Wednesday, 29 October 2008

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.

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:
Answer to life, the universe, and everything42
Attendance at a sporting fixture21,988
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 = anxn + an-1xn-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.

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