DanRollins

asked on

I'm not innumerate, but when the numbers get really large, my eyes glaze over.

Can someone use their common sense and a good calulator's scientifc notation ability to calculate this for me?

If you had a circle the size of the Known Universe -- say 15,000,000 Light Years

in diameter -- you could calculate its circumference by using pi. How accurate

would you need the value of pi to be to get a correct circumference, within say

an inch or two? How about to within a nanometer? How about to within a

Compton radius (a distance comparable to the size of an electron).

Coming at it another way:

If you have a value of pi calculated to 1 million digits, and there was a typo at

digit number 900,000 so that it was off by 1 in that position, how much

would that affect your calculation of the circumferance of astronomical-scale

objects, such as the Visible Universe?

This is just to satisfy my curiousity. I've stated a sort of interesting proposition here:

https://www.experts-exchange.com/questions/22414656/How-could-God-prove-that-He-is-God.html#18627694

And I'm basing my supposition only on gut instinct and a vague memory of something I read *long ago* -- that you need surprizingly few digits of pi to get an accurate measure of circles the size of, say the orbit of Neptune (bonus if you can verify that with some data :-)

-- Dan

Can someone use their common sense and a good calulator's scientifc notation ability to calculate this for me?

If you had a circle the size of the Known Universe -- say 15,000,000 Light Years

in diameter -- you could calculate its circumference by using pi. How accurate

would you need the value of pi to be to get a correct circumference, within say

an inch or two? How about to within a nanometer? How about to within a

Compton radius (a distance comparable to the size of an electron).

Coming at it another way:

If you have a value of pi calculated to 1 million digits, and there was a typo at

digit number 900,000 so that it was off by 1 in that position, how much

would that affect your calculation of the circumferance of astronomical-scale

objects, such as the Visible Universe?

This is just to satisfy my curiousity. I've stated a sort of interesting proposition here:

https://www.experts-exchange.com/questions/22414656/How-could-God-prove-that-He-is-God.html#18627694

And I'm basing my supposition only on gut instinct and a vague memory of something I read *long ago* -- that you need surprizingly few digits of pi to get an accurate measure of circles the size of, say the orbit of Neptune (bonus if you can verify that with some data :-)

-- Dan

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and about 38 digits for a compton radius

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the difference between d-glitches answer and mine is the difference between 15 light years and 15 million light years

ASKER

I'd love to get confirmation of that (so Experts feel free to chime in)... but that's about how I remember it.

Amazing! ... So calculating pi beyond even the 40th digit has no meaning even when comparing the largest thing imaginable to (basically) the smallest particle known.

Do you have a back-of-a-napkin calculation for the same thing based on a cirrcle the size of the orbit of Neptune (roughly 4,500,000,000 km from the sun). I didn't really trust my memory, but perheps its correct: I'm thinking that pi had to be only nine digits or so to be within a few inches of exact.

Amazing! ... So calculating pi beyond even the 40th digit has no meaning even when comparing the largest thing imaginable to (basically) the smallest particle known.

Do you have a back-of-a-napkin calculation for the same thing based on a cirrcle the size of the orbit of Neptune (roughly 4,500,000,000 km from the sun). I didn't really trust my memory, but perheps its correct: I'm thinking that pi had to be only nine digits or so to be within a few inches of exact.

ASKER

Dang.. the Universe is said to be about 15 Billion (not million) Light Years in diameter. How much does that change things?

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calculate everything yourself

If C = circumference and d = diameter

C = d . pi

dC = d . dpi

dC/C = (d . dpi)/(d.pi)

dC/C = dpi/pi

that is, the percentage uncertainty i n C is the same as the percentage uncertainty in pi. This holds true for any C.

If C = circumference and d = diameter

C = d . pi

dC = d . dpi

dC/C = (d . dpi)/(d.pi)

dC/C = dpi/pi

that is, the percentage uncertainty i n C is the same as the percentage uncertainty in pi. This holds true for any C.

or dC/d = dpi

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>>>Pi is used for many high-precision and iterative calculations that require accuracy in excess of that needed to make real-world measurements.

what are they as a matter of interest?

I've heard of pi being used for random numbers, since no one could find any intrinsic 'non-randomness' in the sequence.

what are they as a matter of interest?

I've heard of pi being used for random numbers, since no one could find any intrinsic 'non-randomness' in the sequence.

ASKER

I'd also be interested in that.

Of course "real world measurments" might just include needing a precise enough value of pi to calculate the curvature of the earth for plotting a missle trajectory -- surely no more than 5 digits would be accurate enough to hit a building.

It's hard to imagine an application that could benefit from knowing pi to, say, 100 decimal places.

Of course "real world measurments" might just include needing a precise enough value of pi to calculate the curvature of the earth for plotting a missle trajectory -- surely no more than 5 digits would be accurate enough to hit a building.

It's hard to imagine an application that could benefit from knowing pi to, say, 100 decimal places.

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ASKER

I had not heard about the google calculator/converter. Thanks NovaDenizen!

As with many otherwise intelligent people, my eyes glaze over when thinking about really large numbers. In this case, it's (now) obvious to me that the value of pi can be treated as 1 for the purposes of this question and the end result will be within an order-of-magnitude of being correct.

So the google search string of:

15,000,000,000 light years in nanometers

yields

1.41907926 × 10^35 nanometers

And the answer to my question is "on the order of 35 decimal places" for accuracy in nanometers. An electron is about 10^-15 m (a femtometre) in width (disregarding quantum-mechanical tomfoolery [thanks NovaDenizen] :-) that's 6 additional orders of magnitude (15-9) and so....

41 or 42 decimal places of pi

would surely provide enough accuracy to determine the circumferance

of the Visible Universe in units the width of an electron

My original "hunch" was correct... that decimal places after the 900,000-th position are completely "lost in the noise." In fact, one could probably say that about the 100-th or even the 50-th decimal place (at least for purposes of illustration, at the level of my "need to know").

Also, the bonus question: Decimal places needed to calculate the circumference of the orbit on Neptune, with accuracy of a few inches is

14

(as provided by ozo)

I'm going to split the points to most participants but with the majority going to the answers I personally found most enlightening.

=-=-=-=-=-=-=-=-=-=-=

For continued discussion, it might be interesting to consider a variation on the original thought that got me here:

Although altering some arbitrary digit of pi makes no sense, think of other measured forces, such as, say Gravity or the electro-weak force...

Assume that you know the value to of say, G, to 1 million decimal places. If the least significant digit of that value were to magically change (that is, an alteration by a factor of 10^-1,000,000) would such a change possibly have a macroscopic effect on the universe, changing Life As We Know It in any way?

As one additional sidebar:

I searched the first 1,000,000 digits of pi and found my own Social Security Number and Date of Birth. Does this mean that my identity is woven into the VERY FABRIC OF THE UNIVERSE???

:-)

Thanks for participating!

As with many otherwise intelligent people, my eyes glaze over when thinking about really large numbers. In this case, it's (now) obvious to me that the value of pi can be treated as 1 for the purposes of this question and the end result will be within an order-of-magnitude of being correct.

So the google search string of:

15,000,000,000 light years in nanometers

yields

1.41907926 × 10^35 nanometers

And the answer to my question is "on the order of 35 decimal places" for accuracy in nanometers. An electron is about 10^-15 m (a femtometre) in width (disregarding quantum-mechanical tomfoolery [thanks NovaDenizen] :-) that's 6 additional orders of magnitude (15-9) and so....

41 or 42 decimal places of pi

would surely provide enough accuracy to determine the circumferance

of the Visible Universe in units the width of an electron

My original "hunch" was correct... that decimal places after the 900,000-th position are completely "lost in the noise." In fact, one could probably say that about the 100-th or even the 50-th decimal place (at least for purposes of illustration, at the level of my "need to know").

Also, the bonus question: Decimal places needed to calculate the circumference of the orbit on Neptune, with accuracy of a few inches is

14

(as provided by ozo)

I'm going to split the points to most participants but with the majority going to the answers I personally found most enlightening.

=-=-=-=-=-=-=-=-=-=-=

For continued discussion, it might be interesting to consider a variation on the original thought that got me here:

Although altering some arbitrary digit of pi makes no sense, think of other measured forces, such as, say Gravity or the electro-weak force...

Assume that you know the value to of say, G, to 1 million decimal places. If the least significant digit of that value were to magically change (that is, an alteration by a factor of 10^-1,000,000) would such a change possibly have a macroscopic effect on the universe, changing Life As We Know It in any way?

As one additional sidebar:

I searched the first 1,000,000 digits of pi and found my own Social Security Number and Date of Birth. Does this mean that my identity is woven into the VERY FABRIC OF THE UNIVERSE???

:-)

Thanks for participating!

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