The
official publication of the New Mathwright Library
and Café:
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you wish to unsubscribe to this Newsletter, just reply to this message with
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1] New subscription and membership policy in the Library
2] Some new WorkBooks
A
full-year subscription and membership in the Library is now only $15.00 (USD).
Three-year subscriptions are also available for $35.00. Once you subscribe,
you may download your free Mathwright Player 2000 at the Library
Store using a password that you receive online. After that, you may download
any WorkBooks in the Stacks during the period of your subscription. WorkBooks
in the 3D Gallery may be downloaded with or without a subscription, but you
will need Mathwright PlayScapes to read them. When your subscription period
expires, you will be reminded to renew it at the Library Store only if you try
to download a WorkBook after that. Until that time, there are no restrictions
on what WorkBooks you may download or when you may download them.
We
have added 37 new WorkBooks to the Library (including a new College
Algebra Course) in January, 2001. You may read about each of these WorkBooks
if you go to What's New?. Here, we highlight
a few of those WorkBooks, along with another recent WorkBook called "Cubic
Equations."
We
usually learn in High School how to solve quadratic equations, both by formula,
and by a method called "completing the square." And we may suspect that there
is a similar formula and method for solving "cubic equations." There is such
a formula, and you may find it on page 4 of this 12-page WorkBook. If we reproduced
it here it would require 7 or 8 full lines of text, and it would be very difficult
to interpret in your word processor. It is a long formula. No wonder we don't
learn that formula in High School. Most of us don't learn it in college, either!
Now
it happens that there is no need to learn that formula. Gerolimo Cardano and
Nicolo Tartaglia developed in the 16th century, a method for solving cubic equations
that is remarkably efficient and effective, and is equally mysterious. In our
WorkBook we show that there is a general procedure that may be used to solve
quadratic, cubic, and even quartic equations which places them all in a unified
context.
This
procedure relies on the use of complex numbers, in particular, on the geometry
and algebra of the roots of unity. It discloses an interesting relationship
between polynomial equations and roots of unity, and it leads to a very easy
way to learn (and remember) how to solve cubic and quartic equations.
The
WorkBook makes use of a system of numbers that is somewhat larger than the complex
numbers. We call that system of numbers: Cubic Numbers, and the WorkBook is
full of experiments that develop the properties of those numbers both geometrically
and algebraically. Using these numbers, Cardano's method (which the WorkBook
illustrates by applying it, step-by-step, to solve cubic equations that you
propose) becomes completely transparent. Certain geometric symmetries concerning
the real roots of cubics with real coefficients come to the surface in experiments
in the WorkBook that are by no means apparent from Cardano's method.
This
WorkBook summarizes and illustrates the work in a recently published paper by
Jim White and Dan Kalman.
This
WorkBook is designed to assist the beginner to learn how to solve linear equations
in one variable. These equations may contain any number of variables, but of
course, the reader must specify one of them to be solved in terms of the others.
Fractions are written in the usual form: 2/3, for example. It generates linear
equations or allows the reader to make them up.
But
it does not tell her the answer. Instead, it follows her steps as she transforms
and simplifies the equation and tells her whether each step is correct, or whether
it is mistaken. In fact, if she makes an incorrect transformation, the WorkBook
gently points that out, and if the transformation is correct, it congratulates
her.
At
any time, the learner may ask "advice" from the Assistant, and it will suggest
a next goal to try. Thus if she got stuck trying to solve for X in the equation:
2X + 7 = 5, it might suggest: "Try to get 2X on one side by itself" but would
not tell her how to do it.
There
are five levels of study. In each level, the WorkBook creates problems for the
learner and then allows the student To attempt solutions. The student attempts,
step-by-step, to transform the equation until the variable is alone on one side.
The Assistant also keeps track of the student's progress in the session, and
when she "checks out" it prepares a table and a bar graph showing how she did.
She may save these to disk and recall them later to gauge her progress as she
moves through the levels of the WorkBook.
This
course, consisting of 43 interactive pages, and taught for college credit as
a distance learning course by its author, Samad Mortabit, contains 7 WorkBooks
entitled:
For
example, Book 6, Exploring Quadratic Functions,
while improved and expanded here, was a great favorite in the earlier Library.
That WorkBook is an introduction to quadratic functions, and quadratic growth.
It is full of experiments that will help readers understand the ideas, beginning
with Galileo's law of freely falling bodies. Other experiments include tossing
a ball up with various velocities, and applying the brakes to stop a car moving
at various speeds. Different representations for quadratic functions are explored,
and the relations of those representations to their graphs is developed through
interaction. Solving quadratic equations both by graphing, and algebraically
are explored next. For this, it generates quadratic equations randomly, or allows
readers to make them up. In either case, it draws the graph and so displays
the solution graphically, then explains, step-by-step, how to solve the quadratic
equation algebraically. It does this for readers' problems also!
This
is followed by a similar treatment of quadratic inequalities. Whether readers
use an inequality it generates, or they make one up, it graphs the inequality,
and shows which interval(s) in the line provide(s) the solution, by drawing
that interval(s). Next, it gives a step-by-step explanation of how the inequality
can be solved algebraically.
You
may download the entire course at once, but then you have to extract each lab
in the directory created for it, and There is no convenient button on the Start
menu to open each one separately. It is probably better to download each lab
separately.
Consider
a cubic polynomial equation that has three distinct and non-collinear complex
roots. As a specific case, consider the polynomial equation p(z) = (z-a)(z-b)(z-c)
= 0, where a, b, and c are fixed complex numbers, and z is a complex variable.
We
know that p(z) = 0 has roots at a, b, and c, which we visualize as three points
in the complex plane. If those points are not collinear, they define a triangle
within which it is possible to inscribe a unique ellipse, tangent to the sides
at their midpoints. Marden's theorem says that the foci of the ellipse are precisely
the roots of the derivative, p'(z).
In
this WorkBook, you may experiment with this, and a more general version of Marden's
Theorem. And you may review properties of complex numbers and the complex plane,
both visually and algebraically. This WorkBook will also give you the opportunity
to explore the geometry of ellipses inscribed in triangles. This is a "hands
on" opportunity to investigate some beautiful correlations between algebra and
geometry.
At
every step, the book invites you to ask questions, and to see for yourself what
the answers to those questions are. You will enjoy this piece of work by Mathwright
author Dan Kalman.
Space
Shuttle is a 12-page Interactive Mathematics/Science WorkBook that celebrates
the role of mathematics in Rocket Science with multimedia and vivid sprite animation.
It
is designed to be used by students ages 14 and older for private self-directed
study and recreation. Students learn the 'Rocket Equation' that correlates such
factors as payload weight, fuel weight and type, fuel specific impuse, and gravitational
acceleration into the determination of a Rocket's performance.
The
first half of the WorkBook consists of a series of experiments and calculations
designed to place the shuttle in orbit. The shuttle must attain an eventual
speed of 0.7*(orbital velocity) or roughly 4.5 km/sec. at a height of about
226 Km above the surface of the Earth.
After
a two-minute burn, the SRBs (Solid Rocket Boosters) are discarded, and the remaining
climb to orbital velocity at 7.75 Km/sec at 250 Km is done by hand using the
Shuttle engines. That is where the fun begins. Two interactive simulations illustrate
this task. The objective is to attain nearly circular orbit, and that requires
a balance of centripetal acceleration and gravity. The first simulation guides
the reader to balance these forces by displaying, at each instant of flight,
the desired velocity and attitude of the shuttle. This gives a real-time simulation
from the viewpoint of the pilot in the cockpit, who sees the Earth move below
him as he adjusts the attitude and velocity of the craft. If he did not succeed
in solving the Rocket Equation correctly, and cannot muster enough thrust, he
will inevitably come hurtling back to Earth in a simulated crash and burn. What
better motivation to 'do the math' !
The
next simulation is quite challenging. The player must "dock" with a satellite
that is in circular orbit. During this Exercise, he watches the satellite as
he approaches it and attempts to match his velocity with the satellite. This
is not easy! The simulations display in an almost realistic way the difficulties
in doing this by hand. But they are also quite entertaining, and in the end,
challenging to try.
James E.
White, Ph.D.
Library
Director