The rules of the game...
The lessons in this course are organized around a series of illustrations. The
text describes what the image is intended to convey. Included in the text you
will find links to the interactive displays from which the images were taken.
This arrangement allows the pages to be rendered on the screen rather quickly,
with the gist of the story intact. It is the interactive displays though, that
make this online material different than a printed text. I urge you to take the
time to follow each link on the pages and play around with the displays. It is
there that much of the learning takes place. At the bottom of each page, with
the site navigation links, is a link to a glossary to help with unfamiliar terms.
In due course we will discover exactly what the image at the left is and how
it was made. This image was actually created in an interactive display
that comes up in the offfline version of the Order course. The online
version has its limitations.
Let's begin this journey
together more or less at the beginning with a quick review of
numbers and arithmetic. Certain numbers are called
"real", which seems to mean that there are others,
possibly unreal. Real numbers may be thought of as lying along a
straight line with zero in the middle and extending as far as
imaginable to the right for positive numbers and as far as
imaginable to the left for negative numbers. Included are whole
numbers (integers), fractions (rational numbers) and decimals not
expressible as fractions (irrational numbers).
The image at the right illustrates a
short segment of the real number line, between -20 and +20. The
indicated example here is an irrational number. Click on the link for the
Real Number Line
interactive display from which this example was taken.
Real numbers may be added, subtracted, multiplied and divided
in the ordinary way. In this program we will use the following
symbols for mathematical operations. For addition +, for
subtraction - , for multiplication *, and for division /. Raising
numbers to powers, sometimes called exponentiation, we place the
exponent above the number and to the right, like 34,
or we use the ^ symbol like 3^4. Both mean 3 multiplied by itself
4 times. Numbers may be raised to non-integer powers like
2.70.5. This is the square root of 2.7. Applying any
combination of addition, subtraction, multiplication and division
to real numbers results in another real number. Raising real
numbers to powers also gives a real result in many cases. We will
cover the interesting exception later.
Sometimes mathematicians like to make up rules that produce
one number from another, like take a real number and divide it by
two to get another real number. Or take a number, multiply it by
itself and add a second number to it to get another number. The
possibilities are nearly endless. In this program we will deal
with some of these rules for combining numbers which in
themselves are fairly simple but which under the conditions we
will explore, produce wonderfully complicated results. Don't
worry, the complications are wonderful but not fearful.
This making up of rules is more than a means of filling the
idle hours. It turns out that many physical phenomena may be
described by rules like this. Rules of the sort we cover here are
called "functions". The number produced is said to be a
function of the number "taken" at the beginning. In
particular these are called a single valued functions. Each
number in, gives one number out. The "taken" number is
called the "independent variable" the produced number
the "dependent variable". We talk about a function as
"returning" the dependent variable.
Frequently the independent variable is called "x"
and the dependent variable called "y". We will refer to
x and y in that in the rest of this discussion. Some functions
have restrictions on the allowed values of x. The set of all
values of x which are allowed for the function is called the
"domain" of the function. The set of all values of y
which the function returns is called the "range" of the
function. The word "set" as we will use it just means a
collection of related objects or numbers. The mathematical
definition is very close to ordinary usage.
Functions of the sort we are talking about may be written out in
mathematical terms as an equation or formula. For ease in reading
we will use a distinctive color for mathematical expressions. For
is such a function. This says that to get a value for y take any
x and add 2 to it. You can see that the function relates a y to
every x. The number 2 in the function is neither the independent
variable nor the dependent variable. Numbers like this in
functions are known as "parameters". If added in the
function the parameter is called a "constant", if
multiplied in the functions it is called a
"coefficient". You might imagine a function to be a
"black box" into which you put values of the
independent variable and out of which pop values of the dependent
Function Black Box
display illustrates the idea for the function
y = x*(10-x) .
In addition to the arithmetic operations of addition,
subtraction, multiplication, division and exponentiation; there
are other defined functions that operate on real numbers. In
particular there are two, which we will use in our examples later.
The sine function, symbolized as
takes any real number and returns a number between 1 and -1. The
symbol pi (said pie) is the pi of "pie are square"
fame. The exponential function, symbolized as
takes any real number and returns a positive real number.
Specific use of these functions is covered later.
Sometimes it is useful to look at the way
that y depends on x by plotting the relationship on a graph. We
plot the x value along the horizontal direction, called the
"x-axis". For each value of x the corresponding value
of y is plotted over x in the vertical direction, called the
"y-axis". The series of y locations traces out a curve
which represents the function. Recognize that any graph
represents only a sampling of points covered by the function. It
is an accurate representation of the function only if the
function has no bizarre behavior between sample points.
display we will
demonstrate how a function may be represented by a graph.
For purposes of introducing the graphing concept, marking the
y values with the actual number was OK, but as a practical matter
we will not want our graphing illustrations peppered with
numbers. We could replace the y numbers with some more convenient
marker and move the values to the y-axis as we did with the x
values. If this were a paper graph that is what we would do. Then
to determine the x and y values called "coordinates" of
any point on our graph we would have to estimate the values from
the axis scale.
Fortunately we are going to work with
computer graphs rather than paper ones. On the computer screen we
will take advantage of the cursor provided to us by our operating
system. The coordinates of the point on our graph pointed to by
the cursor can be displayed and updated as the cursor is moved.
The x and y values are displayed in the bottom margin.
display to see
the rudimentary graph produced in the Graphing display replaced
with a better version.
We draw graphs by selecting a few points along the x-axis,
calculate the corresponding y values, plot those points and then
connect the dots to fill in all the other uncalculated points
representing the function. What we assume in doing this is that
the function is fairly smooth and behaves on a small scale pretty
much the same way it does on the larger scale covered by the
points which were calculated.
There are three functions that we will use later to demonstrate
some fundamental principles in this program. There are many which
could be used but these meet the requirements and are not too
complicated. One is called a quadratic function. It is
spoken: y equals gain times x times, 1 minus x. In symbols understandable
to computers and programmers it is written:
This function is known as the "logistic" equation. It
might be used to represent the growth of an animal population in
the presence of a limited food supply.
The symbols x and y are the independent and dependent
variables as we mentioned previously. The parameter g which we
identified earlier as a coefficient, is just a multiplier. We use
the symbol g and call it the "gain" because the action
of this parameter on the function is similar to that of the term
known as gain in electronic circuits. Rather small variations in
the gain will produce radically different behavior of the
function under certain conditions as we will see. This particular
function is prominent in the mathematics of chaos.
Another function we use is called sinusoidal. It is spoken:
y equals gain times the sine of pi times x
Written it is:
We have already met x, y and g. The symbol pi is the same as
the as ratio of the circumference to the diameter of a circle,
about 3.1415926. The symbol sin stands for the sine function
which converts the angle represented by pi*x into a number
between -1 and 1 according to certain rules. It is from this
symbol that the sinusoidal function gets its name.
The last function is a little more complicated in form but is
commonly used in statistics and all sciences. The name is the
Gaussian function, after Mr. Gauss no doubt. It is the famous
bell shaped curve that describes the statistical distribution of
just about everything. This one is said:
y equals g times e to the minus, x minus 1, squared.
The 2 means that the stuff in parentheses just before
it is raised to the power 2, in other words multiplied by itself
one time or "squared". The symbol exp( ) represents the
exponential function in which one raises the number e,
2.7182818284..., to the power equal to the stuff in its
parentheses. These three functions have something in common.
Their graphs have one or more high spots with a low value on
either side of the high. This makes them suitable for the
demonstration of the principles we want to explore.
You may show the three functions in turn, in the next display.
The functions equation will be displayed along with the graph.
The parameter g is set to a value of 4.0 in quadratic and
Gaussian examples and to 1.0 in the sinusoidal case. We will have
an opportunity to vary it later. The graph displayed is drawn as
a continuous line as we discussed. It is the curve we think we
would get if we took every possible value of x between our chosen
limits, calculated the corresponding y and plotted the points. In
fact the computer plots it by dividing the display area
into steps whose number is equal to the number of pixels in the
width of the drawing area and plotting that many points.
Also plotted on the graph with the
selected function is a straight line representing the set of all
points on the graph where
. It is
included here to demonstrate the fact that two independent
functions may be displayed on the same graph and to show one way
to find the points on the graph which fulfill the conditions of
both functions. Use the cursor to find approximately the x and y
values where the selected function and the function
both hold true. We say these points satisfy
the simultaneous equations represented by the two graphs. Play
around with these
This concludes the Numbers, Functions and Graphs discussion. The
next topic is called Extending Graphing Concepts.
To view this lesson click on the Next link below.
Are there any questions?