Why dimensions can be treated as algebraic quantities?

[0000]

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?

As always, excuse me if my english isn't very clear.

[calculus_jy]

asking such question, i assume you are like me, a person fusing about how everything tiny detail functions,:approve:

yes it is rite that they can be treated as algebraic quantities,
in physics, a letter is usally used with the inclusion of units
we say, the force is donated by F, not F N as in F Newton

the apple case is rite, but we dont see ourselfs multiple apples by apples do we,we can say, i bought 6 sets of 4 apples, that implies 4 * 6 applies ie 24 apples but we dont see ourselves multiplying apples by applies

we can treat them as algebric quanties as we always express them as rates (unit 1) / (unit 2), such as in velocity (m/s) but we find it better to treat units as algebric qualites as exampled in the case of acceleration
a= (v_f - v_0) / t where v_f is the final velocity and v_0 is the intial velocity which is denoted by the unit (m/s) and time in s
then a's unit must be (m/s * 1/s)

we donate this unit
(m * 1)/ (s*s) which is m s^-2

lets give it a numberic value as well
the acceleration of a car is 2 metres per second per second or 2 metres per second squared ( to the north) . then in a 1 sec period, the cars speed changes m s^-2 * 1s =2 metres per second or (2m/s)/s * 1s = 2 metres per second

so really there is not sec^2 in the world but it exists as sec and is usually expressed as the rate of something ( unit 1 (eg. Joule)/ sec) and then it follows that the rate of change of that rate is ( unit 1/ sec / sec) which is better expressed as (unit 1 s^2) and as seen in the acceleartion case multple this by a value of time will result in the change if the other rate in that time period!

I hope this is suppose to help you more than to add confusion!
PS my english is not so perfect either!

[malawi_glenn]

Yes that is true calculus_jy , the units like J*m/s^3 dont exists, they just help us to see what a specific physical formula tells us. And also to make sure that what is on the left side is the same as the right side. You can't compare a quantity that has the units J/T with another one that has m/s^3

[0000]

Thank you for your answers.