Why Model?

Models represent information needed to answer a question.

Transforming one model of reality into another can transform a question that is difficult to answer in the original model into one that is easy to answer in the transformed model.

For example, "What is 1,234,567 x 98,765,432 ?" is more difficult for a human to answer than "What is 14.0262308592797 + 18.4082582229538 ?"

Here is a mathematical transformation (using exponents and logarithms) that converts the first question "What is x times y?" into the second "What is u plus v?":

f(x) = u, g(u) = x
f(y) = v, g(v) = y

Define inverse functions f and g such that:

f(xy) = f(x) + f(y), x,y ≠ 0
g(u+v) = g(u)g(v) = xy

f(1) = 0
g(0) = 1

f(x) = log_e(x) = u
g(u) = e^u = x

So, if:
x= 1,234,567
y = 98,765,432

then:
u=14.0262308592797
v=18.4082582229538

u+v = 32.4344890822335

xy = g(u)g(v) = g(u+v) = e^{32.4344890822335}
= 121,932,543,087,944

Of course, the transformations from one model to the other and back are not so easy for humans without the aid of a calculator, which could have easily solved the original problem without transformation.