How Do Mathematicians ‘make up’ or perhaps more correctly ‘derive’ equations

Question

I know this is definitely a thing otherwise we wouldn’t have equations.

What are the methods for making up an equation from scratch and from making up an equation from an existing equation?

The only way I have been able to do this is using dimensional analysis and I’d like to learn other ways, if you want to see an example of me using dimensional analysis to ‘make up an equation’ see my answer to this question.

How for example can you make up the following without experimentation:

Speed equation: $$\text v=\frac{\text d}{\text t}$$

Density equation: $$\text p=\frac{\text m}{\text v}$$

After seeing some answers, how do you make up an equation for a definition? @littleO tells me the velocity equation is a definition and I’d like to know how the guy who made the equation went about it.

Or how do you look at a square and come up with a quadratic equation?

Are there any equations you can make up without experimentation?

If this is too general, specific examples are welcome.

Answer 1

You always need to remember, that "equations" are nothing more than formalized language. Before people "invented" equations, they argued with words, which today are a pain to read. Here is Newtons Second Law (summed up nicely in $\vec{F}=m\vec{a}$):

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

So math, as you may learned it in school may be terrifying because much teaching is done using the formal set of rules, rather than words. However, what is important are the ideas that are conveyed either with words, or with the equations. To go from one equation to another, we simply use clever tautologies, definitions of certain variables, that replace other variables (in the $v$ example, $\frac{d}{s}$)

Let's do a basic example: Let's assume all you own is a bag of apples, of unknown quantity. Now let's say I give you an apple. How many apples do you have?

Clearly, you have 1 apple, plus the amount of apples in the bag, no? So now let's define some variables:

• $A$ is the amount of apples you have after I give you an apple
• $b$ is the amount of apples there are in the bag.

Going off of these definitions, we can say the following:

$$A=1+b$$

This makes sense, no? You definitely have one apple, in addition to those that are in the bag. Furthermore, you cannot have more apples, because we assumed that you don't have anything else (note how we are assuming things). Et voilà, you have an equation with certain dimensional values!

Now let's get funkier. Let's say that the amount of apples in the bag is equal to the amount of apples my grandmother has. So if:

• $g$ is the amount of apples my grandmother has

then you know

$$g=b$$

Now you see, that there is no need for you to know explicitly and only how many apples are in the bag. If you know the amount of apples my grandmother has, you know the amount of apples in your bag. To formalize this in equation form, this: $$A=1+b$$

becomes this: $$A=1+g$$

because $g=b$. Now you have another equation. Clearly, you can own trucks with apples, divisions of apples, square root of the amount of apples that the CEO of ExampleFirm makes in USD per year, per store location, etc. You can formalize what I'm saying, but as you can tell, just because you have an equation, doesn't mean it's correct. That's why you need to assume that certain things are the case, in order for the equation to make any real world sense.

You can graph virtually any equation. It might be a boring graph (all variables defined), but remember: Even a graph is just a tautology, i.e. a way to convey the information that is in the equation (which in turn, is the information that is given linguistically). So when you ask "How do I make up an equation", what you are... really asking... is "How do I form words" or "How do I make an argument" or "How do I say something that can be quantified", and the answer, as far as I can tell, is "You quantify, and then say what you are quantifying".

Answer 2

Do an experiment. Now, do it again, but change just one of the parameters (e.g., the temperature, or the amount of substance, or the volume) while keeping all the other parameters the same. Now do it fifty more times, recording the results of all these experiments. Now look at your table of all the different results you got, each one with its corresponding input value. Draw a graph showing the dependence of the output on the input. Compare it to all the graphs you know about. If the graph fits one of the ones in your arsenal, congratulations! You have discovered a law of Nature.