Lol, my philosophy is exactly yours. Allow simplification as necessary, because to do otherwise is a pointless uphill battle. Only use as much accuracy as you really need.
In this case, it doesn’t matter if pi is 3 or 5 or 30. It’s just for teaching purposes. You would need critical thinking to determine how much simplification you can do, which is much better taught by simplifying things differently as you need, rather than just keeping pi as 3 and saying that works everywhere.
I get it now. I was taking exception to your characterization of 3 and 5 being equally inaccurate in the sense of how close they are to the actual true value, which, of course, can never be known, except in every more accurate approximations.
In that case, I guess we still have a difference of opinion. I think that using approximations that are closer to their true value are more useful in teaching, despite (and maybe because of) the greater difficulty. If the student is not yet ready for that level of difficulty, then perhaps a different problem should be presented.
To that end, I actually think that there are several things to teach. That PI is not 3 or 3.14 or any other decimal expansion. That 3 is close enough for most casual encounters outside school. That 3.14 is close enough for most engineering work. That 3.1416 is close enough for most scientific work. That 15 decimal places is close enough for rocket scientists. That 37 decimal places are enough to calculate the circumference of the universe to within the diameter of a hydrogen atom. (https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ is my reference for the last two items. The others are just wild-ass guesses.)
What I mean is, if you’re using 3, you’re approximating, heavily. If you do anything critical using that value, it’s as bad as using 5 really, imo. Is it really the case that 3 can be used casually? Like in what, workmanship, crafting or something else?
Personally, I would say that pi should be presented as 3.14 and calculators should be used, there’s no reason to fear less than elegant numbers xD. And no, that’s not close enough for most engineering work, as an engineer we don’t usually approximate that much despite the memes, since you have to reduce the margin of error as much as practical. You generally don’t even approximate, just leave it as pi the symbol for the most part since in the end you won’t calculate it manually. The errors stack up the more you use the value. Eg, multiply an inaccurate value of pi by pi and the error you get is exponential.
That aside, I think 5 is more elegant than 3 so if youre approximating to avoid the cumbersome numbers why not go for elegance instead of accuracy? xD
When I’m figuring the buoyancy of a 20 litre pail or, alternatively, how much it’ll weigh when filled with sand, 3 is easier to work in my head for off-the-cuff estimates so I know about how many pails I need.
That said, I do typically use the π button on my calculator when it comes time to actually execute on the project. :)
Lol, my philosophy is exactly yours. Allow simplification as necessary, because to do otherwise is a pointless uphill battle. Only use as much accuracy as you really need.
In this case, it doesn’t matter if pi is 3 or 5 or 30. It’s just for teaching purposes. You would need critical thinking to determine how much simplification you can do, which is much better taught by simplifying things differently as you need, rather than just keeping pi as 3 and saying that works everywhere.
I get it now. I was taking exception to your characterization of 3 and 5 being equally inaccurate in the sense of how close they are to the actual true value, which, of course, can never be known, except in every more accurate approximations.
In that case, I guess we still have a difference of opinion. I think that using approximations that are closer to their true value are more useful in teaching, despite (and maybe because of) the greater difficulty. If the student is not yet ready for that level of difficulty, then perhaps a different problem should be presented.
To that end, I actually think that there are several things to teach. That PI is not 3 or 3.14 or any other decimal expansion. That 3 is close enough for most casual encounters outside school. That 3.14 is close enough for most engineering work. That 3.1416 is close enough for most scientific work. That 15 decimal places is close enough for rocket scientists. That 37 decimal places are enough to calculate the circumference of the universe to within the diameter of a hydrogen atom. (https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ is my reference for the last two items. The others are just wild-ass guesses.)
What I mean is, if you’re using 3, you’re approximating, heavily. If you do anything critical using that value, it’s as bad as using 5 really, imo. Is it really the case that 3 can be used casually? Like in what, workmanship, crafting or something else?
Personally, I would say that pi should be presented as 3.14 and calculators should be used, there’s no reason to fear less than elegant numbers xD. And no, that’s not close enough for most engineering work, as an engineer we don’t usually approximate that much despite the memes, since you have to reduce the margin of error as much as practical. You generally don’t even approximate, just leave it as pi the symbol for the most part since in the end you won’t calculate it manually. The errors stack up the more you use the value. Eg, multiply an inaccurate value of pi by pi and the error you get is exponential.
That aside, I think 5 is more elegant than 3 so if youre approximating to avoid the cumbersome numbers why not go for elegance instead of accuracy? xD
When I’m figuring the buoyancy of a 20 litre pail or, alternatively, how much it’ll weigh when filled with sand, 3 is easier to work in my head for off-the-cuff estimates so I know about how many pails I need.
That said, I do typically use the π button on my calculator when it comes time to actually execute on the project. :)
That’s interesting. I imagine if you do that long enough you’ll just approximate with pi intact as well intuitively.
Anyways nice conversation.
Yes! Thank you. It’s always nice to see other perspectives.