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## Newton, Raphson and Numerical Methods

I have recently taught Numerical Methods to my FP1 class. The topic itself is not one of my favourites, but I think there are definitely places that fun can be had when teaching it.

I have uploaded the notebook presentation and exported PowerPoint here.

Interval bisection is fairly dry, and there are always many questions on “Why do we have to do this, can’t a computer do it?” my usual answer here is that computers can do it, but someone needs to tell them how, so if they re going to be able to program computers to do this they need to understand it.

Linear interpolation is more fun, and it is fairly easy with a bright class and effective prompting to get them to work out how to do it themselves, which they always find satisfying.

The part of the topic I like the best is Newton-Raphson method. It is the bit which requires the most maths, rather than just an ability to substitute. I also find that it is the bit that generates most discussion. I pose questions such as “Why is it sometimes divergent?”, which encourages deeper thinking.

I also like to discuss Newton and Raphson, my students are well aware of Newton by the time the topic comes round. They have always, of course, heard the apple anecdote (be it true or not) and know about gravity. If I have taught them any calculus topics previously then they know about his great work in that field, his battle with Leibnitz and his fractious relationship with Hooke. But they don’t know a thing about Joseph Raphson.

Why would they? When I first taught the topic I spend ages trawling the internet to see if I could find a picture to include, but there was nothing. About all that is known of the poor fellow is the date he was admitted to the royal society, his signature, that he attended Jesus College and coined the word pantheism. What makes it worse is that the so-called “Newton-Raphson” method isn’t a method they came up with at all, but rather it’s entirely Raphson’s work! The reason for the name is that 50 years or so after Raphson published his method, Newton published his own more complex hence inferior version of the method. So his most notable achievement is a victim of Stigler’s Law.