Introducing polynomials

Below is my first draft of my section introduction. Hopefully it introduces polynomials from a perspective that everyone can understand. I welcome any comments from my fellow authors.

Polynomials

We have previously, on page XX (fragment 24), introduced some basic algebra and straight line graph plotting. In this section we will introduce some simple mathematical curves, and in doing so, show that all numbers may be represented by a point on a given curve or curves.

We will start by looking at some every day numbers, and show how they can be represented by algebraic notation. If you are already familiar with such concepts, you may wish to skip this section.

From numbers to algebra

We all use numbers in our daily lives. Some examples of numbers we use are:

Height	1.54m
Weight	53kg
Price	£4.99
Answer to life, the universe, and everything	42
Attendance at a sporting fixture	21,988
The units of measurement are unimportant for this exercise.

Most of the time we give very little thought as to what these numbers mean from a mathematical perspective.

What does a number like 21,988 actually mean?

Firstly, we know that it is a count of the number of people who attended a sporting fixture. Counting is one of the first skills we learn, often using our fingers. It is no coincidence that our counting system revolves around the number of fingers, or digits, that we have. A count of something is a one-to-one correspondence between the number and the physical thing.

We can break 21,988 down into its constituent parts:

21,988 = 20,000 + 1,000 + 900 + 80 + 8

We call each of the numbers being added together in this equation terms.

We can break each term down a little further:

21,988 = (2 × 10,000) + (1 × 1,000) + (9 × 100) + (8 × 10) + (8 × 1)

The brackets, or parenthesis, around the 2 × 10,000 means "do this calculation first".

100 is 10 × 10, or 10 squared. We can also write this as 10². Likewise, 1,000 is 10 × 10 × 10, or 10 cubed. We can also write this as 10³. We call these powers of 10. For consistency, we can extend this logic to all other powers of 10, although, some may not be obvious, for instance 10¹ is 10, and 10⁰ is 1.

We can now rewrite our breakdown of 21,988:

21,988 = (2 × 10⁴) + (1 × 10³) + (9 × 10²) + (8 × 10¹) + (8 × 10⁰)

This example is very specific to the number 21,988. What if we want to write a generic formula for any number?

Firstly, we can replace all instances of 10 with x, and 21,988, with y:

y = (2 × x⁴) + (1 × x³) + (9 × x²) + (8 × x¹) + (8 × x⁰)

We now need to replace the digits 2, 1, 9, 8 and 8. We will replace them with an a, but as they are all different, we need to distinguish between them. We can do this by using a sub-scripted number, which for clarity, will be the same as the power of x. We call these coefficients, and we call the number attached to the coefficient an index:

y = (a₄ × x⁴) + (a₃ × x³) + (a₂ × x²) + (a₁ × x¹) + (a₀ × x⁰)

We can tidy this equation up a bit. We stated above that 10¹ is 10, and 10⁰ is 1. We can be more generic and state that x¹ is x, and x⁰ is 1. This simplifies the last two terms of the equation to:

(a₁ × x) + a₀

We can also remove the ×, by either using a . in its place (i.e. a₁.x), or, as we will do here, just implying multiplication. Because multiplication has a higher precedence than addition, we can also remove the parenthesis. Applying this to the entire equation then gives:

y = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀

This is fine, for a number of 5 digits or less (i.e. a₄ = 0 cancels out x⁴, and hence will give a 4 digit number), but what about numbers consisting of an arbitrary number of digits? We can replace the index number of most significant term with n, and the index number of the next most significant term with n-1:

y = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀

We call this type of equation, a polynomial, where poly means many, and nomial means term.

We call the most significant index the degree of the polynomial. In the last equation, the degree is n, and in the 21,988 example, the degree is 4.

It will be noted, that because our least significant index number is 0 (that is we count from 0 and not 1), n will always be 1 less than the maximum number of digits in the number.

We now have an abstract representation of any number.

To bring all of this full-circle, if we set x to 10, a₄ to 2, a₃ to 1, a₂ to 9, a₁ to 8 and a₀ to 8, we get:

y = (2 × 10⁴) + 10³ + (9 × 10²) + (8 × 10) + 8

In our 21,988 example, x was always 10. This does not always have to be the case. On page XX (fragment 77), when we introduce bits, we will be dealing with powers of 2 and 16.

We shall now look at a number of different polynomial equations, by plotting the resultant value of y for various values of x.

Formatting of code fragments on blogs

I have just left a comment on Darrel's blog regarding formating of code fragments within a blog. Unfortuantly, the blog commenting facility doesn't allow use of the HTML pre tag, and hence the point I was trying to make was somewhat lost! For clarity, I repeat it here.

Use of the HTML pre tag leaves white space intact, allows direct use of > and <, and uses a courier font. E.g.:

setup(){
  int x = 0;
  if (x > 0){
    doSomething();
  }
}

Notation, notation, notation

Upon re-reading my last post, it occurred to me that the notation chosen was likely to be confusing.

To illustrate, we will take the 4 equations previously stated, and line up all of the terms:

Constant	y =							a
Linear	y =					ax	+	b
Quadratic	y =			ax2	+	bx	+	c
Cubic	y =	ax3	+	bx2	+	cx	+	d

It will be noted that the name of equivalent coefficients changes in each equation.

If we were to use a generic form for the equation, then this name changing goes away:

y = a_nxⁿ + a_n-1x^n-1 + ... + a₃x³ + a₂x² + a₁x + a₀

Whilst this looks more complex, if we just concentrate on equations of the cubic order and lower, we get:

y = a₃x³ + a₂x² + a₁x + a₀

If a₃ is 0, then we get a quadratic equation, and so on:

y = a₂x² + a₁x + a₀

I think this is potentially less confusing than having the coefficient names change with each order of equation. It also provides for a cleaner and clearer mapping between the underlying mathematics and Java arrays, if we were to chose that mechanism to store the coefficients.

Initial thoughts

Most of the chunks that touch on mathematical subjects are going to be challenging to write. Our task is to write a book that demonstrates how the various facilities in the Processing language can be used to create pictures. Out task is not to teach mathematics. That said, it is difficult to use the various facilities effectively if one does not have some basic knowledge as to the underlying theory. Getting this balance right is potentially one of the biggest challenges we will face.

Obviously I need to start with what a polynomial is, and what they look like, in terms of their generic form. I'd like to use the old joke that a polynomial is a hungry parrot, but I suspect that's a bit too Pythonesque for most people!

Being restricted to quadratic and cubic equations simplifies the task slightly. I can see these equations being a little scary to some:

• y = ax² + bx + c (quadratic)
• y = ax³ + bx² + cx + d (cubic)

I will probably start off with constants and straight lines, and build up from there:

• y = a (constant)
• y = ax + b (linear)

Apart from showing the effects of altering the various coefficients (a, b, c and d), and a very brief explanation of power notation, I can see little to be gained by going into the maths any further.

I would like, where ever possible, to highlight real world examples of these curves. For example, the parabola, as described by a quadratic equation, is formed as the result of a conic section, and is also used for reflectors with a focal-point, such as satellite dishes and light reflectors.

Whilst waiting for my book to arrive, I will do some background reading.

Following other mass writing blogs

I've noticed that other authors are already following this blog. So far, I am only following the 3 blogs created by Darrel. Trying to follow 85 blogs is likely to be virtually impossible. I will therefore only follow those blogs written by authors whose subject matter is a prerequisite for my chuck. I hope that the other authors won't be upset by this decision.

Polynomials

I sent an e-mail to Darrel earlier today listing my 3 preferred chunks to write. I got my third choice, which is fine. So, my brief is:

CHUNK 55
TITLE Polynomials
DESCRIPTION Describe the general form of quadratic and cubic polynomials and show how they can be drawn using the techniques in Greenberg 262--267. Describe a simple program that shows the various curves that are generated by such polynomials
OUTCOME

• Describe the general form of a polynomial
• Develop code that draws a particular polynomial.

REFERENCE Greenberg 262-267
HINT Keep the programs simple.
PROGRAM No program required.

Book availability, or rather lack thereof

Just popped into my local Waterstone's to order a copy of Ira Greenberg's book, only to be told that is currently reprinting, and won't be available until the end of November. Waterstone's have just 3 copies in the country, so if your local store is Kingston University, Scarborough or Worcester, you may be in luck.

There are 2 suppliers on eBay with over 30 copies between them (here and here), but I've just ordered a copy from Amazon for £28 including p&p.

Please note that I am not connected in any way to any of the suppliers mentioned above.

In the beginning ...

... I, along with many other Open University students, was sent an e-mail by Professor Darrel Ince seeking volunteers to join a mass writing project. The aim? Very simple! Take approximately 85 individuals, who prior to signing up for this project, in the main, are unknown to each other, and get them to write a book about Processing, a Java-based computer-art environment.

OK, so why am I doing this?

Well, a number of reasons:

1. It is a good opportunity to explore new ways of working; collaborating with individuals mainly via the Internet has many potential advantages and disadvantages. This is an ideal opportunity to gain experience first hand.
2. To get used to academic writing. I will (hopefully) be writing a dissertation for my MSc in a few years time, so any writing experience I can get in the mean time will be useful. I just need to finish the PGDip first!
3. To attempt to explain technical and mathematical subject matter in manner that those with little or no technical or mathematical knowledge can understand. Techies are notoriously bad at being able to explain themselves in simple language, and I include myself in that.
   
4. To learn a new technology; I had never even heard of Processing prior to this project.
5. I am hoping to tie this project in to the next unit of my PGDip, which is Project management. I need to e-mail my course tutor to check if he deems this to be a suitable project, in terms of the course requirements and objectives.
   
6. To have some fun!

So, what next?

1. Download, install and play with the Processing environment. Done.
   
2. Order Ira Greenberg's book that the OU is very kindly buying for each of us. Done.
   
3. Choose a chunk to write. Done.
   

Obviously, there is loads of other stuff to do (like write some words and code), but the above will keep my busy for a while.