# The Newton-Raphson method, explained in a Platonian dialogue

As you all know, I like the Newton-Raphson method. And so, explaining it covers quite a few articles on this website. Still, there is more we can do to explain this method. There is always more. To do that, I once again enlisted my academic colleague ChatGPT, and asked for this:

## The Prompt

Write a step-by-step instruction for the Newton-Raphson method, but in the style of a Platonian dialogue

## The Platonian dialogue explaining the Newton-Raphson method

Socrates: My dear friend, have you ever pondered upon how one might find the root of a complex equation?

Glaucon: Indeed, Socrates, it is a task that often vexes me.

Socrates: Fear not, for I shall guide you through the steps of a method known as Newton-Raphson, a path that will lead us to the root.

Glaucon: I would be most grateful, Socrates.

Socrates: First, we must imagine that we have an equation, let us call it $$f(x)$$. The root we seek is a value of $$x$$ where $$f(x)$$ equals zero. Do you follow?

Glaucon: Yes, Socrates, that seems clear.

Socrates: Very well. To start our journey, we must first make a guess as to where the root might be. This we shall call $$x_0$$. Like a blind man seeking light, we must take a step, even if we do not yet see the path.

Glaucon: I see, we are stepping into the unknown. What next?

Socrates: We then seek guidance from the derivative of the equation, which we shall call $$f'(x)$$. The derivative, like a wise mentor, shows us the way by revealing the slope of the curve at a certain point.

Glaucon: I understand, Socrates. The derivative guides our path.

Socrates: Precisely, Glaucon. Now, with the aid of our mentor, we apply the following formula:
$x_1 = x_0 – f(x_0)/f'(x_0)$

Just as a student learns from his teacher, our new guess, $$x_1$$, benefits from the wisdom of the old, $$x_0$$.

Glaucon: I see, so the new guess is closer to the root?

Socrates: Indeed, Glaucon. But one lesson is seldom enough to grasp the whole truth. Just as we must repeatedly question our beliefs, we must iterate this formula:
$x_{i+1} = x_i – f(x_i)/f'(x_i)$

With each iteration, we draw ever closer to the root, just as through questioning, we draw ever closer to the truth.

Glaucon: And when do we stop this process, Socrates?

Socrates: We stop when we can come no closer to the root, when further questioning adds no more to our understanding. Then we have found our answer.

Glaucon: Thank you, Socrates, for this enlightening discourse. The path to the root, much like the path to the truth, is indeed a fascinating journey.

## More Newton-Raphson Style Variations

If you like this sort of thing, i.e., fun versions of scientific explanations, then you might be interested to read one of the following as well. At least, once you are through them all, you should be able to explain the Newton-Raphson method to absolutely anybody

Andreas

I am a nerd and trained in theoretical physics (with a PhD and everything ...). My interests are life as a human being and how computing can teach us valuable things about that as well as about nature in general. Read more about me.