If Pi is such an important number...

所有讨论 > Steam 论坛 > Off Topic > 主题详情

Lobster Moth 10 月 14 日上午 11:42

Why is it only ever calculated to 2 or 3 decimal places? Surely all the answers will be wrong.

引用自 Lobster Moth：
引用自 fractalgem：
...are you trying to inject math trolling flavored humor into the forums or are you legitimately confused by the subject?

My answer will change depending on which it is. I'm reasonably sure it's the former...but man, sarcasm does NOT carry well through text.

I'm not confused by it. I'm just wondering how mathematicians justify imprecise answers.when maths is about precision. If you said 2+2=6, you would be wrong. So why is it not wrong to use 3.14x2 instead of 3.14e70000Xx2 when the difference in the answer is 70,000+ digits imprecise? Which might be fine if you're trying to measure the circumference of a coffee cup and you can afford to be 70,000 nanometers imprecise, but not if you are trying to measure the circumference of the universe.

Also. I'm bored.

When mathemeticians want to give out Pi related numbers as an exact value, they will leave it in as a constant. E.G. writing π r^2 rather than 3.14 r^2. The sum of the recipricol squares can either be given as a numeric approximation or written as (1+1/4+1/9+1/16+1/25+...)=(π^2)/6.

Some such infinite series don't have a known way to relate their exact value to π (or one of its close cousins, like e). The only exact value for these is to leave the series in its original form. You better hope you don't need to multiply these series together, or that if you do, that they'll be one of the rare cases where terms quickly cancel and the calculation becomes weirdly tractable, letting you actually compactify the multiplication rather than just leaving it as "series1xseries2".

If you are transforming the exact expression into a practical application, then it's on whoever is doing the practical application to figure out how many digits or terms they should even bother with.

Finally, the importance of each digit decreases the further past the decimal you go. Each digit is only worth 1/10th the proceeding digit. as such, 100 digits of PI isn't well described as being "innacurate by 69900 digits. Rather, it's best described as "off by a factor of 1 in 10^100".
https://simple.wikipedia.org/wiki/Planck_length

The plank length is about 1.616255×10^−35 m. I will approximate as 1x10^-35 for my sanity

there are about 9.5x10^15 meters in a light year. I will approximate as 10^16 for my sanity.
The visibile universe is about 93 billion light years wide. I will approximate as 10^11 for my sanity.

So basically, the universe is about 10^62 plank lenghts wide.

If you have 100 digits of pi, you're looking at a precision levevel of 1 in 10^100.

Errors in measuring the universe stemming from using an approximation of PI will be thus on the order of about 10^-38th...of a plank length

The universe itself will go "NO U" if you try to actually measure stuff smaller than a plank length. The universe-enforced uncertainty dwarfs the error stemming from using a PI approxim atin by a factor of 10^38.

Hope that helps.

< >

正在显示第 1 - 13 条，共 13 条留言

Tonepoet 10 月 14 日上午 11:46

It's an irrational number, so you can't calculate it perfectly, at least not in the decimal system, and 3.14 is close enough for most practical purposes, although if you calculate it farther out, you can increase the accuracy. A big part of the question is how far out do you need to calculate it for it to be good enough for all practical applications, and that turns out being the 37th decimal place[www.jpl.nasa.gov].

最后由 Tonepoet 编辑于; 10 月 14 日上午 11:48

Lobster Moth 10 月 14 日上午 11:47

引用自 Tonepoet：
It's an irrational number, so you can't calculate it perfectly, at least not in the decimal system, and 3.14 is close enough for most practical purposes, although if you calculate it farther out, you can increase the accuracy.

What about the impractical purposes?

Tonepoet 10 月 14 日上午 11:50

引用自 Lobster Moth：
引用自 Tonepoet：
It's an irrational number, so you can't calculate it perfectly, at least not in the decimal system, and 3.14 is close enough for most practical purposes, although if you calculate it farther out, you can increase the accuracy.

What about the impractical purposes?

Then you can calculate it further out if you want. Some people have memorized 70,000 digits of it.

Lobster Moth 10 月 14 日上午 11:52

引用自 Tonepoet：
引用自 Lobster Moth：

What about the impractical purposes?

Then you can calculate it further out if you want. Some people have memorized 70,000 digits of it.

But as there are 70,000+ decimal places, the answers are still incorrect?

fractalgem 10 月 14 日上午 11:55

...are you trying to inject math trolling flavored humor into the forums or are you legitimately confused by the subject?

My answer will change depending on which it is. I'm reasonably sure it's the former...but man, sarcasm does NOT carry well through text.

bAd a!m 10 月 14 日上午 11:56

引用自 Lobster Moth：
Why is it only ever calculated to 2 or 3 decimal places? Surely all the answers will be wrong.

We have a more precise value for Pi than we have for our own body: your height, weight, etc are never that accurate.

Also, for most things in the REAL life at the REAL scale we need as humans, using 6 to 10 decimals is more than accurate enough. For you, there's no difference between 0.0000001 and 0 only at a very small factor that matters.

Jetstream Seum 10 月 14 日下午 12:00

As Tonepoet says, it's pointless to go beyond 3-5 digits for most purposes.

It is a very interesting number, though. So arbitrary, given its significance. If you were creating any kind of system, up to an including a universe, you would want your constants to be easily divisible, factorable and/or computable. It's obviously not the case in our reality.

Lobster Moth 10 月 14 日下午 12:01

引用自 fractalgem：
...are you trying to inject math trolling flavored humor into the forums or are you legitimately confused by the subject?

My answer will change depending on which it is. I'm reasonably sure it's the former...but man, sarcasm does NOT carry well through text.

I'm not confused by it. I'm just wondering how mathematicians justify imprecise answers.when maths is about precision. If you said 2+2=6, you would be wrong. So why is it not wrong to use 3.14x2 instead of 3.14e70000Xx2 when the difference in the answer is 70,000+ digits imprecise? Which might be fine if you're trying to measure the circumference of a coffee cup and you can afford to be 70,000 nanometers imprecise, but not if you are trying to measure the circumference of the universe.

Also. I'm bored.

Triple G 10 月 14 日下午 12:07

It´s only a crutch anyways, because a circle is no triangle - but apparently we´re too stupid to calculate circles. But two digits seem to be clever enough to call us intelligent.

Lobster Moth 10 月 14 日下午 12:07

引用自 bAd a!m：
引用自 Lobster Moth：
Why is it only ever calculated to 2 or 3 decimal places? Surely all the answers will be wrong.

We have a more precise value for Pi than we have for our own body: your height, weight, etc are never that accurate.

Also, for most things in the REAL life at the REAL scale we need as humans, using 6 to 10 decimals is more than accurate enough. For you, there's no difference between 0.0000001 and 0 only at a very small factor that matters.

Sure, but that's down to the inaccuracy of our instruments rather than the value being impossible to calculate.

For instance. I may be 95.987986412312878931cm in height and the measure can only show 95.9, but the height is still an exact number, we just don't bother calculating it. It is impossible to find the accuracy of Pi because it has infinite accuracy, or maybe infinite inaccuracy. I'm not entirely sure.

#10

Abaddon the Despoiler 10 月 14 日下午 12:07

I remember the first 11 decimals like a phone number, that's close enough.

Nothing is perfect, margin for error.

最后由 Abaddon the Despoiler 编辑于; 10 月 14 日下午 12:08

#11

该讨论串的作者已表示此帖子解答了原先的主题。

fractalgem 10 月 14 日下午 12:34

引用自 Lobster Moth：
引用自 fractalgem：
...are you trying to inject math trolling flavored humor into the forums or are you legitimately confused by the subject?

My answer will change depending on which it is. I'm reasonably sure it's the former...but man, sarcasm does NOT carry well through text.

I'm not confused by it. I'm just wondering how mathematicians justify imprecise answers.when maths is about precision. If you said 2+2=6, you would be wrong. So why is it not wrong to use 3.14x2 instead of 3.14e70000Xx2 when the difference in the answer is 70,000+ digits imprecise? Which might be fine if you're trying to measure the circumference of a coffee cup and you can afford to be 70,000 nanometers imprecise, but not if you are trying to measure the circumference of the universe.

Also. I'm bored.