Summary: General ideas about the broad project of
mathematics, motivated often by a desire for beauty and out
of curiousity. Curiousity being the search for truth
(which mathematicians believe exists, at least in the realm of pure
mathematics, although applications to the real world are fraught with
many assumptions and approximations), the way we go about this is by
simplifying as much as possible, and only accepting statements
and explanations which are absolutely crystal clear and have withstood
careful examination by the whole mathematical community.
Detailed contents:
We got curious about how tall were the rocks at Garden of the
Gods.
We had few tools to make measurements, but could at least use a
friend of known height and could pace off some (approximate)
measurements on the ground.
We made as simplified a diagram as possible, whereupon doodling
a bit, we noticed some things from basic geometry.
Definition: Two triangles are said to be similar if
there is a single number $k$ with the property that each side of
the second triangle is $k$ times as long as the correspond side of
the first triangle.
The way to think of similar triangles as that they look very much
the same, only one of them has been run through a zoom lens, so
segment in it is $k$ times as long as it used to be.
We can use similar triangles to learn something new as follows:
Say $\Delta ABC$ is similar to $\Delta DEF$ and
we know the lengths of the sides $\overline{AB}$ and
$\overline{DE}$.
The "zoom factor" must be
$k=\frac{length(\overline{DE})}{length(\overline{AB})}$ —
a number we can calculate from the lengths we know. So think of
this $k$ as a known number from now on.
The rations of the other side lengths must be equal to that
same zoom factor
$$\frac{length(\overline{EF})}{length(\overline{BC})}=k$$
If we know $length(\overline{BC})$, that means that we can
calculate $length(\overline{EF})$ by cross-multiplying that
last equation to get
$length(\overline{EF})=k\cdot length(\overline{BC})$, the
right-hand side of which is the product of two known numbers!
Theorem: If all three angles of one triangle equal the
corresponding angles of a second triangle, then the two triangles
are similar.
A theorem is a mathematical statement for which there is
a careful, complete explanation of why it is true. This explanation
is called a proof.
Theorem: The sum of the three angles in any triangle equals
$180^{\circ}$.
This theorem only applies to flat ["planar"] triangles, not, for
example, trangles written on the surface of a globe. It also tells
us that once we know two the values of two of the angles in a
triangle, the third one is forced to be whatever angle is left to
make the total equal $180^{\circ}$. This means, for example, that
we only need to match two paris of corresponding angles in
the above theorem about similar triangles, since the third triangle
for both triangles will fixed with the same value in both triangles.
Another thing we noticed in our simple diagram of GotG rocks and
assistant standing tall, was that the two triangles we drew both had
a right angle in them.
Definition: An angle is called a right angle if its
value is $90^{\circ}$. Equivalently, if the two rays which form the
sides of the angle are on two perpendicular lines.
Definition: A triangle is called a right triangle
if one of its angles is a right angle. The two sides of a right
triangle which meet at that right angle are called its legs,
while the third side, the one opposite the right angle, is called
the hypothenuse.
The most famous fact about right triangles, which was actually
already known hundreds of years before Pythagoras was born, is:
The Pythagorean Theorem: Suppose the sides of a triangle
have lengths $a$, $b$, and $c$. Suppose also that the angle
between the sides of lengths $a$ and $b$ is a right angle —
meaning that $a$ and $b$ are the lengths of the legs and $c$ is the
length of the hypothenuse of this triangle. Then
$$a^2+b^2=c^2$$
If you are every asked to state the Pythagorean Theorem, you have
state (almost) all of the above. If you just give the equation, as
many students do, the concerns are "what are those $a$, $b$, and
$c$?" and "you never mentioned that the triangle must have a
particular property [of being a right triangle]."
Notice that if you are going to use the Pythagorean Theorem to
solve a problem, you need to know the lengths of two of the sides
of a triangle — which must be a right triangle — and
then you can solve for the third side length.
In particular, for our question "How tall are the GotG rocks?"
we would need to know, maybe, the distance from the observer to the
highest point of the rock — which would be the length of the
hypothenuse of a right triangle in our diagram — and the
distance from the observer to the base of the rock — being
the length of one of the legs of the triangle. The hypothenuse
length is hard to get without something like a high-tech "laser
rangefinder."
Dr. Montoya pointed out that there is a right triangle whose
sides have lengths 3, 4, and 5. (This nice right triangle was
actually known to the Babylonians.) Your math instructor then
proposed a conjecture (a conjecture is a theorem that some
people think is true, but no one yet has been able to prove) that
this is the only right triangle whose sides are all whole
numbers.
We then noticed that we could multiply all the sides by the same
number (making a new triangle similar to the old one!) and
get other right triangles with sides of whole-number lengths,
such as 6, 8, 10 or 9, 12, 15. So a more modest conjecture might
be that all right triangles with all sides having whole number
lengths must be multiples of (similar to) the 3, 4, 5
example.
Unfortunately, this is not true, either. In fact, the famous
philosopher Plato (who did a little math; in fact, he spent a
little time in a Pythagorean community) described a way to
generate an infinite number of whole-number-sided right
triangles, non of which is simlar to any other one.
The French mathematician (and lawyer, actually) Pierre de
Fermat conjectured in 1637 that there are no whole number
solutions if we use cubes instead of squares in a Pythagorean
Theore-type equation. That is, he claimed that there are no
triples of whole numbers $a$, $b$, and $c$ for which
$a^3+b^3=c^3$ — or, in fact, for any power higher than 2!
— but he claimed not to have room for the proof in the
margin of the book where he wrote the claim. This is sometimes
called Fermat's Last Theorem, although it really should
have been called a conjecture for the next 357 years, at
the end of which the British mathematician Andrew Wiles (who was
then a professor at Princeton University) was able to find a
proof [building upon the work of dozens of other mathematicians].
Summary: A little review of yesterday, and more discussion of
mathematics as the search for beauty and truth (of a purely mental
kind). Starting to talk about lines,
Cartesian coordinates, equations of lines, and therefore
slope and $y$-intercept.
Detailed contents:
The hypothesis is the assumption you must make in a
theorem in order for the conclusion to be true. Many theorems
have the form "If $A$, then $B$," in which case $A$ is the
hypothesis and $B$ is the conclusion.
Definition: The points in the plane can be described by
Cartesian coordinates in the form $(x,y)$, where the $x$
describes how many units to move along the horizontal ($x$-) axis
and the $y$ tells how many units to move parallel to the vertical
($y$-) axis. The origin is the point with Cartesian
coordinates $(0,0)$ (so: right where the axes cross).
Starting with an equation which has the variables $x$ and $y$,
we say that the graph of that equation is the set of all
points in the plane whose Cartesian coordinates $(x,y)$ make the
equation true when plugged in.
Definition: An equation of the form $y=mx+b$ is an equation
for a line. The $m$ is the slope of the line and the $b$ is
the $y$-intercetp.
If a line has slope $m$, that means that anywhere along the
line, if you move 1 unit to the right (increase $x$ by 1), the
line moves $m$ units up (increases the $y$ by $m$). Or, if you
move (run) any distance, call it $run$, in the $x$ direction, the
line goes up (rises) by a distance $rise$, where
$$m=\frac{rise}{run}$$
In practice, this is often how we compute the $m$.
If a line has $y$-intercept $b$, that means that it passes
through the point $(0,b)$. In other words, if passes through
the $y$-axis $b$ units above (or below, if $b$ is negative) the
origin.
When you want to find out where two lines intersect, you are
looking for a point $(x,y)$ which satisfies both of the equations.
That means you write them both down, and solve them together for
first one variable and then the other.
Did an example of linear equations for cost of internet service
which had a basic price (which became the $y$-intercept) and
price per MB (which became the slope), for two different providers.
The intersection point told us how many MB would make the two
providers cost the same amount, while fewere MB would make one
provider cheaper and more MB would make the other cheaper.
Another example was of finding the height of the rocks at Garden
of the Gods (again!), by having choosing a coordinate system in
which to work -- meaning: choosing where the $x$- and $y$-axes
were, and where the origin was. Then, using these coordinates, we
wrote the equations of two different sight lines from the over
the top of the head of an assistant and to the top of the rocks,
with the assistant standing in two different places. The
point of intersection of those lines was the top of the rocks,
written in the coordinate system we had chosen from which we could
read off the hight of that point immediately.
No AM math, instead an extra joyful (and long)
Worksheet 3
:
Summary: Going over some issues in recent worksheets. Then
discussing exponential growth, exponential decay, and
some of the arithmetic of exponentiation.
Detailed contents:
Going over Worksheets 1-3.
Writing up your solution to a mathematical problem is about
telling a story, explaining to your reader what you are
doing at every step, why you want to do that, and why you know it
is OK to do that.
Exponential models as very like linear models, only with
multiplication by the base of the exponenet taking the role of
addition of the slope.
Introduction to arithmetic of exponentiation, such as by
repeated doubling — the function is $y=2^x$ — or
compounding interest — the function is $y=1.05^x$, if the
interest rate is 5%.
Summary: Some terminology, then applied problems with quadratic
equations.
Detailed contents:
The conjugate of the complext number $a+bi$ is $a-bi$.
A function is a formula $f(x)$ which has an input and an
output. You can graph a function $f(x)$ by plugging in lots of
numbers for $x$, getting output numbers $y=f(x)$, and putting a
point on the Cartesian plane at location $(x,y)$.
The graph of a quadratic function $y=f(x)=ax^2+bx+c$ is shaped
like a bowl (technically, it's a parabola) which points up
if $a>0$ and down if $a<0$.
The maximum or minimum of a parabola is at the $x$-coordinate
halfway between any two $x$ values whose corresponding $y$ is the
same.
Examples of solving and maximizing quadratic functions from
economics, art, and farming.
Here are some practice problems to
get ready for the quiz tomorrow.
:
Math Quiz 3, on the material from this week,
here is a review sheet for the quiz.
below, it means that math class was videoed and you can get a link by e-mailing
me.