Insights Frequently Made Errors in Equation Handling - Comments

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Orodruin

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Regarding 5 and 6, dimensions and units, early physics education may have to take a lot of blame, at least from what I remember from my high-school textbooks. Several problems were formulated using text such as "Let the wheel mass be ##m## kg and its radius ##R## m" (but in Swedish), in this case making ##m## and ##R## dimensionless quantities. Later, students have problems with dimensional analysis as well as units ... No wonder.
 

haruspex

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Regarding 5 and 6, dimensions and units, early physics education may have to take a lot of blame, at least from what I remember from my high-school textbooks. Several problems were formulated using text such as "Let the wheel mass be ##m## kg and its radius ##R## m" (but in Swedish), in this case making ##m## and ##R## dimensionless quantities. Later, students have problems with dimensional analysis as well as units ... No wonder.
Yes, I had the same thought. I'll add a comment on that, thanks.
 
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Simple but very useful tips for avoiding silly errors. Nice one!
 

Vanadium 50

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I give partial credit only until the numbers go in, and encourage my colleagues to do the same. If someone writes a+sqrt(b/2) when they mean a+sqrt(b)/2, there is a good chance I can figure this out. If they write 9.54317265 when they mean 4.5112121, I'll never figure it out.

And also be annoyed by the likely extraneous precision.
 

Philip Wood

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"One does sometimes come across questions like “A block has mass m kg…”. In this case, the variable m is a dimensionless number. This isn’t actually wrong, but it is poor style and not to be emulated."

I entirely agree. One problem we're up against, in the UK at least, is that in mathematics students are taught to represent quantities as pure numbers. So "A block has mass m kg" is just the sort of wording they meet in mathematics classes, textbooks and exams. Students have to use one convention in Maths and another in Physics; I'm surprised they cope as well as they do.

I'm so keen on the convention of representing a physical quantity as the product of a number and a unit, that I urge students to put in the number with its unit as soon as they replace an algebraic symbol by a given value, just as haruspex recommends in 6 (above)…

"Conversely, when working numerically, units should, in principle, always be included:
Average speed = 120 km / 3 h = 40 km/h"

In more complicated cases, simplifying the units to get a unit for the final answer is an excellent exercise in itself, as well as often providing useful check on preceding algebra.
 

rude man

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Regarding 5 and 6, dimensions and units, early physics education may have to take a lot of blame, at least from what I remember from my high-school textbooks. Several problems were formulated using text such as "Let the wheel mass be ##m## kg and its radius ##R## m" (but in Swedish), in this case making ##m## and ##R## dimensionless quantities. Later, students have problems with dimensional analysis as well as units ... No wonder.
If m and R are left as m and R rather than giving them numbers, I don't see how that makes m and R dimensionless. Did you mean m adn R to be numbers?
 

rude man

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This post is of great importance to anyone not already cognizant of its importance.
Dimensions get even more important when uncommon functions are concerned.
For example: in Laplace transforms we correctly have δ(t) ↔ 1, but then if we're dealing in voltages we say a 1 volt-sec. delta function = 1δ(t). But this is wrong. A 1 V-sec. input voltage should be written as kδ(t), k = 1 V-sec.
Similar observation applies in mechanical dynamics, etc. of course.
 

rude man

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I give partial credit only until the numbers go in, and encourage my colleagues to do the same. If someone writes a+sqrt(b/2) when they mean a+sqrt(b)/2, there is a good chance I can figure this out. If they write 9.54317265 when they mean 4.5112121, I'll never figure it out.

And also be annoyed by the likely extraneous precision.
Yes! That's why I disregard homework questions like "where is my mistake?" unless numbers are avoided 'till the end.
 

Orodruin

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If m and R are left as m and R rather than giving them numbers, I don't see how that makes m and R dimensionless. Did you mean m adn R to be numbers?
Since they explicitly say ##m## kg, ##m## has to be a dimensionless. If ##m## has dimension mass, e.g., ##m = 1## kg, then the mass would have dimension mass squared, which is nonsense as replacing ##m## in the statement with 1 kg would imply "has the mass 1 kg kg".
 

vela

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One pet peeve of mine related to #4 is when students use two variables, like m and M, to represent the same quantity.
 

Vanadium 50

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I have a textbook that uses h for two different things. h is a quantum number (h, j, and k) and also Plack's constant.
 
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Great post, haruspex! A large number of homework problems could be solved without help if everyone would follow those points.

I have a textbook that uses h for two different things. h is a quantum number (h, j, and k) and also Plack's constant.
Writing letters in a different font totally makes them different.
 

Orodruin

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Writing letters in a different font totally makes them different.
This is so true! Obviously ##\mathcal L## is different from ##L##. One is a Lagrangian and one is a fixed length.
 
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Great post, haruspex! A large number of homework problems could be solved without help if everyone would follow those points.


Writing letters in a different font totally makes them different.
Completely true, but problematic in the digital world where rendering of text might mangle the formatting (such as losing the distinction between D and Δ.) Obviously, science and math written traditions originate in the eras of handwriting. Rendering in the digital age is a problem we created, but have not yet adequately solved.
 

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