Why numerical methods?
* They might produce the right answer
* They frequently do
* They are easy to visualize or imagine
* You get used to working with a routine that is both fallible but quite simple and remarkably able to work in a wide variety of situations. This is what machine learning does, but there are more sophisticated routines.
At some point you need to make a decision to go down the road more focused on analysis & modelling vs machine learning & prediction. It's not that the two are exclusive, but they really do seek to address really big forks in the problem space of using a computer to eat up data and -- give me predictions or give me correct answers
Google needs lots of prediction to fill in holes where no data may ever exists. Analysis and modeling can really fall down when there is no data to confirm a hypothesis or regress against.
An engineer needs a really good model or the helium tank in the Falcon 9 will explode one time in twenty vs one time in a trillion. The model can predict, based on the simulation of the range of parameters that will slip through QA, how many tanks will explode. Most prediction methods are not trying to solve problems like this and provide little guidance on how to set up the model.
On the prediction side, you will learn all the neural net and SVM stuff.
On the analysis and modelling side, get ready for tons of probability and Monte Carlo stuff.
They are all fun.
Newton's method and other similar numerical methods are the hello world of a branch of mathematics known as 'numerical analysis' and scientific computing. This is not Machine Learning.