a test run, on my polished table. So I gave it a good wipe with a clean towel, and did the same for the table.
I figured with the possible tiny debris, and oil from the lathe and equipment, that if anything the wheel would roll a little bit further, because of the increase in diameter that the particles would cause.
But to the average fellow the wheel looked like it belonged in an operating room.
I give the wheel a roll, and to my surprise, it rolls to a ratio of 3.14159, this made me weak in the knees. Because everything I had ever gotten from the Grumman guys was right on.
But I am a good sport and good friend. So I figured I would for the sake of science, clean up the wheel, and table with xylene. And then roll it for a real scientific measurement that I would post with an apology to the guys. I actually thought that the wheel would show once cleaned a number less then 3.1459 if it was already at 3.1459.
So I roll the wheel and I admit I was not real happy about doing it. But I roll it, and it rolls long. I mean much longer then before. I can see the first mark on the table and the wheel just rolled past that mark. So my first thought is wow, wishful mis-roll. So I roll it again, it rolls right on the new longer roll mark. I do it many times. And then I get very confused. And take a break.
I for the life of me cannot figure out what took place. If anything in my mind at the time, the wheel should have rolled longer with the dirt on it. How could it have rolled longer with the dirt off of it. It couldn't, right?
Well a couple hours later, after I drew some cadd drawings of a wheel and a bump, and then rotated objects on the screen again and again. It hit me like a ton of bricks. A dirty wheel simulates a dirty bumpy road. And everyone knows that a dirty bumpy road is not the shortest distance between two lines.
So with a 70-75 T-8 aluminum wheel rolled on the same material, flat rolling surface. I was able to conclude that pi is larger then 22/7 and closer to 3.14308 by actual testing.
The idea of the computer people using a polygons sides, combined cumulative length, to determine the length, of a circles circumference, is probably the least mathematical thing I have ever seen. It should be the new definition for comparing apples and oranges.
But then the original computer people decided to replace the division key with the fraction symbol. And most know they have two different meanings in mathematics.
Sincerely,
William McCormick
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| serpicojr |
| Posted: Thu Apr 03, 2008 10:22 pm Post subject: |
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| I'm not sure where you got your anecdote about Archimedes, William, but Wikipedia offers another story: Archimedes showed that π < 22/7 by circumscribing a regular 96-gon about a circle. I'd love to continue this discussion with you, and I'd really love to convince you that your estimate of π is close but wrong. |
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| Cuntinuum |
| Posted: Fri Apr 04, 2008 2:11 am Post subject: |
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| bit4bit |
| Posted: Fri Apr 04, 2008 5:22 am Post subject: |
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| How do we know they didn't just start making them up after the first 5 digits or so? |
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| sunshinewarrior |
| Posted: Fri Apr 04, 2008 5:29 am Post subject: |
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| bit4bit wrote: |
| How do we know they didn't just start making them up after the first 5 digits or so? |
Because if you have a good enough computer, you can calculate many of them yourself? |
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| bit4bit |
| Posted: Fri Apr 04, 2008 7:51 am Post subject: |
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I know, I was joking. It's not like anyone can recite the first million digits of pi off the top of their head.  |
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| William McCormick |
| Posted: Fri Apr 04, 2008 5:52 pm Post subject: |
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Joined: 03 Apr 2008
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| serpicojr wrote: |
| I'm not sure where you got your anecdote about Archimedes, William, but Wikipedia offers another story: Archimedes showed that π < 22/7 by circumscribing a regular 96-gon about a circle. I'd love to continue this discussion with you, and I'd really love to convince you that your estimate of π is close but wrong. |
I had gone to Freeport public schools. At the time they were the math capital of the world, and I was a mathematician and honors student.
I was taught that Archimedes wanted a perfect ratio of a wheels diameter to circumference.
I was taught that he created a rather large wheel and rolled it on a rather large solid marble walkway.
There was even debate as to what size wheel he created to do the experiment. It was stated that the wheel was as tall as he was. At the time Greece used a different form of measurement, then we do today. If indeed at the time we were correct, Archimedes stood almost seven feet tall. And so did his wheel.
I have heard today that they claim that the cumulative measurements of the sides of an inscribed polygon, and the cumulative measurements of the sides of a circumscribed polygon, somehow average out to pi.
And although most college students will never even have a chance to see the huge difference that having the correct ratio of pi can do for you in life. It is a rather small difference, only a couple thousandths difference in whatever unit you are measuring in.
Most machinery and equipment compensates for this discrepancy by adding yet another corrective factor. This is unnecessary if you have the correct ratio of pi.
Rules like "And a little bit more" really do not cut it in the real world. When someone from college comes in and demands that the machinery is calibrated to the 3.1459 standard, it gets a bit comical. If not embarrassing.
As I mentioned pi is actually closer to 22/7 by actual measurement, and in day to day life use, it comes out to be 3.14308.
Sincerely,
William McCormick
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| serpicojr |
| Posted: Fri Apr 04, 2008 6:57 pm Post subject: |
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| William McCormick wrote: |
| I had gone to Freeport public schools. At the time they were the math capital of the world, and I was a mathematician and honors student. |
I guarantee you that Freeport (and I'm not sure which one you're talking about, because there are quite a few, and none is distinguished above any other) was never the math capital of the world. But since we're talking math credentials, let me give you mine: I'm a math PhD student at a leading university on the East coast, so I'm surrounded by a lot of good math and could argue I'm smack dab in the middle of the math capital of the world.
| Quote: |
I was taught that Archimedes wanted a perfect ratio of a wheels diameter to circumference.
I was taught that he created a rather large wheel and rolled it on a rather large solid marble walkway.
There was even debate as to what size wheel he created to do the experiment. It was stated that the wheel was as tall as he was. At the time Greece used a different form of measurement, then we do today. If indeed at the time we were correct, Archimedes stood almost seven feet tall. And so did his wheel. |
He may have done this, but he also realized that this is not the mathematical way to go about finding π. As I stated above, he used pure geometric arguments to show that π < 22/7.
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| I have heard today that they claim that the cumulative measurements of the sides of an inscribed polygon, and the cumulative measurements of the sides of a circumscribed polygon, somehow average out to pi. |
Close. It's the limit, not the average, of perimeters that can be taken to find π. And it's not that surprising. The argument is pretty simple--inscribed polygons have perimeter shorter than the circumference, circumscribed polygons have perimeter greater than the circumference, and as you let the number of sides increase to infinity, the perimeters or circumscribed and inscribed polygons approach the same number, π. In particular, this shows, as I stated above, that π < 22/7. If you don't understand this argument, I'd be happy to go into it deeper.
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| And although most college students will never even have a chance to see the huge difference that having the correct ratio of pi can do for you in life. It is a rather small difference, only a couple thousandths difference in whatever unit you are measuring in. |
I don't know, the math curriculum at my university stresses error, and I'm going to assume that the science and engineering departments do, too. So I'm not sure where you're pulling this from.
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| Most machinery and equipment compensates for this discrepancy by adding yet another corrective factor. This is unnecessary if you have the correct ratio of pi. |
You're the self-proclaimed machinist, so I'll have to believe you here. However, I'm going to assume that some sort of measurement bias is occurring--the way you measure radius or circumference (or both) must be slightly inaccurate. I believe the error probably creeps in when you calculate the length by rolling the circle and measuring on the flat surface.
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| Rules like "And a little bit more" really do not cut it in the real world. |
Very true. I can calculate π to arbitrary precision. Can you do that by rolling circles around?
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| When someone from college comes in and demands that the machinery is calibrated to the 3.1459 standard, it gets a bit comical. If not embarrassing. |
Yes, I am sure he goes back to school and has a good laugh with his colleagues at your expense. Either that or he goes to the wig shop after he pulls all his hair out.
| Quote: |
| As I mentioned pi is actually closer to 22/7 by actual measurement, and in day to day life use, it comes out to be 3.14308. |
No. You're not right. Perhaps for your applications, your assumptions are slightly off and this approximation is actually better than the real value of π, but this is not the value of π. In real life applications, say solving differential equations for engineering calculations, you'd run into a whole slew of problems if you used your value for π. |
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| William McCormick |
| Posted: Fri Apr 04, 2008 7:45 pm Post subject: |
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As a side note, in my time we used to calculate out 22/7 on paper to see if at some point the pattern of numbers would change. That is how this game started. There were no calculators, other then mechanical ones that would only go out a couple places.
Although the pattern looked pretty good, someone had speculated that at some point it would change. I just got a repeating number, but I never went out that far. Someone speculated that out a great number of places that the pattern might change.
I don't know if that was just to keep our minds off of, blowing up the school, or if they really believed that. But that was how all this started.
Then in the fifth grade they bought the honor students a one million dollar main frame computer. That we started to work on. It did not have a division key that I recall, but it would not perform in line problems like it was supposed to be able to do.
Being that the whole premise of computers and the grants, was that the computer was going to advance in line mathematical problem solving, it seemed like the computer company missed the boat.
As early as 1928 they used the in line division symbol in electrical and telephone training manuals.
The division symbol means to take everything on the left and divide it by everything on the right. That is not what a fraction symbol does. The computer effectively destroyed mathematics. But a lot of money was on the line.
The in line division key in conjunction with the proven fastest order of math, would allow the input of in line math formulas with almost no parenthesis in real applications of math to build real things. It was a tested and proven superior order.
The order went addition, subtraction, multiplication and then division. That meant that the division symbol and the multiplication symbol could be placed anywhere in the formula and they would take everything on the left and multiply or divide by everything on the right. Because addition and subtraction had already been done.
Years ago there were no, negative exponents. And there would not be today, if there was a division key on the keyboard. We used to divide by an exponent to get or show a fraction of a whole.
This allowed for the very different fraction symbol, to have its very own unique roll in the formula. Without difficult to enter parenthesis. In real life you enter fractions into a formula all the time. Fractions simplify entry.
http://www.Rockwelder.com/Electricity/Capacitor.pdf
Sincerely,
William McCormick
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