Why are we allowed to cancel units?

[Mr Davis 97]

As a simple example, when we do chain-link conversion, we are allowed to cancel units in order to obtain the correct answer. However, units are not numbers, so why is this allowed?

 

[Dale]

How many meters per meter are there? What is the dimensionally of one meter per meter?

 

[Mr Davis 97]

How many meters per meter are there? What is the dimensionally of one meter per meter?

Okay, well that helps with cancelling units, but I have another question about units. What does it mean physically when we say something line Newton-meters (as an example)? Division of units is easy to see because it's one quantity per another quantity. But when we multiply two units, what is the significance of the outcome (N * m for example)?

 

[HomogenousCow]

Okay, well that helps with cancelling units, but I have another question about units. What does it mean physically when we say something line Newton-meters (as an example)? Division of units is easy to see because it's one quantity per another quantity. But when we multiply two units, what is the significance of the outcome (N * m for example)?

It doesn't mean anything "physically". At the base level, physics is just figuring out what mathematical model happens to reflect the behavior of the real world. Multiplication is something you do to the numbers which you've taken to abstractly represent some aspect of the physical world. It just so happens that the multiplication operation has some nice features which match the behavior of nature.

 

[Mr Davis 97]

It doesn't mean anything "physically". At the base level, physics is just figuring out what mathematical model happens to reflect the behavior of the real world. Multiplication is something you do to the numbers which you've taken to abstractly represent some aspect of the physical world. It just so happens that the multiplication operation has some nice features which match the behavior of nature.

So there is no inherent reason why division yields a comprehensible explanation (such as 2 meters PER second of travel) while multiplication does not?

 

[Nugatory]

But when we multiply two units, what is the significance of the outcome (N * m for example)?

It sort of depends on the units you're multiplying...

Two common examples:
- If I have a roll of cloth X meters wide, and I unroll Y meters from it... How much cloth do I have? How do I capture the fact that if the cloth is twice as wide, or I unroll twice as much, I get twice as much cloth; but if the roll is twice as wide and I unroll twice as much, I get four times as much? I have to multiply the width of the roll (X meters) by the amount unrolled (Y meters) to find that I have XY meters² (and of course meters² is just another way of saying "meters times meters").
- The battery that starts my car every morning is rated in units of Ampere-hours. That means (with many simplifying assumptions, because real-life lead-acid batteries behave in very complicated ways) that it it can deliver X amps for Y hours, or 2X amps for Y/2 hours, or X/2 amps for 2Y hours... You should see the analogy between amps and hours in these calculations, and the width in meters and the length in meters in the first example.

 

[RMalayappan]

So there is no inherent reason why division yields a comprehensible explanation (such as 2 meters PER second of travel) while multiplication does not?

Well the reason why 2 meters PER second works so well is that you are measuring the number of meters it takes you to go in a second and the way you do that mathematically is by dividing the number of meters by the number of seconds. And who says multiplication doesn't yield anything meaningful? We have obviously meaningful units that come from multiplication like the light-year and the square meter.

 

[Nugatory]

So there is no inherent reason why division yields a comprehensible explanation (such as 2 meters PER second of travel) while multiplication does not?

It does lend itself to a comprehensible explanation. You are right that division of units generally is interpreted as a PER relationship, and we see these all the time. Multiplication of units is a bit less common in daily life, but when you do see it, it can be interpreted as a FOR relationship: From my example above, one Ampere-hour is the amount of electricity moved by a one-amp current flowing FOR one hour.

 

[Dale]

What does it mean physically when we say something line Newton-meters (as an example)?

As Nugatory mentioned this would be one Newton of force applied for one meter of distance.

 

[Mr Davis 97]

As Nugatory mentioned this would be one Newton of force applied for one meter of distance.

How is saying "one Newton of force applied for one meter of distance" different than saying "one Newton of force applied per one meter of distance"?

 

[Mr Davis 97]

I think that it is clear that if two sides of an equation are equal, their dimensions must be equal. But when doing dimensional analysis, why are we allowed to cancel dimensions when there is a ratio containing two of the same dimensions (such as time in x = (1/2)at^2). I see how we can cancel units, since they are just specified amounts of a physical quantity, and thus follow the rules of arithmetic and algebra, but when we are doing dimensional analysis, I don't see how we can just cancel dimensions. For example, in the above equation, what if I have acceleration in meters per minute squared and t^2 in hours. The units are not the same so we cannot cancel. However, the dimensions are the same. But since the units are not the same, how are we able to cancel the dimensions? Do we assume that units are the same while doing dimensional analysis?

 

[RMalayappan]

I think "for" isn't the best word, something like "in" or "over" would be better. A Newton meter(joule) is equivalent to the change in energy that results from applying a force of one Newton over a distance of one meter. A Newton/meter would be applicable if you were pushing an object by adding force to it over a certain distance and the number of Newtons/meter would be the average rate of force increase. For example, in the case of the N/m, if you started off pushing an object with a force of 1 N and ended up pushing it with a force of 5 N after a distance of two meter, the average rate of force increase would be 2N/m, since you increased the force by an average of 2N for every meter you traveled. The first represents accumulation, while the second represents a rate of change. If you're familiar with calculus, this corresponds to the ideas of integration and differentiation.

 

[DrClaude]

For example, in the above equation, what if I have acceleration in meters per minute squared and t^2 in hours. The units are not the same so we cannot cancel. However, the dimensions are the same. But since the units are not the same, how are we able to cancel the dimensions? Do we assume that units are the same while doing dimensional analysis?

When doing dimensional analysis, you don't care about the numbers. In your example, min²/h² is a pure number, so you can discard that. That's why the actual units are not important, only what kind of units.

 

[Dale]

How is saying "one Newton of force applied for one meter of distance" different than saying "one Newton of force applied per one meter of distance"?

One Newton per meter would be something like the force constant of a spring. It would denote a linear increase in force as distance increases. It is completely different than applying a Newton for a meter.

 

[Heliosphan]

Some results are just unit-less and act as a pure number result.

 

[Astronuc]

As a simple example, when we do chain-link conversion, we are allowed to cancel units in order to obtain the correct answer. However, units are not numbers, so why is this allowed?

In some cases, unit factors represent an equivalence as expressed by proportional relationship, e.g., inch/cm or cm/inch.
http://www.chem.tamu.edu/class/fyp/mathrev/mr-da.html

We also like to work with changes in physical quantities as expressed by rates, e.g., distance/unit time, e.g., m/s, or ft/hr. Find distance over some period of time involves integration of the speed (or velocity) over some duration of time.

Other examples have been given. The force * distance is interesting because a unity like N.m can reflect a force applied over a distance (i.e. work), where the force is applied in the direction or tangent to the distance or path, or it can refer to a force applied normal to the distance (moment) between a pivot and point of application on a moment arm.

In material, we refer to a stress, which is similar in a sense to pressure and is expressed as units of force divided by the area over which the force is applied, and it can be expressed as an energy per unit volume, e.g., strain-energy density.
http://www.me.mtu.edu/~mavable/MEEM4405/Energy_slides.pdf

 

[sophiecentaur]

How is saying "one Newton of force applied for one meter of distance" different than saying "one Newton of force applied per one meter of distance"?

Actually, I would read 1N/m as a sort of Linear pressure - say under the blade of a cutter. If the blade thickness were constant then the Force per linear piece of blade could be stated in N/m. But it would hardly be a useful universal unit. The familiar units are familiar because they are frequently used and perhaps they are more or less 'acceptable', depending on familiarity more than anything else. We often find familiar things easier to accept but that could be said to be irrational.

 

[Andy Resnick]

As a simple example, when we do chain-link conversion, we are allowed to cancel units in order to obtain the correct answer. However, units are not numbers, so why is this allowed?

This is a fairly subtle and sophisticated question. At bottom, the use of units is a significant conceptual discriminator between physics and math. There are few few good responses already, I'll simply add that your question falls under the general topics of "ratio reasoning" and "scaling", both often under-taught in the sciences. As pointed out, sloppy use of complex/compound units can lead to confusion: (N*m) can either be units of work or of torque, for example. Similarly, it's not always correct to 'cancel' the units: units of strain are sometimes expressed in units like millimeters/meter and drug doses in mg/kg, both of which would seem to be 'unitless'.

I introduce ratios of units using a familiar example from the grocery store: 5 oranges/$2. Most people intuitively understand what this means (for example, it does not mean you must buy 5 oranges) and are comfortable with separately considering the number (5/2) and the compound unit (oranges/dollars). Similarly, most people feel comfortable with the concept of $2/5 oranges.

Ratio reasoning and use of units is also an important problem solving technique: besides being able to check your answer for 'reasonableness', you gain insight: for example, anything with units of length/time is a velocity.