Frequently Made Errors in Equation Handling - Comments

[haruspex]

haruspex submitted a new PF Insights post

Frequently Made Errors in Equation Handling

Continue reading the Original PF Insights Post.

 

[Orodruin]

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]

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.

 

[Greg Bernhardt]

Well done haruspex!

 

[PWiz]

Simple but very useful tips for avoiding silly errors. Nice one!

 

[Vanadium 50]

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]

"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]

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]

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]

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]

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]

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]

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.

 

[mfb]

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]

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.

 

[anorlunda]

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.