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Today's Pi () Day

iMpathetic said:
Aw....... Meanie.
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yeah, a little bit. 😱 I made it (not totally from scratch but still, I heavily altered it to my liking) and have used it on here before though so that entitles me to a bit of stinginess about it, I reckon. 😛

But thanks for asking, I take it as a compliment, of sorts. 🙂
 
iBlue said:
yeah, a little bit. 😱 I made it (not totally from scratch but still, I heavily altered it to my liking) and have used it on here before though so that entitles me to a bit of stinginess about it, I reckon. 😛

But thanks for asking, I take it as a compliment, of sorts. 🙂
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Well, you're welcome. As you can see, I have gone from about four hundred to 961 posts in the past month and a half, so I haven't really settled into a good avatar yet.
 
Ok, this is strange. Q wouldn't normally miss this sort of thing by so many hours. It's just not like him to miss this.

He must be passed out from the excessive partying and drinking in celebration of this day.
 
Abstract said:
Ok, this is strange. Q wouldn't normally miss this sort of thing by so many hours. It's just not like him to miss this.

He must be passed out from the excessive partying and drinking in celebration of this day.
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Doesn't he already have a thread on this that's like six pages?
 
Math fun

I'm glad you missed me. I was partying nonstop on Pi Day, and enjoying a re-read of last year's thread. Remember, if you celebrate pi day for 365 days a year (366 this year), then you'll always be happy!

I will give you an amazing pi fact in honor of this year's Pi Day. Can you compute an integer using roots, multiplication, and exponentiation of irrational numbers, including our friend pi? It sounds impossible unless you somehow cancel out the irrationals, right?

Well, try these steps using Calculator.app:
  1. Select View->Scientific (or press Command-2).
  2. Press MC to clear memory.
  3. Press 640320 then X^3 then + then 744 then = to compute 640320^3 + 744.
  4. Press M+ to store that number, which is obviously an integer, in memory.
  5. Enter 163. This is a prime number.
  6. Press square root. The square root of 163 is an irrational number.
  7. Press x then pi then = to multiple by pi, another irrational. The result is also irrational.
  8. Press the e^x key to exponentiate. You have computed e^(sqrt(163) x pi).
  9. That's clearly irrational. Or is it?
  10. Press M- to subtract this result from the integer in memory.
  11. Take a deep breath.
  12. Press MR to recall the result.
  13. What do you see? Flat 0! An irrational raised to the power of an irrational times an irrational, minus an integer!
Nothing canceled out the irrationals (see Note below), and it's not a bug in Calculator. I'm sorry if you will now be up all night worrying that the laws of mathematics have been repealed and, if not, whether poor Doctor Q will be arrested for breaking them.

Note: It's true that e^(pi x i) = -1, so using complex powers of e can "cancel out" irrationality, but e^pi is known to be irrational.
 
*wooosh* <- that was the sound of all that maths stuff going right over my head. 😱 😛

iMpathetic said:
Well, you're welcome. As you can see, I have gone from about four hundred to 961 posts in the past month and a half, so I haven't really settled into a good avatar yet.
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Well I like the homestar runner one. 🙂
 
Doctor Q said:
I'm glad you missed me. I was partying nonstop on Pi Day, and enjoying a re-read of last year's thread. Remember, if you celebrate pi day for 365 days a year (366 this year), then you'll always be happy!

I will give you an amazing pi fact in honor of this year's Pi Day. Can you compute an integer using roots, multiplication, and exponentiation of irrational numbers, including our friend pi? It sounds impossible unless you somehow cancel out the irrationals, right?

Well, try these steps using Calculator.app:
  1. Select View->Scientific (or press Command-2).
  2. Press MC to clear memory.
  3. Press 640320 then X^3 then + then 744 then = to compute 640320^3 + 744.
  4. Press M+ to store that number, which is obviously an integer, in memory.
  5. Enter 163. This is a prime number.
  6. Press square root. The square root of 163 is an irrational number.
  7. Press x then pi then = to multiple by pi, another irrational. The result is also irrational.
  8. Press the e^x key to exponentiate. You have computed e^(sqrt(163) x pi).
  9. That's clearly irrational. Or is it?
  10. Press M- to subtract this result from the integer in memory.
  11. Take a deep breath.
  12. Press MR to recall the result.
  13. What do you see? Flat 0! An irrational raised to the power of an irrational times an irrational, minus an integer!
Nothing canceled out the irrationals (see Note below), and it's not a bug in Calculator. I'm sorry if you will now be up all night worrying that the laws of mathematics have been repealed and, if not, whether poor Doctor Q will be arrested for breaking them.

Note: It's true that e^(pi x i) = -1, so using complex powers of e can "cancel out" irrationality, but e^pi is known to be irrational.
Click to expand...

Standard calculators don't do arbitrary precision calculations, so using Mathematica,

640320^3 + 744 = 262537412640768744

and e^(sqrt(163) x pi) =

2.625374126407687439999999999992500725972*10^17

so the difference is (more or less)

7.499274028*10^-13

One thing I find interesting about Pi though is that 6 consecutive 9s occur round about the 760th decimal. Imagine the first person calculating this and thinking "oh it's rational after all" and then "Oh damn!"
 
Thanks Loge. I was certain on first inspection it had to be a rounding error, (or more precisely, an issue of insufficient significant digits) but couldn't really be bothered to go and actually prove it.

Er. Too many pies! Yes, that's my vaguely on topic excuse.
 
Loge said:
One thing I find interesting about Pi though is that 6 consecutive 9s occur round about the 760th decimal. Imagine the first person calculating this and thinking "oh it's rational after all" and then "Oh damn!"
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That would make one awesome YouTube video if they actually caught it on tape.
 
Pssst! Loge's phone number, Loge's National Insurance number, and the numeric part of Loge's street address occur consecutively within the digits of pi! What a huge security breach!
 
Doctor Q said:
Pssst! Loge's phone number, Loge's National Insurance number, and the numeric part of Loge's street address occur consecutively within the digits of pi! What a huge security breach!
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The NI number might still be safe - it contains letters 😛
 
JML42691 said:
But 22/7 isn't pi, 22/7 is 3.142857142857143, pi=3.14159265358979323846
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Another approximation is 355/113 which gives:

3.141592920

PI being:

3.141592653 (with no rounding)

For a difference of:

0.000000267

Not too bad for a quick approximation, and a bit more accurate than 22/7.

Pi talk is fun! Especially when it is an Apple one! 😀
 
355/113 is a worse approximation than just remembering the digits.

"355/113" is seven characters to remember.

Assuming that you can remember 3.14, the next seven characters (with rounding) are 1592654, which you could remember as "159 to 654"

"355 by 113" gives 3.141592920 which is accurate to 6 decimal places.

"159 to 654" gives 3.141592654 which accurate to 9 decimal places, giving you 3 places more accuracy for the same memory effort.

😀
 
RedTomato said:
355/113 is a worse approximation than just remembering the digits.

"355/113" is seven characters to remember.

Assuming that you can remember 3.14, the next seven characters (with rounding) are 1592654, which you could remember as "159 to 654"

"355 by 113" gives 3.141592920 which is accurate to 6 decimal places.

"159 to 654" gives 3.141592654 which accurate to 9 decimal places, giving you 3 places more accuracy for the same memory effort.

😀
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Huh, 355/113 is 6 digits to remember. Just like 22/7 is 3 digits to remember.

Anyhow, 113/355 is more accurate than 22/7 which is all I was saying. I was not saying it was easier to remember, or easier to enter into a calculator. Just more accurate.

Personally, I just remember the digits since it is easy.

Usually, 3.1415926535 which is good enough for what I need.

Otherwise, there is always the Internet for more digits. 🙂

P.S. I get the point of your post. I would suggest that everyone remembers things differently. YMMV. 🙄
 
Still more math fun

The New Yorker had an interesting article about how our brains do math.

It describes some fascinating experimental results. For example, we (at least in the western world) apparently compare numbers by thinking of them in a line from left to right, with larger numbers to the right of smaller numbers. So we think of 4 as being to the right of 3. From the article:
A few years ago, while analyzing an experiment on number comparisons, Dehaene noticed that subjects performed better with large numbers if they held the response key in their right hand but did better with small numbers if they held the response key in their left hand. Strangely, if the subjects were made to cross their hands, the effect was reversed. The actual hand used to make the response was, it seemed, irrelevant; it was space itself that the subjects unconsciously associated with larger or smaller numbers.
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In other words, having the trigger mechanism to the right of your body makes you faster at recognizing the "greater than" condition, while having the trigger to the left of your body makes you faster at recognizing "less than."

Other facts from the article:

* Our brains can recognize "three things" or "four things" as patterns without counting them mentally, but we have to count to identify larger quantities.

* The number of digits we can easily memorize depends on the language we speak, e.g., more digits for Chinese speakers because the words for the digits are shorter. We aren't memorizing the digits themselves, but the mental sound of reciting them. If you've got some two-digit numbers to memorize, it's better to use Cantonese than French.
 
Doctor Q said:
It describes some fascinating experimental results. For example, we (at least in the western world) apparently compare numbers by thinking of them in a line from left to right, with larger numbers to the right of smaller numbers. So we think of 4 as being to the right of 3.
Click to expand...

Nothing surprising about that. It's how almost all numbers are taught in the western education system. Children are taught the number line, with minus infinity off to the left, and +infinity off to the right. This then forms the basis for teaching graphs and charts, via co-ordinates and x and y. Later in college, the number line is extended with +i upwards and -i downwards.

The number of digits we can easily memorize depends on the language we speak, e.g., more digits for Chinese speakers because the words for the digits are shorter. We aren't memorizing the digits themselves, but the mental sound of reciting them. If you've got some two-digit numbers to memorize, it's better to use Cantonese than French.
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Interesting. And how does that apply to sign language users who conceive of numbers as handshapes rather than words? I wonder if users of sign languages or dialects with more complex handshapes for certain numbers find it harder to remember sequences of numbers?
 
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