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On The Significance Of Numbers
Saturday, 14 March 2015 13:56Numerology.
As a species, we are very good at finding patterns in things, even when they aren't really there. We take solace in patterns of numbers -- your mother's phone number, a friend's birthday, an old house number -- and recognize when we see them elsewhere. Back in the early 70s, when Wendy and I both went and applied for Social Security Numbers -- she needed one for a part-time job as a grocery store checker and I was sent along, too -- I was able to remember my number because it was associated with three Santa Fe railroad locomotive numbers. Yeah, I was always a geek.
And as a night owl I go to bed late, and I often amused by the digital clock displaying "magic times" -- 12:34, 1:23, 2:34, 3:45 -- and of course Pi Time, 3:14am. Not only are these numbers memorable in some way, but I also notice them because my schedules are such that I often actually make it to the side of the bed at the same clock times. We notice patterns and are creatures of habit.
Of course this is all because of Pi Day. And not just any Pi Day, but the most amazing Pi Day ever! Or at least to hear about it, or the makers of about ten different Pi Day t-shirts that have been showing up in my Facebook feed for months.
Okay, we get it. π = 3.14, March 14th is 3/14. And it's 2015, so π = 3.1415926, so March 14, 2015 becomes 3/14/15 9:26 am/pm. Amazing! Incredible!
Well, not really. Remember, this is in the United States, and despite the usual American narcissism towards the rest of the 95% of the world, most of the world doesn't use 3/14 for March 14th. The European model is 14.3 or 14.3.2015 for 14 March 2015. That would make Pi Day as 3.1 (3 January) or 3.1415 (3 January at 4:15) or 3.14 (3 January 2004) or 3.141 (3 January 2041). In other words, 14.3.2015 at 9:26 just isn't a big deal.
But we see patterns -- and then insist that everyone else see them, too.
Of course, as a number, π is pretty amazing. Much more than just the ratio of the Circumference to the diameter. π shows up all over the place. And π is an irrational number -- the symbol π represents the number, but it cannot be written down completely, as it is a non-terminating, non-repeating number.
Much folklore holds that the digits "must" represent all the writing ever written -- sort of like the million monkeys on a million typewriters for a million years theory of randomness. There was a guy on NPR's Here and Now on Friday mentioning that in the first million digits of π, it isn't even an even frequency of the numbers 0-9. It's like the most common number is 5, with 0, 6 and 9 in shorter supplies -- or something like that.
My good friend from ISP at Northwestern, the late Steve Houdek, spent most of freshman year refining a program that could calculate thousands of digits of pi. It didn't NEED to be done, having been done before, but it provided a really great programming and research project, running from BASIC to FORTRAN to Pascal and later C, I believe.
Here's the first 41 digits of π, or 40 decimal places:
Those million or more digits of π? That's not irrelevant, but it is mathematics and not Physics. Just saying. And I, for one, am all for purely intellectual pursuits, so I'm not knocking mathematics theory.
The Significance of Numbers
Those of you who have ever taken a Dr. Phil Physics course, know that I spend a lot of time talking about Significant Figures. Because in an introductory Physics class, the equations and numbers represent real things which are either measured or calculated.
But, as I tell my students, most scientific calculators are set up to display up to 10 digits and calculate to 12. That doesn't mean you need to use all of them. In fact, whenever I see a student dumping all ten digits in an answer, I know this is an attempt to snow me with numbers, because the student clearly doesn't know what they're doing.
v = d / t = 2.31 meters / 5.34 seconds = 0.4325842697 m/s ?
Nope. Can't justify all those digits.
Most of our typical measurements have three significant figures -- 3 sig. fig. -- which is how many numbers are there, not decimal places, and not counting zeroes which are merely placeholders. So 123 and 1.23 and 0.123 and 123,000 and 0.00000123 are all 3 sig. fig. numbers. Multiply or divide 3 sig. fig. numbers together, and you can only justify 3 sig. figs. in your answer -- though I typically keep an extra "fuzzy" digit to help with the decimal representation of a fraction, or what I would call 3+1 sig. figs.
2.31 meters / 5.34 seconds = 0.433 m/s or 0.4326 m/s. That's it.
The carpenter is told to "measure twice, cut once". In class I hold up a meter stick. It is divided into 100 centimeters and each centimeter is divided into ten millimeters -- or 1 meter = 1000 mm. A millimeter is comparable to the width of a saw blade, so cutting a meter stick to a length of 31.7 cm is reasonable.
I like to point out that I don't know how to make any 10 sig. fig. measurement that costs less than a million dollars to make, so the carpenter building your house is not going to do it.
To get a 10 sig. fig. measurement, we'd need to take that one meter stick and divide it into 1010 or 10 billion pieces. 1 x 10-10 meters is an Angstrom (1 Å = 0.1 nm = 0.1 nanometers). 1 Å is about the size of a hydrogen atom. So... if you want to measure your cut to 10 sig. figs., you have to measure from the last atoms of the last molecule of lignon or cellulose poking out on either ends of the wood. AND have it not move during the course of the repeat measurements or cuts.
Uh-uh. Not going to happen.
Why does the calculator give you 10 digits and calculate to 12 if you can't use them all? Because calculators are intrinsically stupid machines and have no idea what you are doing. So it gives you what it can best do, and expects you -- the person with the brain -- to figure out how much or what part you really need. Because let's face it. Four people sharing a pizza for $18.29 plus 6% sales tax and a 1/6th tip totals $22.6186333 or $5.65465833 per person. And you can't make change from a penny for 0.465833¢.
Why do some people get it when splitting a check -- and not in a Physics problem? (evil-grin)
The universe is some 14 billion light years across. A light year is 1 LY = 9.429 x 1015 meters or 9.429 quadrillion meters. An atom is 1 Å wide, 1 x 10-10 meters. The nucleus of an atom is about 1 femtometer = 1 fm wide, 1 x 10-15 meters. If we were to divide a nucleus into a thousand parts, like we did with the meter stick, they'd be 1 attometer = 1 am wide, 1 x 10-18 meters. That's the smallest SI metric prefix I've seen used.
That makes a light year, 1 LY = 9.429 x 1030 femtometers. So to calculate the circumference of the universe, 132 billion x 1030 femtometers wide, using C = πD = 2πr, or the area, A = πr2 = πD2/4 we would need to know π to 41 digits.
Usually less. Far less. Really, far less.
Like I said, 41 decimal places of π:
Something to think about on this made up Pi Day...
http://mail.colonial.net/~hkaiter/aa_newest_images/happy-pi-day1.jpg
So... what kind of pie are you going to have on Pi Day? We're going to have a Key lime pie... mmm.
What? You thought I wasn't going to celebrate Pie Day with some π?
Are you crazy?
Looking for another way to "celebrate" the Once In A Lifetime Most Epic Pi Day Of All Time (or at least this century, for certain values of Once In A Lifetime and Epic)? Do a Google search on "pi day" and click on the Images tab. Scroll and scroll and scroll. There's a lot of fun out there...
Dr. Phil
As a species, we are very good at finding patterns in things, even when they aren't really there. We take solace in patterns of numbers -- your mother's phone number, a friend's birthday, an old house number -- and recognize when we see them elsewhere. Back in the early 70s, when Wendy and I both went and applied for Social Security Numbers -- she needed one for a part-time job as a grocery store checker and I was sent along, too -- I was able to remember my number because it was associated with three Santa Fe railroad locomotive numbers. Yeah, I was always a geek.
And as a night owl I go to bed late, and I often amused by the digital clock displaying "magic times" -- 12:34, 1:23, 2:34, 3:45 -- and of course Pi Time, 3:14am. Not only are these numbers memorable in some way, but I also notice them because my schedules are such that I often actually make it to the side of the bed at the same clock times. We notice patterns and are creatures of habit.
Of course this is all because of Pi Day. And not just any Pi Day, but the most amazing Pi Day ever! Or at least to hear about it, or the makers of about ten different Pi Day t-shirts that have been showing up in my Facebook feed for months.
Okay, we get it. π = 3.14, March 14th is 3/14. And it's 2015, so π = 3.1415926, so March 14, 2015 becomes 3/14/15 9:26 am/pm. Amazing! Incredible!
Well, not really. Remember, this is in the United States, and despite the usual American narcissism towards the rest of the 95% of the world, most of the world doesn't use 3/14 for March 14th. The European model is 14.3 or 14.3.2015 for 14 March 2015. That would make Pi Day as 3.1 (3 January) or 3.1415 (3 January at 4:15) or 3.14 (3 January 2004) or 3.141 (3 January 2041). In other words, 14.3.2015 at 9:26 just isn't a big deal.
But we see patterns -- and then insist that everyone else see them, too.
Of course, as a number, π is pretty amazing. Much more than just the ratio of the Circumference to the diameter. π shows up all over the place. And π is an irrational number -- the symbol π represents the number, but it cannot be written down completely, as it is a non-terminating, non-repeating number.
Much folklore holds that the digits "must" represent all the writing ever written -- sort of like the million monkeys on a million typewriters for a million years theory of randomness. There was a guy on NPR's Here and Now on Friday mentioning that in the first million digits of π, it isn't even an even frequency of the numbers 0-9. It's like the most common number is 5, with 0, 6 and 9 in shorter supplies -- or something like that.
My good friend from ISP at Northwestern, the late Steve Houdek, spent most of freshman year refining a program that could calculate thousands of digits of pi. It didn't NEED to be done, having been done before, but it provided a really great programming and research project, running from BASIC to FORTRAN to Pascal and later C, I believe.
Here's the first 41 digits of π, or 40 decimal places:
π = 3.14159 26535 89793 23846 26433 83279 50288 41972This is more π than you'll ever need for doing Physics problems.
Those million or more digits of π? That's not irrelevant, but it is mathematics and not Physics. Just saying. And I, for one, am all for purely intellectual pursuits, so I'm not knocking mathematics theory.
The Significance of Numbers
Those of you who have ever taken a Dr. Phil Physics course, know that I spend a lot of time talking about Significant Figures. Because in an introductory Physics class, the equations and numbers represent real things which are either measured or calculated.
But, as I tell my students, most scientific calculators are set up to display up to 10 digits and calculate to 12. That doesn't mean you need to use all of them. In fact, whenever I see a student dumping all ten digits in an answer, I know this is an attempt to snow me with numbers, because the student clearly doesn't know what they're doing.
v = d / t = 2.31 meters / 5.34 seconds = 0.4325842697 m/s ?
Nope. Can't justify all those digits.
Most of our typical measurements have three significant figures -- 3 sig. fig. -- which is how many numbers are there, not decimal places, and not counting zeroes which are merely placeholders. So 123 and 1.23 and 0.123 and 123,000 and 0.00000123 are all 3 sig. fig. numbers. Multiply or divide 3 sig. fig. numbers together, and you can only justify 3 sig. figs. in your answer -- though I typically keep an extra "fuzzy" digit to help with the decimal representation of a fraction, or what I would call 3+1 sig. figs.
2.31 meters / 5.34 seconds = 0.433 m/s or 0.4326 m/s. That's it.
The carpenter is told to "measure twice, cut once". In class I hold up a meter stick. It is divided into 100 centimeters and each centimeter is divided into ten millimeters -- or 1 meter = 1000 mm. A millimeter is comparable to the width of a saw blade, so cutting a meter stick to a length of 31.7 cm is reasonable.
I like to point out that I don't know how to make any 10 sig. fig. measurement that costs less than a million dollars to make, so the carpenter building your house is not going to do it.
To get a 10 sig. fig. measurement, we'd need to take that one meter stick and divide it into 1010 or 10 billion pieces. 1 x 10-10 meters is an Angstrom (1 Å = 0.1 nm = 0.1 nanometers). 1 Å is about the size of a hydrogen atom. So... if you want to measure your cut to 10 sig. figs., you have to measure from the last atoms of the last molecule of lignon or cellulose poking out on either ends of the wood. AND have it not move during the course of the repeat measurements or cuts.
Uh-uh. Not going to happen.
Why does the calculator give you 10 digits and calculate to 12 if you can't use them all? Because calculators are intrinsically stupid machines and have no idea what you are doing. So it gives you what it can best do, and expects you -- the person with the brain -- to figure out how much or what part you really need. Because let's face it. Four people sharing a pizza for $18.29 plus 6% sales tax and a 1/6th tip totals $22.6186333 or $5.65465833 per person. And you can't make change from a penny for 0.465833¢.
Why do some people get it when splitting a check -- and not in a Physics problem? (evil-grin)
The universe is some 14 billion light years across. A light year is 1 LY = 9.429 x 1015 meters or 9.429 quadrillion meters. An atom is 1 Å wide, 1 x 10-10 meters. The nucleus of an atom is about 1 femtometer = 1 fm wide, 1 x 10-15 meters. If we were to divide a nucleus into a thousand parts, like we did with the meter stick, they'd be 1 attometer = 1 am wide, 1 x 10-18 meters. That's the smallest SI metric prefix I've seen used.
That makes a light year, 1 LY = 9.429 x 1030 femtometers. So to calculate the circumference of the universe, 132 billion x 1030 femtometers wide, using C = πD = 2πr, or the area, A = πr2 = πD2/4 we would need to know π to 41 digits.
Usually less. Far less. Really, far less.
Like I said, 41 decimal places of π:
π = 3.14159 26535 89793 23846 26433 83279 50288 41972More π than you'll ever need for doing Physics problems.
Something to think about on this made up Pi Day...
http://mail.colonial.net/~hkaiter/aa_newest_images/happy-pi-day1.jpg
So... what kind of pie are you going to have on Pi Day? We're going to have a Key lime pie... mmm.
What? You thought I wasn't going to celebrate Pie Day with some π?
Are you crazy?
Looking for another way to "celebrate" the Once In A Lifetime Most Epic Pi Day Of All Time (or at least this century, for certain values of Once In A Lifetime and Epic)? Do a Google search on "pi day" and click on the Images tab. Scroll and scroll and scroll. There's a lot of fun out there...
Dr. Phil
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