what’s the physical meaning of mathematical operations?

Question

Let’s say you have a division. You’ve got 3/2, where 3 might be acres and 2 might be brothers so from there you could deduce that the division just means that the 3 acres of land will be distributed between two brothers. Summation and multiplication are quite straight forward, but what the hell does one do with multiplication and exponentiation? If I got 20 N-m of torque, wtf does the multiplication tells me there? If something follows a square root?

Please feel free to add any other operations to these. Thanks!

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Answer 1

The beauty of math is that it works the same with or without any physical meaning. Its logically self consistent as abstract symbol manipulation.

Now when it comes to applying math, of course you need to match some parts up with physical meanings as needed, but you can treat a lot of it as black box internals of which you know are logical and self consistent.

As for some examples for practical use of math concepts.

Multiplication is when something such as torque is proportional to several things such as force applied and length of the leverage.

Division is often inverse proportionality, such as speed is proportional to distance traveled and inversely proportional to time taken.

Exponentation describes growth very well, processes with feedback. If you reinvest your earnings hour earnings grow even faster.

Many things can have polynomial relations, reflected radar signal is proportional to 4th power of distance for example.

Complex numbers line up very well with oscillations where energy is coupled between two different things such as tension and inertia or electric and magnetic fields.

Matrix operations and linear algebra lines up well with coordinate transformations, inherently pure mathematics, but describes physical space and motion very well.

But just because a mathematical concept is often used in one physical meaning, doesn’t mean it can’t be used to describe completely different physical meanings. Mathematics is just abstracted logic, with whatever type of logic you understand some physical phenomena, you can transform it into a mathematical model and for there manipulate it as such.