<p>1 km/s is roughly 1 pc per million years. This is roughly a consequence of there being <span class="math-container">$\pi \times 10^7$</span> seconds in a year and roughly <span class="math-container">$\pi$</span> light years in a parsec. Useful.</p>

CC BY-SA 4.0
2024-02-26T12:56:39.177Z
2024-02-26T12:56:39.177Z
ProfRob
<p>Since the Sun is 8 light minutes away, and the speed of light is a foot per nanosecond, the <em>diameter</em> of the Earth’s orbit is a trillion feet.</p>

CC BY-SA 4.0
2024-02-27T11:29:14.653Z
This is only useful to Americans though…
pela
CC BY-SA 4.0
2024-02-27T13:03:25.210Z
Diameter of earth's orbit is ~1000 light seconds.
David
CC BY-SA 4.0
2024-02-27T14:34:57.997Z
Similarly, the diameter of the earth's orbit is $3 \times 10^{11}$ meters if you work in metric - the diameter of the orbit is $10^3$ seconds and the speed of light is $3 \times 10^8$ m/s.
Michael Lugo
CC BY-SA 4.0
2024-02-27T16:30:18.443Z
@pela Geocentric => Heliocentric => Americocentric
Barmar
CC BY-SA 4.0
2024-02-28T01:18:13.880Z
The americans should just change their basic length unit to nanolightsecond. This would please the science nerds, and wouldn't change much for people accustomed to feet.
Paŭlo Ebermann
CC BY-SA 4.0
2024-02-28T04:10:33.800Z
The rest of us use "the Canadian system".
Peter Mortensen
CC BY-SA 4.0
2024-02-27T06:22:08.747Z
2024-02-27T06:22:08.747Z
Martin Kochanski
<p>In mathematics there are approximations to pi such as:</p>
<p><span class="math-container">$\pi\approx\dfrac{355}{113}$</span></p>
<p>that have extraordinary accuracy and are useful ways of remembering or being able to calculate a value close to pi that may be used in calculations.</p>
<p>I noticed the the ratio of the diameter of the moon to the earth seems unreasonably close to <span class="math-container">$\frac{3}{11}$</span>.</p>
<p>Is my observation correct?</p>
<p>And are there other useful ways of remembering approximate metrics of various heavenly bodies with surprising economy?</p>

CC BY-SA 4.0
2024-02-26T12:51:32.170Z
1 year is $\simeq \pi \times 10^7$ seconds and see https://xkcd.com/1047/ for other examples of numerology.
ProfRob
CC BY-SA 4.0
2024-02-26T14:18:37.003Z
Fun fact: No rational approximation to π is really useful, since the number of digits to be remembered will never be smaller than the number of decimals in π to which it is accurate. For instance, 355/113 involves six digits, but is only correct to 5th decimal in π.
pela
CC BY-SA 4.0
2024-02-26T18:30:32.400Z
@ScienceSnake I can't give you the proof, unfortunately, but I learned it as an undergrad. I think it's true not only for π, but for all irrational numbers, and I seem to recall the proof involving [continued fractions](https://en.wikipedia.org/wiki/Continued_fraction) and [the Diophantine approximation](https://en.wikipedia.org/wiki/Diophantine_approximation).
pela
CC BY-SA 4.0
2024-02-26T18:33:55.323Z
can't be true for *all* irrational numbers  eg consider 1 + pi/10^9, the rational number 1/1 involves two digits, but is accurate to the 8th decimal place.  @pela
James K
CC BY-SA 4.0
2024-02-26T19:14:57.703Z
The error in a good rational approximation (eg from the continued fraction) $p/q$ is $<1/q^2$. See https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem 355/113 is ~3.14159292, so it gives you pi (~3.14159265) to 7 digits.
PM 2Ring
CC BY-SA 4.0
2024-02-26T19:47:56.770Z
500 light-seconds is about 1/500th longer than one astronomical unit.
notovny
CC BY-SA 4.0
2024-02-26T20:03:06.427Z
You can "upgrade" pi calculations that use 22/7 by subtracting 0.04%, since (22/7)×(1-0.0004) = 3.1416 exactly.
PM 2Ring
CC BY-SA 4.0
2024-02-26T23:30:49.100Z
@PM2Ring that reminds me of the maths class back at uni, when things like finite Taylor series had a remainder term at the end. Applying that concept here: 
"1.234 + x" is an accurate representation of pi up to the nth digit for a sufficient "x"
Apfelsaft
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2024-02-27T11:21:08.487Z
@JamesK Yes, you're right. Maybe "all irrational numbers that are not too irrational"? Well, good thing I'm not a mathematician :)
pela
CC BY-SA 4.0
2024-02-27T11:24:18.957Z
@PM2Ring Did both you and I count wrong? I said "5", you said "7", but only the _6_ digits "141592" are correct, right?
pela
CC BY-SA 4.0
2024-02-27T11:34:05.953Z
@Pela I'm also counting the leading "3". To 7 digits (i.e., 6 decimal places), we have "3.141593"
PM 2Ring
CC BY-SA 4.0
2024-02-27T15:23:01.767Z
@PM2Ring Ah okay. I was thinking "number of decimals". Anyway, in astronomy π = 3. Everything after that is just pernicketiness.
pela
CC BY-SA 4.0
2024-02-28T06:56:15.507Z
@pela But 355/113 is unsually easy to remember due to the repeating digits.
Russell Borogove
CC BY-SA 4.0
2024-02-28T13:41:01.000Z
@RussellBorogove I mix them up all the time, until I start thinking, but when I start thinking, I realize that "π = 3" is just fine…
pela
CC BY-SA 4.0
2024-02-29T09:55:39.523Z
@pela: As an approximation to pi, 355/113 is in error by less than one part in 12 million. Equivalently using seven characters (or six digits), 3.14159 is in error by more than one part in 1.2 million: an error ten times the size.
John Bentin
CC BY-SA 4.0
2024-02-29T13:17:34.430Z
@JohnBentin Right, but I wasn't counting the "3.", only decimals. Including one more decimal makes the difference of the same order. I guess my point was that there really isn't any reason to learn rational approximations by heart. If you'd ever want to learn a better approximation, you'd need a whole new fraction, whereas as learning decimals, you'd just need one more.
pela
CC BY-SA 4.0
2024-03-01T05:09:32.033Z
@pela How accurately can you tell me the size of the moon without having to look it up?
David
CC BY-SA 4.0
2024-03-01T10:01:11.667Z
@David To within a few tens of thousands of kilometer?
pela
CC BY-SA 4.0
2024-03-01T10:02:34.470Z
@David Ah sorry, you said _size_, not distance  Probably a bit more accurately then.
pela
<solar-system>
Unreasonable ratios in astronomy as ways of remembering
David
# Unreasonable ratios in astronomy as ways of remembering

(Asked by: username_0 on 2024-02-26)

In mathematics there are approximations to pi such as:

$\pi\approx\dfrac{355}{113}$

that have extraordinary accuracy and are useful ways of remembering or being able to calculate a value close to pi that may be used in calculations.

I noticed the the ratio of the diameter of the moon to the earth seems unreasonably close to $\frac{3}{11}$.

Is my observation correct?

And are there other useful ways of remembering approximate metrics of various heavenly bodies with surprising economy?

**Tags:** solar-system

> **Comments:**
> **username_1**: 1 year is $\simeq \pi \times 10^7$ seconds and see https://xkcd.com/1047/ for other examples of numerology.  
> **username_2**: Fun fact: No rational approximation to π is really useful, since the number of digits to be remembered will never be smaller than the number of decimals in π to which it is accurate. For instance, 355/113 involves six digits, but is only correct to 5th decimal in π.  
> **username_2**: @ScienceSnake I can't give you the proof, unfortunately, but I learned it as an undergrad. I think it's true not only for π, but for all irrational numbers, and I seem to recall the proof involving [continued fractions](https://en.wikipedia.org/wiki/Continued_fraction) and [the Diophantine approximation](https://en.wikipedia.org/wiki/Diophantine_approximation).  
> **username_3**: can't be true for *all* irrational numbers  eg consider 1 + pi/10^9, the rational number 1/1 involves two digits, but is accurate to the 8th decimal place.  @username_2  
> **username_4**: The error in a good rational approximation (eg from the continued fraction) $p/q$ is $<1/q^2$. See https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem 355/113 is ~3.14159292, so it gives you pi (~3.14159265) to 7 digits.  
> **username_5**: 500 light-seconds is about 1/500th longer than one astronomical unit.  
> **username_4**: You can "upgrade" pi calculations that use 22/7 by subtracting 0.04%, since (22/7)×(1-0.0004) = 3.1416 exactly.  
> **username_6**: @username_4 that reminds me of the maths class back at uni, when things like finite Taylor series had a remainder term at the end. Applying that concept here:  "1.234 + x" is an accurate representation of pi up to the nth digit for a sufficient "x"  
> **username_2**: @username_3 Yes, you're right. Maybe "all irrational numbers that are not too irrational"? Well, good thing I'm not a mathematician :)  
> **username_2**: @username_4 Did both you and I count wrong? I said "5", you said "7", but only the _6_ digits "141592" are correct, right?  
> **username_4**: @username_2 I'm also counting the leading "3". To 7 digits (i.e., 6 decimal places), we have "3.141593"  
> **username_2**: @username_4 Ah okay. I was thinking "number of decimals". Anyway, in astronomy π = 3. Everything after that is just pernicketiness.  
> **username_7**: @username_2 But 355/113 is unsually easy to remember due to the repeating digits.  
> **username_2**: @username_7 I mix them up all the time, until I start thinking, but when I start thinking, I realize that "π = 3" is just fine…  
> **username_8**: @username_2: As an approximation to pi, 355/113 is in error by less than one part in 12 million. Equivalently using seven characters (or six digits), 3.14159 is in error by more than one part in 1.2 million: an error ten times the size.  
> **username_2**: @username_8 Right, but I wasn't counting the "3.", only decimals. Including one more decimal makes the difference of the same order. I guess my point was that there really isn't any reason to learn rational approximations by heart. If you'd ever want to learn a better approximation, you'd need a whole new fraction, whereas as learning decimals, you'd just need one more.  
> **username_0**: @username_2 How accurately can you tell me the size of the moon without having to look it up?  
> **username_2**: @username_0 To within a few tens of thousands of kilometer?  
> **username_2**: @username_0 Ah sorry, you said _size_, not distance  Probably a bit more accurately then.  

---

## 2 answers

### Answer by username_1 on 2024-02-26. Score: 9

1 km/s is roughly 1 pc per million years. This is roughly a consequence of there being $\pi \times 10^7$ seconds in a year and roughly $\pi$ light years in a parsec. Useful.


---

### Answer by username_9 on 2024-02-27. Score: 2

Since the Sun is 8 light minutes away, and the speed of light is a foot per nanosecond, the *diameter* of the Earth’s orbit is a trillion feet.

> **Comments:**
> **username_2**: This is only useful to Americans though…  
> **username_0**: Diameter of earth's orbit is ~1000 light seconds.  
> **username_10**: Similarly, the diameter of the earth's orbit is $3 \times 10^{11}$ meters if you work in metric - the diameter of the orbit is $10^3$ seconds and the speed of light is $3 \times 10^8$ m/s.  
> **username_11**: @username_2 Geocentric => Heliocentric => Americocentric  
> **username_12**: The americans should just change their basic length unit to nanolightsecond. This would please the science nerds, and wouldn't change much for people accustomed to feet.  
> **username_13**: The rest of us use "the Canadian system".  

---

