For example, how did a group of scientist comes up with the formula of “Delicious cake= sqrt(Eggs + flour^3)+constant C”. And is it possible for one to develop and implement the thought process into their daily life?

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Essentially two ways:

1) Derive it – i.e. take a bunch of equations we already know, combine and adapt them to the situation you’re trying to describe and apply mathematical operations until you end up with a solution for your problem.

2) Empirically – conduct a lot of experiments, look at your data, think what kind of equation could describe the data, then fit it.

Generally, if people describe a completely new phenomenon they start with (2) – although eventually someone might come up with a way to derive it from other, more fundamental laws.

E.g. the classical laws governing gravity were found by people who measured movement of planets, then basically tried out equations to explain them, and then tried out different values for the gravitational constant until it fit.

Most of the basic formulas aren’t really complex and built step by step. Volume of a cube is side^(3), that’s simple. Mass is density*volume, so mass of a cube would be side^(3)*density. A momentum of a moving thing is mass*speed, so a momentum of a thrown cube would be side^(3)*density*speed. And so on.

The thought process for most things isn’t really that complex: thing A depends on things B, C and D, A gets four times bigger if you double B, two times bigger if you double C and two times smaller if you double D, so it’s highly likely that A = B^(2)*C/D.

More complex things appear when you apply mathematical operations like integration but that’s still applying a well-defined operation to a simplier formula. Every part has a meaning and they combine into one large equation.

Richard Feynman has a really good video lecture on this (https://youtu.be/0KmimDq4cSU). Basically scientists make a guess, then check if it was right. The guess isn’t completely random; it’s based on known data, and they try to find an equation/model that fits that data. This has become easier with tools like excel, Matlab, etc where you can directly plug in data points and have it find a curve that fits the data.

Alternatively, many useful equations can be derived directly from previously known equations, strictly by using math like trig, calc, diff eq, etc.

Other answers here are good in the abstract, but let’s try an applied example relevant to every student at some point in their lives.

Suppose a class has a final grade that is 40% the final exam, 40% homework, and 20% the midterm. You’re at the end of the course: you took the midterm and got a 70, and averaged a 75 on the homework. You want to earn a B for the course (that is, at least an 80%). How well do you have to do on the final to ultimately get your B?

More generally, what if you’d done worse on the midterm, or better on the homework? What grade would you need then?

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A useful way to think about this kind of problem is to imagine that you *had* what you’re trying to figure out, and think about how you would compute the things you already know.

So. If we *knew* what we got on our final (say, we got a 95), could we compute our grade? Sure.

(Another rule of thumb: it’s often helpful to recast percentages in terms of actual amounts.) Suppose the class has 100 total “points” possible. The final is worth 40 points, the homework is worth 40 points, and the midterm is worth 20. We want to know how many points we got. Well, we got 70% of the midterm points, and 70% of 20 is 14. We got 75% of the homework points, and 75% of 40 is 30. And if we got a 90 on the final (we didn’t yet, but we’re using this as a way to think about *what operations we want to do*), we’d get 95% of its 40 points for a total of 38. Then our total points would be 14 + 30 + 38 = 82 points – so a 95 is enough, at least!

Now, rather than actually finding this number, let’s think about *how* we got it.

70% of 20 is the same thing as (70/100) times 20. 75% of 40 is the same thing as (75/100) times 40. 95% of 40 is the same thing as (95/100) times 40. We computed all these numbers, then added them up. Let’s write that out explicitly, keeping the operations instead of computing them (because we wouldn’t be able to compute them if we didn’t know what one of the numbers actually was):

(70/100) * 20 + (75/100) * 40 + (95/100) * 40 = 82

(So cool, it looks like a 95 would be enough! We *can* still get our B!)

And now let’s try replacing the numbers with where they came from:

(midterm score) * (midterm portion of grade) + (homework score) * (homework portion of grade) + (final score) * (final portion of grade) = final grade

*This* is a formula. If we know all but one of these pieces of information, we can now plug them in and compute the last one. Can you use it to figure out the grade you actually need on the final?

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In real world settings, we observe patterns until we understand them well enough to compute with them, and then we just stop writing the numbers and start writing where they come from. So in your example, a baker noticed that you kept computing sqrt(30 + 40)^3 or whatever, then realized that the *30* represented the number of eggs, the *40* represented the amount of flour, and so on.

There are more advanced ways to go about this. If you understand the forces that act on a system, you can use calculus to figure out how it will behave. If you recognize that a system is evolving over time in a way that depends on its current state, you can use the theory of dynamical systems to talk about how it behaves on average. If you recognize that a system has some sort of symmetry to it, that can help you narrow down what equations can describe it (because they must have the same symmetry properties). And so on. At some level, what you’re asking here is “ELI5, literally all of the hard sciences”.

While I’ve never heard this particular formula, formulas are derived either from previous math (or previous mathematical description of similar phenomena) or from experiments.

First, you identify how many variables you have. For example, let’s say that delicious cake only depend on eggs and flour. Eggs and flour can be quantified (number of eggs, amount of floor) so they can be analyzed mathematically. And they are two, so you have two variables.

Then, you change in a controlled way a single variable and see what happens. For example, if you add one egg while keeping the same amount of floor what happens? What if you remove an egg? What if you double the amount of eggs? What if you have no egg? And so on.

Then, you change the other variable and see what happens.

Then, you change all variables together and see what happens.

In the end, you take all these data and you *interpolate*, which means you start from all these observations and you find the mathematical function that best matches your data.