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MATH 1010 COURSE SUMMARY
MIKE WILLS
Contents
- General Comments 1
- Study Guide 2
- The Real Numbers 4
- Linear Things 4
- The Plane 5
- Systems of linear equations 6
- Polynomials and Exponents 6
- Factorizing 7
- Rational Expressions 7
- Radicals 7
- Quadratics 7
- New Functions 8
- Turning Circles 9
- General Comments Mathematics is everywhere. In nature, we observe patterns and try to under- stand them. In every day life, we regularly use things that would not be around without mathematics. For example, automobiles, buildings, computers, television, and cartography (maps) all rely crucially on mathematical insights. In sports, sta- tistics plays a huge role in determining how good a player is. In medicine, we use charts and complicated machinery that require a certain amount of mathematics to use, understand, and interpret. Figures, numbers, data samples, voting results: all appear in media that we are exposed to everyday. Thus, whether we like it or not, mathematics plays a crucial role in how we Americans live our lives. It is possible to argue that you personally don’t need to know mathematics to achieve success in your life. There is of course a lot of truth to this, but immediate utility is not the only reason to study an academic subject. Other reasons to study an academic subject include:
- Increasing general knowledge: the more you know, the better armed you are for what life throws at you. Ignorance is not necessarily bliss.
- Its inherent beauty and its ability to describe the human condition. Although it may not be apparent, mathematics can be very beautiful, in much the same way that art or music can. It takes a bit of training to appreciate it, I suppose, but the beauty is there.
- Many employers like people who have a good mathematical background. 1
2 MIKE WILLS
- This is a bad reason, but I’ll mention it anyway: for matriculation purposes, a given subject may be required to finish a course. This is the worst reason, because if the only reason that you’re studying a subject is because you’re being forced to, then you’re pretty much wasting your own time. With this in mind, it is best to learn the real subject well, rather than simply attempt to mimic the subject enough to fool the instructor into thinking that you know the material. If nothing else, the mimics tend not to fool the instructor. A nonnegative attitude would therefore be highly desirable. Mathematics is a living, breathing subject, which has fascinated many people for millennia. It seems to me that a good question is “What exactly is mathematics?” There is no easy answer to this, but my thoughts are as follows:
- Mathematics is not just about numbers, although most math involves num- bers.
- Mathematics is not about memorizing formulae or equations, any more than history is about memorizing dates.
- Mathematics is the study of ideas, usually abstract. Using logical reasoning, mathematics considers the consequences of these ideas. In that sense, it is a game. The fact that this abstract, almost surreal approach, leads to amazingly fruitful applications is an example of the “unreasonable effectiveness of mathematics.”^1 If you take a point of view along the above lines, I think that you will find studying mathematics more interesting, less difficult, more enjoyable, and more efficient. Moreover, even if you forget the details of the present course, you will retain the ability to think analytically which is an inevitable outcome if you study mathematics effectively. One thing that is probably not emphasized enough at any level of mathematics is that the method matters far more than the outcome. Stop asking the instructor “Is that the right answer?” and start asking “Is this a correct method?” That leads me to the next section, where I will give my own study hints for the final.
- Study Guide There is no panacea for passing a maths class. We all learn a little differently, and we all have different backgrounds. I will tell you what has worked for me:
- Recopy lecture notes. My lecture notes were always untidy when I was a student, and so I made an early commitment to rewrite the notes when I got back to my office or study. I did this for all classes that I took while in college.
- Prioritize: As I mentioned in a class, I spent my summers in Santa Barbara studying. I did this because I decided that getting my degree was more important then drinking margaritas by the beach. I stand by that- I partied fairly hard as an undergraduate, and as a direct result ended up with an inferior degree and no job prospects. Worse, I was in danger of becoming an alcoholic. Naturally, there are many things more important than getting a degree: for example, family and work. However, both family and work will likely benefit if and when you get the degree. Keep that in mind.
- Do as many problems from the book as you can: There is no substitute for practice. Ideally, doing mathematics should not be drudgery. Unfortunately, it often takes repetition to understand an idea, which means that doing dozens of
(^1) Bertrand Russell
4 MIKE WILLS
- The Real Numbers We don’t cover Chapter 1 in this class, but it is nevertheless a fundamental chapter. Since I assume that you know the basics reasonably well, I give here an advanced point of view. An important point in mathematics is to distinguish between notation and the underlying concepts. Notation is essentially convention: for example, the order of operations has the acronym PEMDAS. This is a convention- there’s no fundamental rule of the universe that says that this is the way the order of operations has to be. However, it has turned out that this is a useful order, more useful than other orders. A crucial concept in mathematics is the idea of a set^2. In fact, the vast majority of modern mathematics (including the study of numbers!) is based on the so-called Peano axioms which are basic precepts about sets. There are a number of different ways of representing sets of real numbers. The most general way is set-builder notation; a key advantage of set-builder notation over interval notation is that it generalizes to arbitrary classes of objects. The real numbers are constructed as follows: we start with the natural numbers, N = { 1 , 2 , 3 ,.. .}. We then throw in 0, and the ‘negatives’ of the natural numbers to obtain the set of integers, Z = { 0 , ± 1 , ± 2 , ± 3 ,.. .}. We then define the rational numbers as ‘ratios’ of two integers: that is, Q = { (^) mn | n, m ∈ Z, m 6 = 0}. Finally, we define the real numbers, R, as all numbers that can be well-approximated by a sequence of rational numbers. Intuitively, if we plot all the numbers in Q on a line, not all the points on the line are filled, but given any point on the line, there are infinitely many rational numbers ‘near it’. We define addition and multiplication on the natural numbers in the way that we learned in primary school. We extend the idea of these two operations to all real numbers, in such a way that the extension when restricted to the natural numbers is what we started with. Thus, if a, b, c, d are integers, with b and d non-zero, a b +^
c d =^
ad+bc bd. Subtraction and division are inverses of addition and multiplication respectively. Addition and multiplication on R are associative and commutative. 0 is an ad- ditive identity and 1 is a multiplicative identity. Every real number has an additive inverse (its negative), and every non-zero real number has a multiplicative inverse (its reciprocal). The key property that ties together addition and multiplication is the distributive law. Finally, the real numbers are totally ordered; that is, if a, b ∈ R then one and only one of the three following conditions holds: (i) a < b (ii) b < a (iii) a = b. Moreover, if a < b and c > 0, then ac < bc. This last statement, together with the above remarks, mean that R is what is called in the jargon a totally ordered field.
- Linear Things
Notation 4.1. For the rest of this handout, x, y represent real variables unless otherwise stated.
Definition 4.2. An equation is linear in one variable if it can be written in the form ax + b = c where a, b, c are real numbers and a 6 = 0.
(^2) Technically, a set is undefined, but the concept is one that we should all have an intuition for.
MATH 1010 COURSE SUMMARY 5
Using the basic properties of arithmetic, it is clear that x = c−a b. Of course, one may have an equation that looks linear, but when written in the above form a turns out to be zero. In that situation, one either gets an identity (0 = 0) in which case, x can be any real number, or, if b 6 = c, one gets a contradiction and the original statement has the empty set for a solution set. Many formulae make use of linear equations. For example, the relationship between degrees Celsius and degrees Farenheit can be expressed as a linear equation. Also, many mixture problems involve linear equations. It also makes sense to discuss linear inequalities; that is inequalities that can be written in the form ax + b < c or ax + b ≤ c, again with a 6 = 0. In that case, the solution set will turn out to be an interval, the exact nature of which will depend upon whether the inequality is strict and also upon the sign of a. it is fundamental that multiplying an inequality by a negative number reverses the inequality. (This is a consequence of the fact that the real numbers are a totally ordered field.) In various situations, one may have to bound ax + b above and/or below. In that case, we end up with compound inequalities, which must be handled with care. Compound inequalities can be expressed succinctly using absolute values. However, when solving an inequality using an absolute value, one must take into account the fact that if |a| ≤ |b| then −b ≤ a ≤ b. Equivalently, either b ≤ −a or b ≥ a. Again, this follows from the fact that the real numbers are a totally ordered field.
- The Plane It is often convenient to work with an infinite plane. For convenience, we pick one point, call it the origin, and then draw two perpendicular lines that intersect at the point. By convention, one line is horizontal and the other is vertical. We impose the real numbers on each line with the 0 on each line at the origin, and the numbers increasing from left to right on the horizontal line (the x-axis) and from down to up on the vertical axis (the y-axis). Having done this, any point in the plane can be represented as an ordered pair of real numbers.
Definition 5.1. An equation is linear in two variables, x and y, if it can be written in the form ax + by = c with a^2 + b^2 6 = 0.
The solution set of such an equation can be represented by a graph in the plane. A fundamental result is that the graph will always be a straight line. If b 6 = 0, then y = − ab x + cb. In this situation, the slope of the line is − ab and the y-intercept is (0, cb ). If b = 0, the line is vertical and the slope is undefined. Two non-vertical lines are parallel if and only if they have the same slope. Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. A non-vertical line has an equation that can be written in the form y = mx + b (slope-intercept form). Often, one knows the slope, m, and one of the points that the line passes through, say (x 1 , y 1 ). In that case, the equation of the line will be y − y 1 = m(x − x 1 ). This is the point-slope form. A linear inequality in two variables will typically have a solution set consisting of a half-plane whose boundary line is the graph of the corresponding linear equation. Which of the two half-planes formed by the boundary line actually gives the solution set is most easily discovered by plugging in suitable points.
MATH 1010 COURSE SUMMARY 7
Special cases of polynomials include monomials (one term), binomials (two terms), and trinomials (three terms). We are already familiar with polynomials of degree 1 (linear) and zero (constant).
- Factorizing With care, multiplying two polynomials together is a straightforward application of the rules of arithmetic and exponents. Factorizing a polynomial into a product of polynomials tends to be more difficult, but is often essential in applications. There are certain techniques that will work: for example, a degree 2 polynomial (a quadratic) can often be factored essentially by inspection: e.g. x^2 − 6 x + 8 = (x − 4)(x − 2). Some basic techniques include factoring by grouping and hoping for the best. A common factorization is x^2 − b^2 = (x − b)(x + b). There are similar factorizations for x^3 − b^3 and x^3 + b^3. In general, x^2 + b^2 does not factor over the real numbers. Frankly, the best approach to factorization is experience.
- Rational Expressions Let p and q be polynomials. Let f (x) = p q((xx)). Then f is a rational function with domain all real numbers, x, such that q(x) 6 = 0. One usually wants to write f
in lowest terms; that is, if one has a choice between x
2 x or^
x 1 =^ x, one usually picks the latter. (There are some technical issues here which we gloss over.) Graphing rational functions tends to be difficult. Common techniques to aid in graphing rational functions is to write the function in lowest terms, note that the points not in the domain of the function give rise to vertical asymptotes, find the x and y intercepts, and then plot points away from the intercepts. Unfortunately, finding the x-intercepts (when the numerator is zero) and the vertical asymptotes (when the denominator is zero) explicitly is usually non-trivial and sometimes impossible. Many formulae involve rational expressions; for example, Newton’s Laws of physics.
- Radicals
Definition 10.1. Let n be a natural number. We say that y is an nth root of x if yn^ = x. If x, y > 0 and n is even, then we write y = n
x. If x, y are real and n is odd, then we write y = n
x.
Again, using the fundamental fact about exponents, namely the sum rule, we can now extend the idea of exponents to rational numbers: suppose a is a real number with a real nth root. Then a n^1 = n
a. Hence, a mn = ( n
a)m. It is customary to write expressions involving ratios of radicals so that only the numerator contains radicals. One does this by rationalizing the denominator.
- Quadratics Quadratic polynomials come up in a variety of applications such as distance problems, area problems, and others. Not all quadratics can be factored over the real numbers. For example x^2 + 1 is irreducible.
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For a variety of reasons, it became useful to invent the symbol, i, and define i^2 = −1. After several centuries of work, this symbol was given a rigorous mathe- matically satisfying framework. The set of complex numbers is given by C = {a + bi | a, b ∈ R}. The normal laws of multiplication and addition apply to the complex numbers. However, C is not totally ordered; in particular, the number i is not positive, zero, or negative. Note that if a > 0 , then
−a = i
a. The conjugate of a + bi is a − bi. If I divide one complex number by another, I can write the resulting ratio in standard form by multiplying and dividing the ratio by the conjugate of the divisor: thus
(11.1)
a + bi c + di
(a + bi)(c − di) (a + bi)(a − bi)
ac + bd + i(bc − ad) a^2 + b^2
We can now find the roots of any quadratic equation, ax^2 + bx + c = 0 by completing the square; that is, by observing that for a 6 = 0,
(11.2) ax^2 + bx + c = a
x^2 + b a
x
x + b 2 a
b^2 4 a^2
Once having completed the square, standard algebraic manipulations yield the qua- dratic formula, x = −b±
√b (^2) − 4 ac 2 a. The graph of y = x^2 is a parabola in the plane which is concave up and has vertex the origin. By completing the square, one can write any quadratic function as f (x) = a(x − h)^2 + k. The graph of y = f (x) is the graph of y = x^2 shifted to the right by h units, then scaled by a factor of a, and then shifted vertically up by k units. Notice that the vertex is (h, k). This method of obtaining new graphs from known ones is quite general.
- New Functions We say that a function, f is one-to-one (injective) if f (a) = f (b) forces a = b. A function is one-to-one if and only if its graph intersects each horizontal line at most once. If f is one-to-one, it has an inverse function, f −^1 , whose graph is a reflection of the graph of f through the line y = x. Finding an inverse function is usually non-trivial, but in principal the method is straightforward: switch the roles of y and x and then solve for y. Notice that f (f −^1 (x)) = x. Functions and inverse functions have the same duality that multiplication and division do: one operation undoes the other. The single most important class of functions in mathematics are the exponen- tial functions, f (x) = ax, where a > 0 and a 6 = 1. Exactly how one computes ax for an irrational number, x, is best left for a more advanced course. The graph of f is concave up and has the x-axis as a horizontal asymptote. The y-intercept is (0, 1) and the function is strictly positive. If a > 1, the function is increasing; otherwise it is decreasing. Of particular interest is f (x) = ex^ where e = 1 + 1 + 12 + (^) 3!^1 + (^) 4!^1 +.. .. It turns out that the slope of the graph of f (x) = ex^ at x = a is ea. This is not immediately obvious, and does not generalize to more arbitrary exponential functions. If f (x) = ax, with 1 6 = a > 0, then f is one-to-one, and its inverse function is f −^1 (x) = loga x; that is, f −^1 is a logarithmic function. The domain of this function is the positive real numbers. There are a number of fundamental properties. We list a few here:
