Archive for March 2010

Current Events: Resistance is Futile! (updated)

March 21, 2010

You will be assimilated – or perhaps it is waht you will have to assimilate. Whatever – the introduction to simple direct current (DC) circuits is a very simple topic. What is more difficult is understanding the underlying principles and concepts – What is a current? What causes it start “flowing”? What exactly is “voltage”? What does it do? Where does it go?

These questions are a lot more difficult (and a lot more interesting!) than the level of  DC circuit analysis that we do in unit 1 Physics. However, it is necessary to concentrate on the Circuit analysis before we get back into the fun stuff.

Let us meditate:

Ommmmm

Oommmm

Ohmmmm

Ohmmm’s LAW!

Popular Populations & Power (or a fistful of Fibonacci!)

March 20, 2010

What does you fist have to with exponential functions? Well aside from wishing you could hit your teacher for making you suffer through Exponential (&Transcendental) functions, it shows you an important sequence of numbers – the Fibonacci (Fibonacci was the first person to bring decimal numerals and place value notation to the west!)  sequence. The fibonacci sequence is:

1, 1, 2, 3, 5, 8, 13 ,21 ,34,…

If you look at the side of your fist, you can see this pattern in the knuckles (not knuckle joints).  The first knuckle is 1. The second knuckle is 1. When you fold your finger over, it makes a curl of three knuckles. When you close your fist (without your thumb) it makes a sequence of five sections. When you include your thumb, it goes to eight – in other words, a fistful of fibonacci (1 knuckle, 1 knuckle, 3 sections in a curl, 5 sections in a fist, 8 sections in a fist and thumb). The Fibonacci sequnce shows up everywhere in nature and art as a ratio called the golden mean. It shows up in places as different as the petals of a rose, sections of a pine cone, the shape of a shell, the Athenean Acropolis, Leonardo Da Vinci’s Mona Lisa and Vetruvian Man, the stock marketthe spiral galaxyDNA, even popular music!

Oops… Was that your battleship? (updated!)

March 14, 2010

The space on which linear functions are drawn is sometimes called the X-Y Plane, because it is made up of two number lines at right angles – the horizontal numberline (X -axis) and the vertical numberline (Y-axis), and defines a flat 2- dimensional area (a plane). It is properly known as the Cartesian Plane, named by René Descartes. You may be familiar with games that use a similar system, like Battleship. Battleship isn’t quite a Cartesian Plane, since it uses letters on the horizontal axis. A game that uses the cartesian plan better is Bug hunt – which requires you to find the shortest path (via “manhattan distance” – you don’t have to understand this, but it is interesting!)

But before we can get to playing with the algebra of lines on a cartesian plane, we need to master some simple algebra first – the algebra of expansion and factorisation. We need to be able to deal with distributive law – the rule for multiplying brackets (1, 2, 3) like this:

Divide & Conquer! (updated with factor theorem)

March 14, 2010

Divide and Conquer (Latin: Divide et Impera) is an aphorism meaning to split an enemies strength and defeat each part individually – in more modern times, the same concept is called “Defeat in Detail”. You need to apply the same tactic to your studies in Mathematical Methods – there is a great amount to learn, but each individual part is not that difficult. You need to be writing your bound reference to take advantage of these separate skills, organising them so that they are easy to find. Any student who doesn’t organise their bound reference is failing to plan (and thus, planning to fail…).

But this post is not merely about study tips – it is all about conquering the polynomial long division algorithm. When we are dealing with polynomial expressions of order higher than two (i.e. highest power term is greater than or equal to 3), such as a cubic or quartic we cannot use the general solution to the quadratic equation (in most situations). We need an alternative method to factorise these expressions – Polynomial Long Dvision is the way to deal with these tricky functions.

A Shocking Lesson

March 14, 2010

Static electricity is an electrifying experience! Sometimes, when you come in contact with a piece of metal, you can receive an electric shock, particularly if you are wearing rubber shoes (like sneakers or runners). Why?

Well, the reason is that you are storing an unbalanced charge. If you remember from our previous studies of the atom, most atoms have no net charge; they have the same number of protons as electrons, and thus have an overall charge of zero. How do you get an excess of charge? Well, that means you have more or less electrons than protons (you can’t gain more protons without changing your nuclear structure, so you must gain or lose electrons.)  and thus you are charge imbalanced. As soon as you touch (or even come close enough that a spark can jump to) another object, the charge will move so that it is more evenly balanced across the two objects. This is called static discharge and can be quite painful! In fact, the average human body can store approximately 3.5 kV of static charge!

“Elementary, my dear Watson”

March 3, 2010

Detective Sherlock Holmes was known for his great skills in searching for clues and joining the dots together to solve the mystery. A linear graph is exacltly that, plotting the points, and then joining the dots together to create a straight line. What would Sherlock Holmes do to figure out what a linear graph was, and how would he’d recognise if the graph was linear without actually drawing it?

A linear graph is simply a straight graph. To tell if an equation is linear by just looking at it, the feature you look at is that the “X” is not raised to an exponent higher than 1, so you will get the an equation in a form of y= mx + c