Numerical method for finding roots of *f(x) = 0*, by rearranging to *x = g(x)* and iterating

*x*_{n+1} = g(x_{n})

Click the image to link to GeoGebraTube

Designed principally for use with the MEI-OCR C3 Coursework on Numerical Methods

Newton-Raphson is an iterative numerical method for finding roots of equations.

*x*_{n+1 }= x_{n }− f(x_{n})/f′(x_{n})

Click the image to link to GeoGebraTube

Designed principally for use with the MEI-OCR C3 Coursework on Numerical Methods

When taking square roots of both sides of an equation, one should be careful not to turf out the negative result without first considering whether it has a true meaning. When using Pythagoras’ Theorem, the last step is to take square roots. So, can we have a hypotenuse with length -5?

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1. Imagine a cuboid made up of lots of little cubes.

2. Imagine a straight line (or “traversal”) connecting two opposite vertices of the cuboid

3. How many little cubes does the line pass through?

Note: Cubes are only counted if the traversal passes *through* them, so if your traversal goes exactly through a vertex or an edge separating cubes within the cuboid, then it is not considered to pass through the adjacent cubes. In the 2×3×4 example shown, the traversal actually passes through an edge in the centre of the cuboid.

**Problem: How many cubes does a traversal pass through in a 60×120×156 cuboid? (Hint: Start small!)**

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- Draw a rectangle on squared paper.
- Draw a diagonal across your rectangle.
- How many squares does it pass through?

Note: If your diagonal goes exactly through a point, like in the 2×4 example, then it is not considered to pass through either of the diagonally adjacent squares.

**Problem: How many squares does a diagonal pass through in a 190×884 rectangle?****b**

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1. From the WordPress Dashboard, install a plugin that enables iframes to be embedded into posts, for example Embed Iframe. (Simple instructions for installing plugins into WordPress may be found here) Read more…

September 13th, 2009
Micky

In Part I we calculated the probability of winning a single-suit version of the Higher of Lower card game. **The objective of this post is to find the probability of winning the full, unsimplified, 4-suit version of the game** played in pubs across the land. Read more…

September 5th, 2009
Micky

I was recently roped into joining some friends for a pub quiz. I hate pub quizzes. For a start, I don’t watch soaps or football or take any interest in divorcing celebrities. The sum total of my contribution is usually a question about the periodic table and another about some obscure ’90’s one-hit wonder.

At the end of this particular quiz there was a competition to win a pot of dosh. Those optimistic enough to enter first had to be lucky enough to have their number drawn from a hat, and then attempt to win the ominous game of Higher or Lower, or “Play Your Cards Right” as it is sometimes known, via a crudely written flash program running on the host’s laptop. If you win the game, you win the cash. Easy? Apparently not. Two players failed in succession and then the host declared that the contents of the pot would roll over to next week. Almost everyone in the pub had just lost a quid. Would anyone ever win the money? I didn’t know, but I was alarmed and **it was my duty to discover the probability of winning this game**.

### On Friday I will be talking in our school chapel. Here is the penultimate draft of the monologue (the style does lend itself to being read out loud)

Good morning.

I want to talk to you today about our monetary system. Most, perhaps all, of us here at Forest School have benefited from the monetary system. Money affords us food, entertainment, transport, holidays; a place to live; stability. Money creates incentives; it gives us jobs, careers and aspirations; and taxes allow huge investments like high-speed rail links and the Olympic park.

But does everyone benefit from our monetary system?

We’re born into a society of which money seems to be the driving force. Where does money come from? Who gives us it to spend?

** Read more…**

### Visualising imaginary forms…

This post builds further on the concepts discussed in the original 1-in-10 Gay Problem post and its sequel, Find a way to my Heart, and delves **deeper into the abstract** to discover an impressive 3d form hidden in our function. It will be assumed that the reader has studied these preceding posts (10 minutes’ fascinating reading!) Read more…