Why is it, that the simple equation "a = bc" explains so many functions/behaviors in our world? It explains Newtons Second Law, the partial pressure of a soluble gas (Henrys Law), the force of a spring, ph of an acidic/alkaline solution, motion and static equations, it goes on continously...So why does this simple equation explain so many workings in our world? How can the multiplication of two numbers so universal to so many different systems?

The behavior depends on whether b is a variable or a constant. F=ma is a linear relationship between force and acceleration; mass is constant in the equation.

 

It's true because many basic things we observe are linear. If they weren't, we may not have ever figured them out.

Why is it, that the simple equation "a = bc" explains so many functions/behaviors in our world? It explains Newtons Second Law, the partial pressure of a soluble gas (Henrys Law), the force of a spring, ph of an acidic/alkaline solution, motion and static equations, it goes on continously...So why does this simple equation explain so many workings in our world? How can the multiplication of two numbers so universal to so many different systems?

I think it is caused by the way human understanging after Galileo works.

We are making measurements. A balance measures one kilogram, a ruler measures one meter, a clock measures one second, etcetera.

Then people like Newton try to combine the measurements.

How can the multiplication of two numbers so universal to so many different systems?

 

I see [math]a=bc[/math] as more of a matter of division than multiplication (might sound odd, but it's a matter of interpretation), assuming [math]b[/math] is a constant. Now, [math]b[/math] can be interpreted as a conversion factor, something which quantitatively relates [math]a[/math] and [math]c[/math]. So it really reflects the importance of direct proportionality between two variables, the fact that their quotient is always constant regardless of their size.

 

It might be of interest to bring up [math]\pi[/math], since it is the constant equal to the quotient of a circle's circumference to its diameter. No matter how big or small the circle is, [math]C=\pi d[/math] is true.

 

These kinds of expressions are also related to the also important equations of form [math]y=\frac{1}{2}cx^2[/math] via integration. Bringing up the circle example, one can derive [math]A=\pi r^2[/math] from the above identity for circumference.

 

So now the question is... What is the fundamental significance of proportionality?

Edited May 20, 201312 yr by Amaton

What is the fundamental significance of proportionality?

In physics one could see this as finding appropreate units.

So now the question is... What is the fundamental significance of proportionality?

Proportionality comes from geometry. Or the inverse.

 

 

 

In physics one could see this as finding appropreate units.

Some conversion factors are dimensionless, some are not.

Edited May 20, 201312 yr by michel123456

Some conversion factors are dimensionless, some are not.

Sure, but I don't think that really effects the basic idea. In some situations, for example, I meam A and you mean 2A. So, up to overall rescalings one could think of proportinality as finding sensiable units.

From a pure maths perspective you could say that it's a consequence of looking for general rules - the type of relationships that we look for are ones that are symmetrical, invertible etc - which limits the amount of operations that could possibly be used to describe the ones we find.

From relativity: E = mc²

and, QM: E = h \nu\

http://en.wikipedia.org/wiki/Quantum_mechanics#History

Edited May 31, 201312 yr by EdEarl