What is the simplest equation ?

There are a few candidates so please justify your offering.

An identity equation that stays true for any value assigned to its variables.

1 hour ago, MigL said:

An identity equation that stays true for any value assigned to its variables.

Ah but which one ?

You mentioned variables in the plural.

The one I am thinking of has essentially one variable.

Sorry it's been so long off line I've forgotten how to hide spoilers, it doesn't seem automatic.

Edited yesterday at 11:35 AM1 day by studiot

2 hours ago, studiot said:

What is the simplest equation ?

There are a few candidates so please justify your offering.

In science? I might vote for E=hν, on the basis that it is self-evidently very simple and is probably the most important relation for anyone who wants to understand anything about light and quantum theory. But then F=ma in mechanics is equally simple and fundamental to understanding.

I'm not even sure there is any one simplest, though. Perhaps the question is a bit like asking what is the best cheese in the world. 😊

The simplest equation means what? The one with the fewest numbers? The one that's easiest to understand? The one with the fewest operations? Something else?

4 hours ago, npts2020 said:

The simplest equation means what? The one with the fewest numbers? The one that's easiest to understand? The one with the fewest operations? Something else?

That's for you to decide when you compose your answer.

5 hours ago, exchemist said:

In science? I might vote for E=hν, on the basis that it is self-evidently very simple and is probably the most important relation for anyone who wants to understand anything about light and quantum theory. But then F=ma in mechanics is equally simple and fundamental to understanding.

I'm not even sure there is any one simplest, though. Perhaps the question is a bit like asking what is the best cheese in the world. 😊

Well I can't think of a simpler one, but I am trying to promote discussion here.

I did try to put my equation in a spoiler, but couldn't remember how to hide it.

Well for me it's e=mc2. The only variables are e and m, the rest is constant and just tells you the proportion between the two that makes them equal. Simple concept. Only one operation. Just three numbers.

Here's my candidate along with a criterion. Criterion: The simplest equation would be one that holds the key to as much phenomenology as possible without involving more than three (perhaps four?) variables plus a number of universal constants. The ideal-gas equation could be a good candidate.

PV=NkT

Why not E=hx(nu) or E=mc²? Only because I tend to regard these as mere changes of units.

Of course, I'm aware of the subjective element in all this.

1 hour ago, npts2020 said:

Well for me it's e=mc2. The only variables are e and m, the rest is constant and just tells you the proportion between the two that makes them equal. Simple concept. Only one operation. Just three numbers.

Yes, I wondered about that but decided against. The problem with it is it is only half the equation. The full version is E² = (mc²)² + p²c².

1 hour ago, joigus said:

The ideal-gas equation could be a good candidate.

PV=NkT

How about

Pretty universal right?

3 minutes ago, sethoflagos said:

How about

Pretty universal right?

I'd call that an inequality.

But still a very strong candidate.

12 minutes ago, sethoflagos said:

How about

Pretty universal right?

But, as @studiot said, that's an inequality. I take it that it wouldn't have to be a universal relation necessarily. Simplicity being the only desideratum. I would also disregard anything that's too close to just a definition.

\[ dS = \delta Q / T \] looks the part too.

I'd love to know what Studiot's choice is.

Edited 19 hours ago19 hr by joigus

4 minutes ago, joigus said:

But, as @studiot said, that's an inequality. I take it that it wouldn't have to be a universal relation necessarily. Simplicity being the only desideratum. I would also disregard anything that's too close to just a definition.

dS=δQ/T

looks the part too.

I'd love to know what Studiot's choice is.

Well I originally thought of the original notion this morning - it just sort of popped up and I thought it would be a good question to create some activity here.

I was thinking that since frequency is the inverse of time then in a way they really are just one single variable in the equation

Frequency x time = 1

10 hours ago, studiot said:

Ah but which one ?

The simplest identity equation would be something like

x = x or maybe x + 0 = x

which state that the variable is always equal to itself, no matter what number is plugged in for the variable.

and you did ask for the simplest, although I assumed you meant class of equation, not a specific one.

So, how about telling us your choice ...

Edited 18 hours ago18 hr by MigL

5 minutes ago, MigL said:

So, how about telling us your choice ...

I don't have a choice. I'm perfectly happy for someone else to come up with a better offering.

But is x a scientific variable ?

x+0 could be binary.

I have already explained where the idea came from and why I posted it.

18 minutes ago, joigus said:

But, as @studiot said, that's an inequality. I take it that it wouldn't have to be a universal relation necessarily. Simplicity being the only desideratum. I would also disregard anything that's too close to just a definition.

dS=δQ/T

looks the part too.

I'd love to know what Studiot's choice is.

True enough. I did consider just posting the limiting equality, but that falls foul of the 'just a definition' issue.

However, most, if not all property conservation laws (energy, mass, momentum, momergy ets) can be expressed in the form:

ΔE = 0

21 minutes ago, studiot said:

But is x a scientific variable ?

A variable can represent length, mass, time, etc., so ... yeah.

If your criteria is simplicity, I don't see how you can get 'simpler' than an identity equation.
I can offer plenty of reasons why the equations offered by other members are not 'simple'.
I'm asking for your reasoning dismissing the identity equation.

12 minutes ago, MigL said:

A variable can represent length, mass, time, etc., so ... yeah.

If your criteria is simplicity, I don't see how you can get 'simpler' than an identity equation.
I can offer plenty of reasons why the equations offered by other members are not 'simple'.
I'm asking for your reasoning dismissing the identity equation.

An identity like density=mass/volume, or the like? A definition?

39 minutes ago, sethoflagos said:

True enough. I did consider just posting the limiting equality, but that falls foul of the 'just a definition' issue.

However, most, if not all property conservation laws (energy, mass, momentum, momergy ets) can be expressed in the form:

ΔE = 0

This reminds me of Feynman's jocular example of the simplest equation of physics being U=0

U being the amount of 'unworldliness'. = (F-ma)²+(E-mc²)+... etc. Taking all known equations and forming quadratic real scalars from them (thereby positive). The sum must be zero.

And it's as simple as can be. The complexity is in the details, of course.

32 minutes ago, joigus said:

The complexity is in the details, of course.

Like:

j_M = j_H = f/2

Simple form of the Chilton-Colburn Analogy.

Edited 17 hours ago17 hr by sethoflagos
typo

There are a bunch that have just one independent variable. I’m not sure how you argue one is inherently simpler than another

2 hours ago, joigus said:

An identity like density=mass/volume, or the like? A definition?

Isn't every equation a mathematical definition ?
One variable is defined in terms, or changes of, other variables.

An identity simply defines itself, and so is the simplest type.

Edited 14 hours ago14 hr by MigL

6 hours ago, exchemist said:

Yes, I wondered about that but decided against. The problem with it is it is only half the equation. The full version is E² = (mc²)² + p²c².

Now you're complicating things. 😜

I would agree but it's not usually written all the way out like that other than someone actually performing the math. Even professional physicists frequently use the shorthand from what I have seen.

My first stab at it was identity, x=x, but then I wondered if it was too much a tautology. An equation should "say something" if we're speaking in the domain of science. The simple ones do seem to be definitions. V=IR. The complexity comes when we unpack the deeper physical meaning of volt and amp and ohm as those exist in a circuit. (Or that volts are also joules per coulomb, or watts per amp, or electronvolts per elementary charge, etc.) Then we get into complex relationships that are teased out empirically and can't always be handily compared to water flow in some ideal Hagen–Poiseuille setup. For me, a really simple equation would say something that doesn't get you into all that unpacking. It's late here, so I'll have to think on that later.

Wait, okay, I guess water flow in a pipe at its most basic might be one of the simplest, the rate is Q = A × V (Flow Rate equals Area times Velocity).

The OP asked for the simplest equation. Strictly speaking, an equation is an expression to be solved rather than a formula, identity, or definition. Thus, the simplest equation is:

ax + b = 0

to be solved for x.

Edited 12 hours ago12 hr by KJW

I’d offer up

\[\delta S=0\]

because the expression has the simplest form while still capturing the widest phenomenology in physics. After all, this principle underlies most things from QFT all the way up to GR; it is, in a sense, the closest thing we have to a “theory of everything”.

So I’m relating “simplicity” to the form of the equation; other choices will lead to other candidates.