# Frequently Made Errors in Equation Handling - Comments

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• haruspex
In summary: 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.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.
haruspex
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Frequently Made Errors in Equation Handling

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mfb
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.

Orodruin said:
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.

Well done haruspex!

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

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.

rude man
"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.

Orodruin said:
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?

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.

Vanadium 50 said:
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.

rude man said:
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".

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

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.

Great post, haruspex! A large number of homework problems could be solved without help if everyone would follow those points.

Vanadium 50 said:
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.

mfb said:
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.

mfb said:
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.

## 1. What are some common errors in handling equations in scientific research?

Some common errors in handling equations in scientific research include incorrect use of symbols, missing units, and incorrect mathematical operations. It is important to carefully review and double-check all equations to ensure accuracy.

## 2. How can I avoid making errors in handling equations?

To avoid making errors in handling equations, it is important to have a thorough understanding of the mathematical concepts and principles involved. Also, be sure to carefully check all units, symbols, and calculations before using an equation in your research.

## 3. Are there any specific tips for handling complex equations?

Yes, when dealing with complex equations, it can be helpful to break them down into smaller, more manageable parts. Clearly label each step and double-check all calculations to ensure accuracy. It can also be helpful to seek feedback from colleagues or consult with a mathematical expert.

## 4. What should I do if I discover an error in an equation I have used in my research?

If you discover an error in an equation you have used in your research, it is important to address it as soon as possible. First, identify the error and determine the correct equation. Then, update your research to reflect the correct equation and make any necessary adjustments to your findings or conclusions.

## 5. Is it acceptable to use equations from other sources in my research?

Yes, it is acceptable to use equations from other sources in your research, as long as you properly cite the source and ensure the accuracy of the equation. It is also important to use equations from reputable sources and to carefully check for any errors or mistakes before incorporating them into your research.

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