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Digestable Higher Mathematics David Alan Rogers March 23, 2011 Mathematics is an underrated subject. I hope to show that it isn’t just aboutadding and subtracting numbers. Mathematics is a rigorous and formal logic, and assuch, 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 thebooks I recommend will be affordable, AND GOOD (in my humble opinion).1 An Introduction to Set TheoryIn day to day life we often ﬁnd it convenient to compose lists of related objects. Abiologist 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 elementof 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.” Supposewe wish to note several elements in the set Z. We then introduce the notion of asubset. Let’s take a set; N = {1, 2, 3, ...}. Because the set N = {1, 2, 3, ...} is containedwithin 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 theshorthand; N ⊂ Z. In introductory calculus we explore topics in “real analysis.” As such, we willrestrict ourselves to the study of objects constrained by the behavior of the so calledreal numbers. We denote the set of real numbers with the symbol, R. To constructthe so callled real number line (merely a pictoral representation of the set of realnumbers) we start by inserting the integers. 1
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The integers continue on towards positive and negative inﬁnity, as the arrowsindicate. It is clear that the picture must also include the set of whole numbers Nif 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 ofnumbers on this same line. Indeed we can, we deﬁne the set of “rational numbersQ” 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 andassign a length in terms of a fraction. We can use shorthand to describe the set Q. I claim that the symbology I use todeﬁne 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 prelation says; “Q is the set of x, such that x can be written as a fraction q where p andq 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 allthis 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 2
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are said to be “irrational.” The set of irrational numbers I includes π, which has aninﬁnite decimal expansion. The real numbers, physically interpreted as all possible lengths along an inﬁnitenumber 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 daughtersets. Notationally, A ∪ B = C . It is in this form that the set of real numbers is cleanlyexpressed; 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 ingood detail. G.H. Hardy’s A Course of Pure Mathematics discusses the construction of thereal number line with more rigor but does not adopt set notation (unfortunately).2 R2 and Fuctions of a Single VariableI must now carefully introduce the notion of a function. When you were in highschool (maybe middle school) you were taught to graph something like y = x 2 in aplane (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 bymoving 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 youwere anything like me, you probably ﬁlled up the planes with dots for a whole yearwithout understanding a damn thing that was going on. What you were doing in your diaper days was pictorial representing the elementsof a set of points in the set R2 . The set R2 is merely the set R generalized to twodimensions. Instead of consisting of points on a number line, it consists of points ina plane. That is, it is the set of two-dimensional points (aka ordered pairs) expressedin the form (x, y) where x R and y R. That is, x and y only take real values. When wetake an ordered pair, we can pictorially represent it in a plane. The set of all orderedpairs contained in the real plane is the set R2 (pronounced r-two, NOT r-squared). 3
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To graph a point in R2 we look at its x and y component [(x, y) represents the twocomponents] 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 andonly one real unique output for every x in R. Most of you are probably familiar with the “one and only one” constraint in theform of the “straight line test,” that is no line drawn perpendicular to the x-axis 4
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should contain two outputs of a function. If a graph fails this test, it cannot be thegraph 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 thepower of set theory and notation in the following deﬁnition, if you understand thefollowing 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 inCartesian 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 isdeﬁned such that it maps from R to R,” or better said “ f is a map from R to R.” Whatis meant by a map? Our map is the operation or formula we apply to x in order toget f (x). The next statement is ∀x R ∃ f (x) R, such that f (x) = y = y 0 . It is read “forevery 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 thefunction 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 aset C ≡ {(x, f (x)) : , (x, f (x)) R2 }, this set of points is what we are actually drawing inthe Cartesian plane when we graph a function. I encourage you to test it for simplefunction 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). Thislimitation can be removed, that is we can have a function f : A → B (the functionmaps from one arbitrary set to another arbitrary set). However, in real analysis thesesets 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 ifthe boundaries (a and b) are not included. I most often adopt a more convenientnotation; 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, thereal 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 ismuch more rigor in this text. Richard A. Silverman’s Essential Calculus with Applications Really cheap text ifyou buy the Dover publication, I think you can get it used on Amazon for $3-5. It’sdeﬁnitely worth the money. 5
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3 Review of Algebra and Trigonometry3.1 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 careersand equipped with an assortment of tools designed to get the right answer withoutthe 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 ANYMEANS 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 someproblems in arithmetic and see their Algebraic analogs. 2 + 2 =? This was a question you were asked in kindergarten, I hope you know the answeris 5. (Just kidding). In arithmetic you are performing the addition operation between numbers andstating 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 slightlymore 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 withoutperforming arithmetic (i.e. by adding 2 to both sides). One merely needs to look atthe equation and make the assessment that 4 is the only number such that when 2is subtracted from it the remainder is 2. A polynomial is a series of powers in some variable (we take x). I deﬁne thegeneral 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 andwidely used notation to express sums like polynomials: 6
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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 iwhere the summation begins. In our case, we start with x 0 = 1 and c 0 both implyingi = 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 theseries; S(x) = ∞ c i x i is known as a power series. It’s a polynomial of degree inﬁn- i =0ity! Returning to “ﬁnding x,” suppose we want to ﬁnd the values of x that validate anequation 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 thedistributive 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 digestableterms (called “binomials”): x 2 + 2x + 1 = 0 (x + 1)(x + 1) = 0We 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 thereforethe entire left hand is zero because we are then multiplying by 0. The same goes forx = −1. Therefore those are our solutions. The x values that make the polynomialzero are called “zeroes” or sometimes “roots.” The term root comes from the factthat, 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 bx = −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 withhints 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 havecreated to see how well you’ve digested the material. Good luck! 7
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