Why can't we always add quantities with the same dimensions?

[johncena]

According to principle of homogeneity, quantities having same dimensions can be added and subracted...but isn't it false ?
because according to the principle ,
we can add quantities having dimensions M⁰L⁰T⁰
I.E we can add plane angle and solid angle,
we can add angles in different measures, (i.e radian measure + degree measure)without conversion...but what will be the result?
Similarly,
1km + 2m = ?
1s + 1m = ?

 

[CompuChip]

Well, if a = 18 and b = 2pi, then a + b = 18 + 2 pi.
Of course, if a and b are the outcome of a measurement, and the measurement was done in degrees and radians respectively, 18 + 2 pi does not have any meaning. But it is definitely possible to add them.

1 km + 2 m makes perfect sense, since 1 km = 1000 m. So 1 km + 2 m = 1002 m, or 1.002 km.
1s + 1m is not possible, because one has dimension of time and the other has dimension of length.

Also, I never heard the word homogeneity used for this.

Anyway, I don't know if my post is very helpful, but I don't really see what you are trying to do.

 

[Mapes]

According to principle of homogeneity, quantities having same dimensions can be added and subracted...but isn't it false ?

It would be more exact to say that "Quantities with the same units can be added and subtracted with no problem." Quantities with the same dimensions can often be added and subtracted as long as the units are correctly converted, but their are exceptions.

$$\mathrm{1\,km+2\,m=1000\,m + 2\,m=1002\,m}$$

$$\mathrm{1\,s+1\,min=1\,s + 60\,s=61\,s}$$

$$\mathrm{1\,rad+1^\circ=\left(\frac{180}{\pi}+1\right)^\circ}$$

Here's an exception: torque and energy (or work) are both measured in dimensions of ML²T^-2, but the sum of 1 Nm of torque and 1 J of energy isn't generally meaningful. So we write the units differently (Nm vs. J), even though they're dimensionally equivalent, to make the distinction clear.