Why dimensions can be treated as algebraic quantities?

Question

Hi, in my physics book (serway) they say "dimensions can be treated as algebraic quantities" but I don't understand this very well. If I sum meters I get meters, if I multiply meters I think I get meters^2 because the area of a rectangle is b.h. But if, for instance, I multiply seconds.seconds I don't uderstand why I get sec^2.

Algebraically x.x = x^2 but x represent a a number not time, weight, etc.

I see it like 3 apples x 2 apples = 6 apples^2.

Probably I'm getting this wrong, Could you help me to understand this right please?

Excuse me if my english isn't very clear.

Answer 1

This is a perfectly good question. The confusion lies in the term "algebraic quantity". It a vaguely defined term and it is the lack of crispness creates the confusion you have so astutely noticed.

It is true that meters^2 represents an area and so is seconds^2. We associate meters with a units of a dimension such as length and width is makes perfect sense that m^2 represent an area while seconds^2 on the other hand appear to be odd to represent the same. The trick is in ability to realize that a physical area is a particular case of a more abstract algebraic area called A: A=a^2 where a is a dimension of a certain space. So anything can be a dimension. Well...not quite.

The physical interpretation is change of velocity with respect to time a=dv/dt and represented as m/sec^2

Apples on the other hand are not units that represent space they are objects, tangible objects. The idea of having apples^2 seems out of norm.

I hope it put your philosophical bone to rest. If not ... then have fun searching for the truth.

Answer 2

Dimensions are assigned to the physical quantities to remind us their properties,while specifying their magnitudes e.g. in relations invoving various quantities (v=u+at).Convention:-some physical quantities are cosidered as fundamental and have been assigned unique dimensions,and other dimensions are derived logically from them,to avoid burden over our memory!But "apple" is not a dimension,it's a name.It;s not like that 3apples*2apples=6apples^2. It should be like this ''a numerical value * nu. of apples" i.e. 3 * 2apples OR 2 * 3apples=6apples

Answer 3

One word: Relativity.

As you correctly point out, and as normal experience shows, all the dimensions can be thought of as linear, adding algebraically.

But when high velocities or high gravitational fields are present then that is no longer the case and velocities and dimensions add quadratically.