Quote:
12ft^2 = 144ft [says Paul]


You measure area in ft? I wonder how big a 1000ft apartment would be. Did you ever see anyone else anywhere use feet as a unit of area? Ever? I know you won't be able to show me any examples, but can you even remember any?


Quote:

in your mind 5ft^2 is only 5 * ft^2

Yes! You just wrote the exact same thing. Just as 3x = 3*x. There is an implied multiplication between the 5 and the ft.


Paul, there's no way you can do any calculations with physical quantities unless you use some consistent way to operate on them. The easiest is the common convention that nearly everybody has used for hundreds of years.

Treat units like variables

You already treat numbers like variables. You can already say 3*3=3^2 the same way as x*x=x^2. Now just add m*m=m^2. That's really all it is. Just doing that will instantly make all these calculations work exactly the way me and TFF say. You can test it on anything and it will always work exactly the same. Even on the 5x5 grid it will work exactly as MIT says.

Your way leads to measuring area in ft.