1. Digestable Higher Mathematics
David Alan Rogers
March 23, 2011
Mathematics is an underrated subject. I hope to show that it isn’t just about
adding and subtracting numbers. Mathematics is a rigorous and formal logic, and as
such, it’s applications are limitless. With each section we explore (mostly in brevity)
I will provide further readings from my book collection for anyone interested. All the
books I recommend will be affordable, AND GOOD (in my humble opinion).
1 An Introduction to Set Theory
In day to day life we often find it convenient to compose lists of related objects. A
biologist categorizes various species in terms of their kinship with other animals.
For instance, the list (in mathematics we use the word “set”) of mammals;
..., Dog s,C at s, Humans, Monke y s, ...
.
In mathematics, we define a set S in terms of its “elements” (the stuff in the list).
Let us take for example, the set of integers.
Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}
We may point to any element of the set (1 for example) and say: “1 is an element
of the set Z.” To streamline mathematical statements, we adopt a short hand: 1 Z.
Where the symbol epsilon “ ” is taken to mean “is an element of the set.” Suppose
we wish to note several elements in the set Z. We then introduce the notion of a
subset. Let’s take a set; N = {1, 2, 3, ...}. Because the set N = {1, 2, 3, ...} is contained
within Z, we are then at liberty to write Z as Z = {..., −3, −2, −1, 0, N}. And therefore,
N itself is an element of Z. We now say that N is a subset of Z notationally with the
shorthand; N ⊂ Z.
In introductory calculus we explore topics in “real analysis.” As such, we will
restrict ourselves to the study of objects constrained by the behavior of the so called
real numbers. We denote the set of real numbers with the symbol, R. To construct
the so callled real number line (merely a pictoral representation of the set of real
numbers) we start by inserting the integers.
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2. The integers continue on towards positive and negative infinity, as the arrows
indicate. It is clear that the picture must also include the set of whole numbers N
if we have correctly displayed the set of integers Z, and indeed it does (recall N =
{1, 2, 3, ...}).
Now we take the logical next step and ask if we can represent a different class of
numbers on this same line. Indeed we can, we define the set of “rational numbers
Q” as
Q = {x such that x can be written as p/q where p and q are integers}
With such a definition we can now divide the space between two integers and
assign a length in terms of a fraction.
We can use shorthand to describe the set Q. I claim that the symbology I use to
define the rational numbers is the same exact statement as the previous definition:
p
Q= x : x = , (p, q) Z
q
I have introduced the symbol “:” and taken it to mean “such that.” The above
p
relation says; “Q is the set of x, such that x can be written as a fraction q where p and
q are integers (or elements of the set of integers Z).
Because we are at liberty to express integers in the form of a fraction, i.e. 2/2 = 1,
it is clear that the set Q should include the integers. That is, Z ⊂ Q. But even with all
this song and dance we have failed to fill the number line, as there are many num-
bers that exist on such a line that have no fractional representation. Such numbers
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3. are said to be “irrational.” The set of irrational numbers I includes π, which has an
infinite decimal expansion.
The real numbers, physically interpreted as all possible lengths along an infinite
number line can now be constructed in terms of the two sets Q and I. We define the
“union” of two sets to be another set which contains all the elements of the daughter
sets. Notationally, A ∪ B = C . It is in this form that the set of real numbers is cleanly
expressed;
R = Q∪I
Further Reading Martin Liebeck’s A Concise Introduction to Pure Mathemat-
ics is a fantastic (and easy!) read. The book relies on set theory and explains it in
good detail.
G.H. Hardy’s A Course of Pure Mathematics discusses the construction of the
real number line with more rigor but does not adopt set notation (unfortunately).
2 R2 and Fuctions of a Single Variable
I must now carefully introduce the notion of a function. When you were in high
school (maybe middle school) you were taught to graph something like y = x 2 in a
plane (known as the Cartesian plane) by “plugging in” values on the x-axis (-1,0,1,2,etc...)
and evaluating x 2 for the given input. Graphically, you represented this procedure by
moving along the x-axis until you reached your input, and then upwards (in the y di-
rection) until you hit your output. It was there that you put your dot. And if you
were anything like me, you probably filled up the planes with dots for a whole year
without understanding a damn thing that was going on.
What you were doing in your diaper days was pictorial representing the elements
of a set of points in the set R2 . The set R2 is merely the set R generalized to two
dimensions. Instead of consisting of points on a number line, it consists of points in
a plane. That is, it is the set of two-dimensional points (aka ordered pairs) expressed
in the form (x, y) where x R and y R. That is, x and y only take real values. When we
take an ordered pair, we can pictorially represent it in a plane. The set of all ordered
pairs contained in the real plane is the set R2 (pronounced r-two, NOT r-squared).
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4. To graph a point in R2 we look at its x and y component [(x, y) represents the two
components] and find the point in the Cartesian plane that has those same compo-
nents.
Definition 2.1: A real-valued single variable function f (x) associates one and
only one real unique output for every x in R.
Most of you are probably familiar with the “one and only one” constraint in the
form of the “straight line test,” that is no line drawn perpendicular to the x-axis
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5. should contain two outputs of a function. If a graph fails this test, it cannot be the
graph of a function. This is a very elementary notion of what a function is, the def-
inition can be expressed more elegantly in terms of sets. You should appreciate the
power of set theory and notation in the following definition, if you understand the
following you are in good shape.
Definition 2.2: Let f : R → R be defined with ∀x R ∃ f (x) R, such that f (x) = y =
y 0 . Then the set of points C ≡ {(x, f (x)) : , (x, f (x)) R2 } is a non-splitting curve in
Cartesian coordinates.
Let’s analyze this definition step by step. The shorthands used mean:

:
 "such that"


→ "into"



∀ "for every/for all"

∃ "there exists"





≡ "is defined to be"
The first two statements essentially mean the same thing. f : R → R is read “ f is
defined such that it maps from R to R,” or better said “ f is a map from R to R.” What
is meant by a map? Our map is the operation or formula we apply to x in order to
get f (x). The next statement is ∀x R ∃ f (x) R, such that f (x) = y = y 0 . It is read “for
every real x (x R), there exists a real f (x) such that f (x) = y and no other value y 0 .”
That is, f (x) maps to a real output y for every real input x. When we say that the
function has value y and no other value y 0 , we are saying that a single input guaran-
tees a unique output (unique meaning it is one of it’s kind). Finally we construct a
set C ≡ {(x, f (x)) : , (x, f (x)) R2 }, this set of points is what we are actually drawing in
the Cartesian plane when we graph a function. I encourage you to test it for simple
function like f (x) = x 2 .
Note on an assumption that was made in the defintion:
The assumption made on our part is that f (x) is defined for all real x (x R). This
limitation can be removed, that is we can have a function f : A → B (the function
maps from one arbitrary set to another arbitrary set). However, in real analysis these
sets must be subsets of R. The set of inputs for which f (x) is defined, i.e. x R on
−∞ < x < ∞ , is said to be it’s “domain.” The domain of a function is, in most cases,
a certain interval on the real-axis. The domain is written as; a ≤ x ≤ b or a < x < b if
the boundaries (a and b) are not included. I most often adopt a more convenient
notation; a ≤ x ≤ b =⇒ [a, b] and a < x < b =⇒ (a, b). The latter notation is
(in my humble opinion) superior in it’s brevity and convenience. For instance, the
real numbers lying between a and b (a and b included) can be neatly expressed as:
[a, b] R.
Further Reading G.H. Hardy’s A Course of Pure Mathematics Again, there is
much more rigor in this text.
Richard A. Silverman’s Essential Calculus with Applications Really cheap text if
you buy the Dover publication, I think you can get it used on Amazon for $3-5. It’s
definitely worth the money.
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6. 3 Review of Algebra and Trigonometry
3.1 What is Algebra anyways?
Before you run away, think to yourself, “did I ever really understand what I was do-
ing in Algebra?” Many of us were brainwashed in our early mathematics careers
and equipped with an assortment of tools designed to get the right answer without
the slightest bit of thought. “Factor, divide both sides by 2, add 16 to both sides,”
this type of language was adopted to ignore the obvious suggestion “FIND X BY ANY
MEANS NECESARRY” so that the less intelligent primates could pass Algebra with-
out really learning the mathematics. So let’s go back to basics, what is Algebra any-
ways?
Algebra is a hurdle for young students, merely because it introduces the “vari-
able.” When asked to define Algebra for the non-mathematician I merely say “Alge-
bra is arithmetic with unknowns.” In reality, abstract Algebra is the study of so called
“Algebraic structures.” We are not interested in that level of rigor. Let us visit some
problems in arithmetic and see their Algebraic analogs.
2 + 2 =?
This was a question you were asked in kindergarten, I hope you know the answer
is 5. (Just kidding).
In arithmetic you are performing the addition operation between numbers and
stating the result. The analog in Algebra is,
2 + 2 = x, What is x?
x, not quite a “variable” here, is being used to illustrate an unknown. A slightly
more challening problem is the following,
2 = x −2
The question then becomes more apparent, what value of x makes this state-
ment true? Again, the answer is 4, but the solution here can be arrived at without
performing arithmetic (i.e. by adding 2 to both sides). One merely needs to look at
the equation and make the assessment that 4 is the only number such that when 2
is subtracted from it the remainder is 2.
A polynomial is a series of powers in some variable (we take x). I define the
general polynomial of order n as follows;
Definition 3.1.1
Pn (x) = c 0 + c 1 x + c 2 x 2 + ... + c n−1 x n−1 + c n x n
i.e. P2 (x) = c 0 + c 1 x + c 2 x 2 is said to be a polynomial of order 2 (aka a quadratic).
The cs in the equations are the “coefficients (constants that do not vary with x)”
of the x terms. They are placed there for generality. There is a VERY powerful and
widely used notation to express sums like polynomials:
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7. n
Pn (x) = c 0 + c 1 x + c 2 x 2 + ... + c n−1 x n−1 + c n x n = ci x i
i =0
Notice the symbol sigma n=0 has two indices. The bottom index (i) refers to
i
where the summation begins. In our case, we start with x 0 = 1 and c 0 both implying
i = 0. We continue to sum over the integers (i Z) until we reach the top index n,
where we terminate the summation (we stop adding). In higher mathematics the
series; S(x) = ∞ c i x i is known as a power series. It’s a polynomial of degree infin-
i =0
ity!
Returning to “finding x,” suppose we want to find the values of x that validate an
equation involving x 2 .
x 2 + 2x + 1 = 0
We are looking for values of x which will yield zero on the right hand side (RHS).
The most common method of dealing with this type of problem makes use of the
distributive property in algebra. a(b + c) = ab + ac and likewise, (a + b)(c + d ) =
c(a + b) + d (a + b) = c a + cb + d a + d b.
The trick (called “factorization) is to factor the quadratic into two digestable
terms (called “binomials”):
x 2 + 2x + 1 = 0
(x + 1)(x + 1) = 0
We can double check to see if the factorization is correct by distribution.
(x + 1)(x + 1) = x(x + 1) + 1(x + 1) = x 2 + 2x + 1
Now examining the left hand side, we see that if x = −1 we satisfy the equation.
There could have easilly been two solutions, consider for example: x 2 − x −2 = 0.
(x − 2)(x + 1) = 0
We can clearly see that if x = 2, one of the binomials becomes zero and therefore
the entire left hand is zero because we are then multiplying by 0. The same goes for
x = −1. Therefore those are our solutions. The x values that make the polynomial
zero are called “zeroes” or sometimes “roots.” The term root comes from the fact
that, graphically, the roots are the places where the function crosses the x-axis.
In cases where we cannot factor the quadratic, we apply the quadratic formula.
2
b
x = −b± 2a −4ac where the a, b, c are the coefficients in the quadratic equation; ax 2 +
bx +c = 0. The quadratic formula is relatively easy to prove and will be included with
hints as an excersize.
Polynomial of higher orders can often be cumbersome to solve, it is best to ap-
proach them by example.
Before moving on I would suggest that you do the associate problem set I have
created to see how well you’ve digested the material. Good luck!
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