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Occasionally, it might be useful to do as the title here suggests.
Two examples that spring to mind include:
- solving
for
(relative velocity example with
below)
- maximising
without the use of calculus
Note first of all the similarity between:
.
This identity is in the Department of Education formula booklet.
The only problem is that and
are not necessarily sines and cosines respectively. Consider them, however, as opposites and adjacents to an angle in a right-angled-triangle as shown:
Using Pythagoras Theorem, the hypotenuse is and so if we multiply our expression by
then we have something:
.
Similarly, we have
,
where .
Last semester, teaching some maths to engineers, I decided to play (via email) the Guess 2/3 of the Average game. Two players won (with guesses of 22) and so I needed a tie breaker.
I came up with a hybrid of Monty Hall, not too dissimilar to the game below (the prize was €5).
Rules
In this game, the host presents four doors to the players Alice and Bob:
Behind three of the doors is an empty box, and behind one of the doors is €100.
The host flips a coin and asks the Alice would she like heads or tails. If she is correct, she gets to choose whether to go first or second.
The player that goes first picks a door, then the second player gets a turn, picking a different door.
Then the host opens a door revealing an empty box.
Now the first player has a choice to stay or switch.
The second player then has a choice to stay or switch (the second player can go to where the first player was if the first player switches).
Questions:
- What is the best strategy for the player who goes second:
- if the first player switches?
- if the first player stays?
- Should the person who wins the toss choose to go first or second? What assumptions did you make?
- How much would you pay to play this game? What assumptions did you make?
- If there is a bonus for playing second, how much should the bonus be such that the answer to question 1. is “it doesn’t matter”.
Correlation does not imply causation is a mantra of modern data science. It is probably worthwhile at this point to define the terms correlation, imply, and (harder) causation.
Correlation
For the purposes of this piece, it is sufficient to say that if we measure and record values of variables and
, and they appear to have a straight-line relationship, then the correlation is a measure of how close the data is to being on a straight line. For example, consider the following data:
The variables and
have a strong correlation.
Causation
Causality is a deep philosophical notion, but, for the purposes of this piece, if there is a relationship between variables and
such that for each value of
there is a single value of
, then we say that
is a function of
:
is the cause and
is the effect.
In this case, we write , said
is a function of
. This is a causal relationship between
and
. (As an example which shows why this definition is only useful for the purposes of this piece, is the relationship between sales
days after January 1, and the sales,
, on that day: for each value of
there is a single value of
: indeed
is a function of
, but
does not cause
).
This follows on from this post.
Recall the Doubling Mapping given by:
At the end of the last post we showed that this dynamical system displays sensitivity to initial conditions. Now we show that it displays topological mixing (a chaotic orbit) and density of periodic points.
First we must talk about periodic points.
Periodic Points
Consider, for example, the initial state . The orbit of
is given by:
Here we see repeats itself and so gets ‘stuck’ in a repeating pattern:
The orbit of .
The orbit of any fraction, e.g. , must be periodic, because
is either equal to
of
and so the orbit consists only of states of the form:
,
and there are only 243 of these and so after 244 iterations, some state must be repeated and so we get locked into a periodic cycle.
If we accept the following:
Proposition
A fraction has a recurring binary expansion:
,
then this is another way to see that fractions are (eventually) periodic. Take for example,
.
Dynamical Systems
A dynamical system is a set of states together with an iterator function
which is used to determine the next state of a system in terms of the previous state. For example, if
is the initial state, the subsequent states are given by:
,
,
and in general, the next state is got by applying the iterator function:
.
The sequence of states
is known as the orbit of and the
are known as the iterates.
Such dynamical systems are completely deterministic: if you know the state at any time you know it at all subsequent times. Also, if a state is repeated, for example:
then the orbit is destined to repeated forever because
,
, etc:
Example: Savings
Suppose you save in a bank, where monthly you receive interest and you throw in
per month, starting on the day you open the account.
This can be modeled as a dynamical system.
Let be the set of euro amounts. The initial amount of savings is
. After one month you get interest on this:
, you still have your original
and you are depositing a further €50, so the state of your savings, after one month, is given by:
.
Now, in the second month, there is interest on all this:
interest in second month ,
we also have the from the previous month and we are throwing in an extra €50 so now the state of your savings, after two months, is:
,
and it shouldn’t be too difficult to see that how you get from is by applying the function:
.
Exercise
Use geometric series to find a formula for .
Weather
If quantum effects are neglected, then weather is a deterministic system. This means that if we know the exact state of the weather at a certain instant (we can even think of the state of the universe – variations in the sun affecting the weather, etc), then we can calculate the state of the weather at all subsequent times.
This means that if we know everything about the state of the weather today at 12 noon, then we know what the weather will be at 12 noon tomorrow…
In school, we learn how a line has an equation… and a circle has an equation… what does this mean?
The short answer is
points
on curve
solutions
of equation
however this note explains all of this from first principles, with a particular emphasis on the set-theoretic fundamentals.
Set Theory
A set is a collection of objects. The objects of a set are referred to as the elements or members and if we can list the elements we include them in curly-brackets. For example, call by the set of whole numbers (strictly) between two and nine. This set is denoted by
.
We indicate that an object is an element of a set
by writing
, said,
in
or
is an element of
. We use the symbol
to indicate non-membership. For example,
.
Elements are not duplicated and the order doesn’t matter. For example:
.
This post follows on from this post where the following principle was presented:
Fundamental Principle of Solving ‘Easy’ Equations
Identify what is difficult or troublesome about the equation and get rid of it. As long as you do the same thing to both numbers (the “Lhs” and the “Rhs”), the equation will be replaced by a simpler equation with the same solution.
There are a number of subtleties here: basically sometimes you get extra ‘solutions’ (that are not solutions at all), and sometimes you can lose solutions.
Let us write the squaring function, e.g. ,
by
and the square-rooting function by
. It appears that
are an inverse pair but not quite exactly. While
and
,
check out O.K. note that
,
does not bring us back to where we started.
This problem can be fixed by restricting the allowable inputs to to positive numbers only but for the moment it is better to just treat this as a subtlety, namely while
,
… in fact I recommend that we remember that with an
there will generally be two solutions.
The other thing we look out for as much as possible is that we cannot divide by zero.
There are other issues around such as the fact that , so that the equation
has no solutions (no,
is not a solution! Check.). This equation has no solutions.
Often, in context, these subtleties are not problematic. For example, equations with no solutions rarely arise and quantities might be positive so that if we have , only
need be considered (for example,
might be a length).
In this short note we will explain why we multiply matrices in this “rows-by-columns” fashion. This note will only look at matrices but it should be clear, particularly by looking at this note, how this generalises to matrices of arbitrary size.
First of all we need some objects. Consider the plane . By fixing an origin, orientation (
– and
-directions), and scale, each point
can be associated with an ordered pair
, where
is the distance along the
axis and
is the distance along the
axis. For the purposes of linear algebra we denote this point
by
.
We have two basic operations with points in the plane. We can add them together and we can scalar multiply them according to, if and
:
, and
.
Objects in mathematics that can be added together and scalar-multiplied are said to be vectors. Sets of vectors are known as vector spaces and a feature of vector spaces is that all vectors can be written in a unique way as a sum of basic vectors.
A nice little question:
Given a regular pentagon with side length , what is the relationship between the area and the side-length?
First of all a pentagon:
We use triangulation to cut it into a number of triangles:
With in each of the three triangles, there
in those angles around the edges, and, as there are five of them, they are each
.
Next triangulate from the centre. With a plain oul pentagon we might not be sure that such a centre exists but if you start with a circle and inscribe five equidistant points along the circle, the centre of the circle serves as this centre:
As everything is symmetric, each of these triangles are the same and the ‘rays’ are also the same as they are all radii. The angle at the centre is equal to , and furthermore, by symmetry, the rays bisect the larger angles
and so each of these triangles are
.
Using radians, because they are nicer, . Note that, where
is the perpendicular height:
.
A problem for another day is finding the exact value of . It is
Therefore the area of one such triangle is:
,
Therefore the area of the pentagon is five times this:
,
with . It might be possible to simply
further.
Quadratics are ubiquitous in mathematics. For the purposes of this piece a quadratic is a real-valued function of the form
,
where such that
. There is a little bit more to be said — particularly about the differences between a quadratic and a quadratic function but for those this piece is addressed to (third level: non-maths; all second level), the distinction is unimportant.
Geometry
The basic object we study is the square function, ,
:
All quadratics look similar to . If
then the quadratic has this
geometry. Otherwise it looks like
and has
geometry
The geometry dictates that quadratics can have either zero, one or two real roots. A root of a function is an input such that
. As the graph of a function is of the form
, roots are such that
, that is where the graph cuts the
-axis. With the geometry of quadratics they can cut the
-axis no times, once (like
), or twice.
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