| 1 | | here we consider as say prof dr mircea orasanu as prof horia orasanu with School children meet the number line in the early grades. By high school algebra and geometry, the real number line has become a central concept. But really, what is the real number line? Is it a figment of our imagination? How do we define it as something more concrete? |
| 2 | | |
| 3 | | A child’s intuition of the real number line as a straight line in a plane or in space is derived from experience with straight line segments in real life, as the edge of a ruler, the border of a page of paper, the lines on graph paper, the edges of tables, or the lines where the walls meet the ceiling. But what if the line is extended into space, say to Jupiter, or beyond? What happens as the line approaches the outer reaches of space? Even the concept of space itself is based on a precise notion for number line. |
| 4 | | |
| 5 | | And what are the individual real numbers? The child’s intuitive model for a real number corresponds to a dot made with pencil on paper. But each dot really corresponds to a multitude of points, a mound of graphite. Does the heap of graphite represent something other than vacuum? What really are “pi” and “the square root of 2”? |
| 6 | | |
| 7 | | An intuitively appealing construction of the rational numbers is based upon Euclidean geometry. It runs as follows. One starts with a straight line, one marks a point and labels it 0, and one marks a different point and labels it 1. Then one constructs the other integers by marking off steps of equal length, and one constructs the rational numbers by dividing the segments between integers into equal parts. In this model, the real number line, stripped of its arithmetic, is taken as a primitive concept and subjected to the axioms of Euclidean geometry (say Hilbert’s axioms, which are studied in a course on the foundations of geometry; Euclid himself simply proceeded with blind faith that the constructions he performed did not stumble into any holes). And how do we know there is a model of Euclidean geometry? The canonical model for Euclidean geometry is the Cartesian plane consisting of ordered pairs of real numbers, and the verification of the axioms of Euclidean geometry depends on the properties of the real number line. If we follow this route to construct the real numbers from a Euclidean straight line, we find we have traveled in a logical circle. |
| 8 | | |
| 9 | | The circular reasoning that appears in some high school algebra textbooks is not so subtle. In one of them, the rational numbers are defined as quotients of integers, the irrational numbers are defined as the real numbers that are not rational, and then the real numbers are defined as the aggregate of the rational and the irrational numbers. |
| 10 | | |
| 11 | | The book Mathematics for High School Teachers, by Usiskin, Stanley, et al., treats the real numbers in Chapters 2 and 6. In Chapter 2, reference is made to various methods of constructing the real numbers from the rational numbers, without attempting to give a precise definition of the real numbers. Then the authors take a straight line, mark off 0 and 1, represent the rational numbers on the line, and go on to explore in some detail the decimal representation of real numbers. They return in Chapter 6 to the field axioms, and they establish the uniqueness of a complete ordered field. The question of existence is never completely nailed down. Yet they come close, when they say: “In school algebra, real numbers are commonly described as numbers that can be represented by finite or infinite decimals.” |
| 12 | | |
| 13 | | EXERCISE: Suppose a persistent high school student asks you to explain exactly what real numbers are. What explanation would you give the student? |
| 14 | | |
| 15 | | The goal of these notes is to bring you to a point where you can give the student a satisfactory answer to this question. Your answer might be brief, but you should feel confident that you can supply as much detail as the student might insist upon. In particular, you should understand in what sense the real numbers “are” the set of decimals. |
| 16 | | |
| 17 | | THE REAL NUMBER LINE |
| 18 | | |
| 19 | | Rather than specify concretely what a real number is, we will describe the real number line by listing its properties. This is done by defining an axiom system. The primitive concepts in the axiom system are points (real numbers), the operations of addition and multiplication, and an order relation. The list of axioms is quite long, but with one exception they are not difficult to understand. They are familiar properties of the rational numbers. The one exception is the “completeness axiom,” which says that there are no “holes” in the real number line. We refer to any model for the axiom system as “the real number line” or “the field of real numbers.” In other words, the real number line is a set with arithmetic and ordering that satisfies the “real number axioms.” |
| 20 | | |
| 21 | | There are two important facts that justify our use of the expression “the real number line.” First, there is a model for the axiom system. Second, any two models for the axiom system are isomorphic, that is, they can be put in a one-to-one correspondence so that the arithmetic and the ordering correspond. In other words, the real number line exists, and it is unique. We may perform arithmetic operations on the set with confidence, without pausing to consider where the set comes from or where it is going. (The K-12 student is generally happy to perform arithmetic operations on real numbers, oblivious of the defining properties of the real numbers, confident that there is such an entity, and not the least concerned about whether such an entity is unique.) |
| 22 | | |
| 23 | | So what are the real number axioms? The axioms come in three batches corresponding to arithmetic, ordering, and completeness. The axioms taken together assert that the real numbers form a “complete ordered field.” |
| 24 | | |
| 25 | | The construction of the real numbers is usually carried out in a foundational upper division course in analysis (Math 131A at UCLA). The arithmetic axioms, in various combinations, are studied in more detail in upper division algebra courses (Math 110AB and Math 117 at UCLA). The arithmetic axioms assert that the real numbers form a field. The completeness axiom in the form of the Least Upper Bound Axiom is usually introduced in the first calculus course. Completeness is treated in more detail in the foundational analysis course or in a more advanced topology course (Math 121 at UCLA), in the context of metric spaces. The ordering and completeness axioms also appear in some form in Hilbert’s axiom system for Euclidean geometry, which is treated in a course on the foundations of geometry (Math 123 at UCLA). |
| 26 | | |
| 27 | | THE ARITHMETIC AXIOMS |
| 28 | | |
| 29 | | The axioms for arithmetic assert that there are two operations, addition and multiplication, and these operations satisfy certain rules. |
| 30 | | |
| 31 | | There are four axioms for addition. |
| 32 | | |
| 33 | | 1. The associative law, (x + y) + z = x + (y + z), tells us we can perform the operation of addition in any order. Thus the expression x + y + z has an unambiguous meaning. |
| 34 | | 2. The commutative law, x + y = y + x, allows us to switch the orders of the addends. |
| 35 | | 3. There exists an additive identity, denoted by 0, that satisfies 0 + x = x = x + 0 for all x. |
| 36 | | 4. Each x has an additive inverse, denoted by -x, such that x + -x = 0 = -x + x. |
| 37 | | |
| 38 | | We define the operation of subtraction to be addition of the additive inverse, so that |
| 39 | | x minus y, written x – y, is defined to be x + -y. The usual rules for subtraction hold. They are not axioms, but are consequences of the axioms for addition. Subtraction is completely subservient to addition, in the sense that any statement about subtraction can be restated as a statement about addition and additive inverses. |
| 40 | | |
| 41 | | There are four axioms for multiplication, and they are virtually the same as the axioms for addition. |
| 42 | | |
| 43 | | 5. The associative law, (xy)z = x(yz), tells us we can perform the operation of multiplication in any order. Thus the expression xyz has an unambiguous meaning. |
| 44 | | 6. The commutative law, xy = yx, allows us to switch the orders of the factors. |
| 45 | | 7. There exists a multiplicative identity, denoted by 1, that is different from 0 and that satisfies 1x = x = x1 for all x. |
| 46 | | 8. Each x other than 0 has a multiplicative inverse, denoted by 1/x, such that x(1/x) = 1 = (1/x)x. |
| 47 | | |
| 48 | | We define the operation of division to be multiplication by the multiplicative inverse, so that x divided by y, written x/y, is defined to be x(1/y). Note that division by y is defined only for those y’s that have a multiplicative inverse. Division by 0 is not defined. |
| 49 | | |
| 50 | | The usual rules for division hold. They are not axioms, but are consequences of the axioms for multiplication. Division is completely subservient to multiplication, in the sense that any statement about division can be restated as a statement about multiplication and multiplicative inverses. |
| 51 | | |
| 52 | | Finally there is an axiom that guarantees that addition and multiplication are compatible. |
| 53 | | |
| 54 | | 9. The distributive law, x(y + z) = xy + xz, relates the operations of addition and multiplication. |
| 55 | | |
| 56 | | A set with two operations, addition and multiplication, that satisfies these axioms is called a field. Examples of fields abound. The rational numbers form a field. So do the real numbers, and so do the complex numbers. |
| 57 | | |
| 58 | | EXERCISE: Deduce from the field axioms that 0 times anything is 0, so that 0 cannot have a multiplicative inverse. |
| 59 | | |
| 60 | | EXERCISE: Deduce from the field axioms that (-1)(-1) = 1. |
| 61 | | |
| 62 | | EXERCISE: Suppose a high school student asks you why we cannot divide by zero. What explanation would you give to the student? |
| 63 | | |
| 64 | | There are some “funny” fields that do not look at all like the rational numbers. One example of a “funny” field is a field consisting of just two elements, which must be the additive and the multiplicative identities. In this field we define addition by 1+1=0=0+0 and 1+0=0+1=1, and we define multiplication so that 0 times anything is 0, and 1 times 1 is 1. |
| 65 | | |
| 66 | | EXERCISE: Let p be a prime number, and let Zp be the set of congruence classes of integers mod p. The addition and multiplication in Zp is defined to be the usual addition and multiplication mod p. Show that every m in Zp other than 0 has a multiplicative inverse. Remark: Zp is a field with p elements. |
| 67 | | |
| 68 | | A lot of effort in school mathematics goes into defining and interpreting subtraction and division. From a purely mathematical point of view, the definitions are quite simple. “Subtraction of x” is defined to be “addition of the additive inverse of x.” “Division by x” is defined to be “multiplication by the multiplicative inverse of x.” |
| 69 | | |
| 70 | | THE ORDER AXIOMS |
| 71 | | |
| 72 | | The order axioms assert that there is a relation “ < ” defined between certain elements, which satisfies the following rules. |
| 73 | | |
| 74 | | 1. The trichotomy law asserts that exactly one of the relations x<y, y<x, or x=y holds between any two given x and y. |
| 75 | | |
| 76 | | We write x <= y as shorthand for x < y or x = y. Also, we write y > x to mean x < y, and we write y >= x to mean x <= y. |
| 77 | | |
| 78 | | 2. The law of transitivity asserts that if x<y and y<z, then x<z. |
| 79 | | 3. The law of compatibility with addition asserts that if x < y, then x+z < y+z. |
| 80 | | 4. The law of compatibility with multiplication asserts that if x < y and a > 0, then ax < ay. |
| 81 | | |
| 82 | | A field with an ordering that satisfies these axioms is called an ordered field. |
| 83 | | |
| 84 | | EXERCISE: Show from the axioms that -1 < 0 and 0 < 1. |
| 85 | | |
| 86 | | EXERCISE: Show from the axioms that x2 >= 0 for any x in an ordered field. Deduce from this that the complex numbers cannot be ordered to become an ordered field. |
| 87 | | |
| 88 | | EXERCISE: Show from the axioms that in an ordered field, the elements 1, 1+1, 1+1+1, 1+1+1+1, … are distinct. |
| 89 | | |
| 90 | | If the elements 1, 1+1, 1+1+1, 1+1+1+1, … of a field are distinct, we say that the field has characteristic zero. If these elements are not distinct, there is a first positive integer p such that 1+1+…+1 [p summands] is 0. In this case, we say that the field has characteristic p. |
| 91 | | |
| 92 | | EXERCISE: Show that the characteristic of a field is either 0 or a prime, that is, show that the number p above is a prime number. |
| 93 | | |
| 94 | | There is some standard notation that is convenient. In any field, we write 1+1 = 2, 1+1+1 = 3, 1+1+1+1 = 4, and so on. As usual, -n denotes the additive inverse of n. If the field has characteristic zero, we identify these elements with the integers Z, and we regard Z as a subset of the field. Under this identification, addition and multiplication in Z are the same as addition and multiplication in the field. Further, the subfield generated by 0 and 1 (the smallest subfield containing 0 and 1) is isomorphic to the field of rational numbers. In other words, we can regard the rational numbers as being a subset of any field of characteristic zero, and in particular of any ordered field. |
| 95 | | |
| 96 | | EXERCISE: Define the absolute value function by |x| = x if x>= 0, and |x| = -x if x < 0. Show from the axioms that |x+y| <= |x| + |y|. Hint: Consider four cases. |
| 97 | | |
| 98 | | There is another important property of the ordering of the real numbers that cannot be derived directly from the other order axioms. |
| 99 | | |
| 100 | | Archimedean Order Axiom: If a > 0 and b > 0, there is an integer m>0 such that ma > b. |
| 101 | | |
| 102 | | If the ordered field satisfies the Archimedean order axiom, we call it an Archimedean ordered field. By taking a=1 in the Archimedean ordering axiom we see that each b > 0 in the field is bounded above by some positive integer m. Let n be the first integer such that b < n+1. Then n >= 0, and n <= b < n+1. The integer n is the leading entry in the decimal expansion of b. We return to decimal expansions later. |
| 103 | | |
| 104 | | EXERCISE: In an ordered field, let (a,b) denote the open interval from a to b, that is, the set of x in the field satisfying a < x < b. Define [a,b), [a,b], and (a,b] similarly. Show that an Archimedean ordered field is the union over integers n of the semi-open intervals [n,n+1). Show that these semi-open intervals are pairwise disjoint. |
| 105 | | |
| 106 | | The following exercise will be used later, in the discussion of the uniqueness of the real numbers. |
| 107 | | |
| 108 | | EXERCISE: In an Archimedean ordered field, show that if x > 0, there is a positive integer n such that x > 1/10n. |
| 109 | | |
| 110 | | THE COMPLETENESS AXIOM |
| 111 | | |
| 112 | | The completeness axiom for the real numbers is the tersest, yet the most difficult to understand. To state it, we need some preliminary definitions. Let S be a subset of the ordered field. We say that b is an upper bound for S if x <= b for all elements x of S. We say that b is a least upper bound for S if b is an upper bound for S, and |
| 113 | | b <= c for any other upper bound c for S. |
| 114 | | |
| 115 | | EXERCISE: Show that a subset of an ordered field has at most one least upper bound. |
| 116 | | |
| 117 | | One version of the completeness axiom is the least upper bound axiom for a fixed ordered field. |
| 118 | | |
| 119 | | Least Upper Bound Axiom (LUB Axiom): If a nonempty subset of the ordered field has an upper bound, then it has a least upper bound. |
| 120 | | |
| 121 | | We say that a set is bounded above if it has an upper bound. The LUB axiom can be restated simply: a nonempty set that is bounded above has a LUB. |
| 122 | | |
| 123 | | An ordered field that satisfies the LUB axiom is called a complete ordered field. Our goal is twofold. First, we aim to show that there exists a complete ordered field. Second, we aim to show that any two complete ordered fields are isomorphic. This complete ordered field, which is essentially unique, is called the field of real numbers. |
| 124 | | |
| 125 | | Before proceeding to the construction of the real numbers, we state a theorem and give a formal proof to illustrate how the LUB axiom is used. |
| 126 | | |
| 127 | | Theorem: A complete ordered field is Archimedean. |
| 128 | | |
| 129 | | Proof: Fix a > 0. Let S be the set of multiples a, 2a, 3a, … of a. Let c be an upper bound for S. Then (n+1)a <= c for all positive integers n, so that na <= c – a for all positive integers n. Thus c – a is also an upper bound for S, and further c – a < c. We conclude that S does not have a least upper bound. By the LUB axiom, S is not bounded above. Consequently for each b, there is some n such that b < na. Thus the ordering is Archimedean. |
| 130 | | |
| 131 | | EXERCISE: Write out a formal proof of the theorem starting with the lines, “Suppose the field is not Archimedean. Then there are a > 0 and b > 0 such that ma <= b for all positive integers m.” |
| 132 | | |
| 133 | | THREE MODELS FOR THE REAL NUMBER LINE |
| 134 | | |
| 135 | | There are three methods that are often used to construct the real numbers. Each method has its advantages and its disadvantages. Each method leads to a model for the real numbers, that is, a set with addition, multiplication, and ordering that satisfy the axioms for complete ordered field. We shall refer to the three models respectively as the Weierstrass-Stolz model (decimal expansions, the most intuitive model), the Dedekind model (Dedekind cuts, the slickest model), and the Meray-Cantor model (completion of a metric space, the most far-reaching model). |
| 136 | | |
| 137 | | DECIMAL EXPANSIONS |
| 138 | | |
| 139 | | It was Otto Stolz (1886) who pointed out that decimal expansions can be used to define the real numbers. In the Weierstrass-Stolz model, we define the real numbers to be the set of all decimal expansions a = a0.a1a2a3…, where a0 is an integer (positive or negative), and a1, a2, a3, … are integers between 0 and 9, except that we declare a decimal expansion that terminates in all nines to be the same real number as the (terminating) decimal expansion obtained by incrementing the last non-nine term by 1 and replacing the subsequent 9’s by 0’s. Thus for instance we regard 3.2599999… and 3.2600000… as the same real number. (This unfortunate complication is not an essential difficulty, but it does make the verification of the arithmetic axioms into a tedious exercise.) |
| 140 | | |
| 141 | | We think of the decimal expansion a0.a1a2a3… as representing the number a0+(a1/10)+(a2/100)+(a3/1000)+… . For positive numbers this is the usual decimal representation. For negative numbers, it’s not the usual decimal representation, but it is the most convenient for establishing the arithmetic axioms. |
| 142 | | |
| 143 | | EXERCISE: What are the two possible interpretations of the decimal -2.71828? |
| 144 | | How would you respond to a student who asks about the ambiguity? |
| 145 | | |
| 146 | | We define addition and multiplication of these decimals by following the same procedures as we would for finite decimals, adding place by place and carrying if necessary. Checking that this makes sense and that the axioms for addition and multiplication hold is messy, but indeed the arithmetic axioms are satisfied. Defining the order is quite easy, and it is a straightforward task to establish the order axioms and the LUB axiom. The Weierstrass-Stolz model is a complete ordered field. (We should be proud of the model.) |
| 147 | | |
| 148 | | EXERCISE: Define the order relation between two decimal expansions and prove the trichotomy law. |
| 149 | | |
| 150 | | Introducing arithmetic and ordering into the set of decimal expansions is the most intuitive method for constructing the real numbers. It is the model that appeals to school children. It is the model of most comfort to teachers, who can explain with confidence to inquisitive students that real numbers can be defined to be decimal expansions. The disadvantage of the method is that checking the arithmetic axioms is a laborious task. |
| 151 | | |
| 152 | | DEDEKIND CUTS |
| 153 | | |
| 154 | | The subtlest method for constructing the real numbers is due to Richard Dedekind (published in 1872). It is the model that appeared in the first chapter of the first edition of Walter Rudin’s classic textbook, Principles of Mathematical Analysis. Wading through Rudin’s construction of real numbers by Dedekind cuts became trial by fire for many college mathematics majors. The method is so slick that many mathematics majors find it hard to digest; they regard Dedekind cuts as being rather unkind. By the third edition, a kinder and gentler Rudin had relegated Dedekind cuts to an appendix. |
| 155 | | |
| 156 | | For this construction, one begins with the rational numbers. The idea is that a real number x is the right endpoint of a unique semi-infinite open interval (– ∞, x), and this interval is uniquely determined by the rational numbers in the interval. With this idea to guide intuition, one defines a Dedekind cut to be a set E of rational numbers such that (1) if x is in E and y < x, then y is in E, (2) neither E nor its complement is empty, and (3) E does not contain a largest number. |
| 157 | | |
| 158 | | EXERCISE: What sets E of rational numbers satisfy (1) but not (2) or (3)? |
| 159 | | |
| 160 | | In this model, the real numbers are defined to be the set of Dedekind cuts. Order is easy to define. We declare E < F if E is a subset of F. Addition is also easy to define. The sum of the cuts E and F is the set of all sums x + y, where x is in E and y is in F. The product is a little more complicated to define. Once defined, it is straightforward to verify the real number axioms. The main disadvantage of this method is the level of sophistication required to organize and execute these “straightforward” verifications. |
| 161 | | |
| 162 | | In any event, the Dedekind cuts form a complete ordered field. The additive identity in the Dedekind model is the open interval from minus infinity to 0. The multiplicative identity is the open interval from minus infinity to 1. More generally, each rational number r corresponds to the cut (– ∞, r), and this correspondence allows us to identify the rational numbers with a subfield of the Dedekind model for the real numbers. |
| 163 | | |
| 164 | | |
| 165 | | COMPLETION BY CAUCHY SEQUENCES |
| 166 | | |
| 167 | | The most far-reaching method for constructing the real numbers is due independently to Charles Meray (1869, 1872) and Georg Cantor (1872, 1883). Again one begins with the rational numbers. One considers the set of all sequences {xn} of rational numbers such that xn-xm tends to zero as n and m tend to infinity. Such sequences are called Cauchy sequences. We introduce an equivalence relation in the set of Cauchy sequences by declaring two Cauchy sequences {xn} and {yn} to be equivalent if xn – yn tends to zero as n tends to infinity. The real numbers are then defined to be the set of equivalence classes of Cauchy sequences. Addition and multiplication are easy to define. The sum of the equivalence classes represented by two such sequences {xn} and {yn} is defined to be the equivalence class of {xn + yn}, and similarly for the product. It is straightforward to verify the axioms of an ordered field, and a little more complicated to verify the completion axiom. The main disadvantage of the method is the excess labor and the level of sophistication required for working with equivalence classes rather than just sequences. The advantage of the method is that it can be used in a fairly general context to embed metric spaces in “complete” spaces. (A metric space can be embedded as a dense subset of a complete metric space, which is essentially unique.) |
| 168 | | |
| 169 | | OTHER VERSIONS OF THE COMPLETENESS AXIOM |
| 170 | | |
| 171 | | There are several other versions of the completeness axiom that are introduced and used in the calculus course sequence and the basic analysis course. In an ordered field, each of these is equivalent to the LUB axiom. |
| 172 | | |
| 173 | | Every bounded increasing sequence converges. |
| 174 | | |
| 175 | | A decreasing sequence of nonempty finite closed intervals has nonempty intersection. |
| 176 | | |
| 177 | | Every Cauchy sequence converges. |
| 178 | | |
| 179 | | In the context of metric spaces, the latter version of the completeness axiom becomes a definition. We say that a metric space is complete if every Cauchy sequence converges. |
| 180 | | |
| 181 | | EXERCISE: Formulate a definition of a convergent sequence in an ordered field. Use your definition to show that in an ordered field with the LUB axiom, every bounded increasing sequence converges. |
| 182 | | |
| 183 | | UNIQUENESS OF THE FIELD OF REAL NUMBERS |
| 184 | | |
| 185 | | The uniqueness (up to isomorphism) of the field of real numbers is established in outline as follows. We start with a complete ordered field, and we show how to assign to each x in the field a decimal expansion. The first step is to choose an integer a0 such that |
| 186 | | |
| 187 | | a0 <= x < a0+1. |
| 188 | | |
| 189 | | There is then a unique integer a1, 0 <= a1 <= 9, such that |
| 190 | | |
| 191 | | a0 + (a1/10) <= x < a0 + (a1/10) + 1/10. |
| 192 | | |
| 193 | | We continue in this manner, selecting at the nth stage the unique integer an such that |
| 194 | | |
| 195 | | a0 + (a1/10) + … + (an/10n) <= x < a0 + (a1/10) + … + (an/10n) + 1/10n. |
| 196 | | |
| 197 | | Thus each x determines the infinite decimal a0.a1a2a3…. We must show that the correspondence between x and the decimal expansion is a one-to-one correspondence that respects arithmetic and order, so that it is an isomorphism of the complete ordered field and the Weierstrass-Stolz model based on decimal expansions. |
| 198 | | |
| 199 | | If y is different from x, say y > x, then there is n such that y – x > 1/10n. (Recall the exercise based on the Archimedean ordering.) Then x and y do not belong to the same interval of length 1/10n, so the first n+1 entries in the decimal representation of y cannot be the same as those of x, and the decimal representation of y is different from that of x. |
| 200 | | |
| 201 | | Next note that x = a0 + (a1/10) + … + (an/10n), which belongs to the field, corresponds to the terminating decimal a0.a1a2a3…an000…. On the other hand, the correspondence does not yield any decimal that terminates in 9’s. Indeed if y corresponds to the decimal a0.a1a2a3…an999…, where an < 9, and if x is the rational number with terminating decimal a0.a1a2a3…(an+1), then x>y and x – y < 1/10m for all large m, so x = y, contradicting the fact that the decimal corresponding to y terminates with 9’s, not 0’s. |
| 202 | | |
| 203 | | To show that the correspondence between x in the complete ordered field and the decimals in the decimal model is one-to-one, it suffices now to show that each decimal representation that does not terminate in 9’s arises from some x in the field. This step depends crucially upon the completeness axiom. Suppose a0.a1a2a3… is a decimal that does not terminate in 9’s. The set S of elements in the field of the form |
| 204 | | a0 + (a1/10) + … + (an/10n), for n >= 1, is bounded above by a0 + 1. By the completeness axiom, the set S has a least upper bound, call it x. One checks that the decimal corresponding to x is a0.a1a2a3…, as required. |
| 205 | | |
| 206 | | To complete the proof of the uniqueness, we must show that the correspondence preserves the arithmetic and ordering. That the correspondence respects the ordering follows directly from the definition. It is straightforward but somewhat of a hassle to show that the correspondence respects the arithmetic. That does it. |
| 207 | | |
| 208 | | EXERCISE: Sketch an argument to show that an Archimedean ordered field is isomorphic to a subfield of the real numbers. |
| 209 | | |
| 210 | | |
| 211 | | |
| 212 | | EPILOG |
| 213 | | |
| 214 | | We have defined “the real number line” to be something that satisfies the real number axioms, that is, we have defined it to be a complete ordered field. We have sketched the proof that there is a complete ordered field and that it is unique (up to isomorphism). The idea of this approach is quite simple in hindsight, yet it was quite difficult historically for mathematicians to arrive at this point of view. This approach has the effect of divorcing the concept of the real numbers from its geometric origins. This may seem simple, but actually it was quite a difficult step for mathematicians to take (and it is a step that we would not ask school children to take). As mathematicians such as Weierstrass and Dedekind were preparing their calculus lectures, they became ever more acutely aware, over a period of years, that the concept of the real number line was not on a firm footing. Though various ideas had been percolating for some time, the critical year in the historical development of the real number line was 1872, which saw the appearance of Dedekind’s monograph and papers of Meray, Cantor, and Heine (a student of Weierstrass). |
| 215 | | |
| 216 | | The degeometrization of the real numbers was not carried out without skepticism. In his opus Mathematical Thought from Ancient to Modern Times, mathematics historian Morris Kline quotes Hermann Hankel (a brilliant mathematician, died in 1873 at age 34), who wrote in 1867: |
| 217 | | |
| 218 | | Every attempt to treat the irrational numbers formally and without the concept of [geometric] magnitude must lead to the most abstruse and troublesome artificialities, which, even if they can be carried through with complete rigor, as we have every right to doubt, do not have a higher scientific value. |
| 219 | | |
| 220 | | It is not clear that even Dedekind grasped the import of what he had done. According to Kline again, when Heinrich Weber told Dedekind that he should say that an irrational number is no more than the cut, Dedekind responded (in a letter of 1888) that in fact the irrational number is not the cut itself but something distinct, which corresponds to the cut and brings about the cut. |
| 221 | | |
| 222 | | We may compare the divorce of the construction of the real numbers from geometry to the divorce of the foundations of geometry from its origins in the Euclidean geometry of space. Those divorce proceedings lasted through the nineteenth century and beyond with the development and discovery of non-Euclidean geometries and various axiomatic approaches to geometry, including finite geometries.2 Exponential, Logarithmic and Hyperbolic Functions |
| 223 | | Definition 3: |
| 224 | | If , the logarithm to the base a of x is . |
| 225 | | Definition 4: |
| 226 | | The number e is defined by |
| 227 | | . |
| 228 | | Note: |
| 229 | | . |
| 230 | | |
| 231 | | Theorem 5: |
| 232 | | is differentiable for and |
| 233 | | . |
| 234 | | [justifications of theorem 5:] |
| 235 | | |
| 236 | | Theorem 6: |
| 237 | | is differentiable for all x and |
| 238 | | . |
| 239 | | [justifications of theorem 6:] |
| 240 | | |
| 241 | | Note: |
| 242 | | . |
| 243 | | Definition 5 (The hyperbolic cosine and sine functions): |
| 244 | | The hyperbolic cosine function is defined as |
| 245 | | , |
| 246 | | while the hyperbolic sine function is defined as |
| 247 | | . |
| 248 | | |
| 249 | | |
| 250 | | Theorem 7: |
| 251 | | . |
| 252 | | |
| 253 | | Note: |
| 254 | | . |
| 255 | | Note: |
| 256 | | |
| 257 | | since |
| 258 | | |
| 259 | | |
| 260 | | Definition 6 (other hyperbolic functions): |
| 261 | | The hyperbolic cosine function is defined as |
| 262 | | |
| 263 | | |
| 264 | | Hyperbolic Functions |
| 265 | | Hyperbolic cosine of x: |
| 266 | | |
| 267 | | (vi) Sector Area |
| 268 | | |
| 269 | | |
| 270 | | |
| 271 | | |
| 272 | | |
| 273 | | Example For the cardioid , the total area enclosed by the curve will be given by |
| 274 | | |
| 275 | | |
| 276 | | Example For one of the leaves of the four leaved rose |
| 277 | | |
| 278 | | |
| 279 | | (vii) Intersections of curves expressed in polar form. |
| 280 | | Example Find the points of intersection of and . Find also the area enclosed between the two graphs, outside the cardioid. |
| 281 | | |
| 282 | | For intersections |
| 283 | | |
| 284 | | Points of intersection are in the form |
| 285 | | and |
| 286 | | The sketch below helps. |
| 287 | | |
| 288 | | |
| 289 | | |
| 290 | | |
| 291 | | |
| 292 | | |
| 293 | | Shaded area = area of sector of circle |
| 294 | | |
| 295 | | |
| 296 | | |
| 297 | | |
| 298 | | |
| 299 | | |
| 300 | | Hyperbolic sine of x: |
| 301 | | |
| 302 | | Hyperbolic tangent: |
| 303 | | Hyperbolic cotangent: |
| 304 | | Hyperbolic secant: |
| 305 | | Hyperbolic cosecant: |
| 306 | | |
| 307 | | Identities |
| 308 | | |
| 309 | | |
| 310 | | Derivatives |
| 311 | | |
| 312 | | |
| 313 | | |
| 314 | | Integrals |
| 315 | | |
| 316 | | |
| 317 | | |
| 318 | | Useful Identities |
| 319 | | |
| 320 | | |
| 321 | | |
| 322 | | Derivatives of Inverse Logarithm Formulas for Evaluating |
| 323 | | Hyperbolic Functions Inverse Hyperbolic Functions |
| 324 | | |
| 325 | | |
| 326 | | |
| 327 | | Integrals of Inverse Hyperbolic Functions |
| 328 | | |
| 329 | | |
| 330 | | Lesson Summary: |
| 331 | | Students will measure distances and angles in Euclidean and Hyperbolic space on intersecting line segments, circles and triangles to discover the character of hyperbolic space. Students will use this knowledge to construct a triangle and determine whether triangle in hyperbolic space have circumcenters. |
| 332 | | |
| 333 | | Key words: |
| 334 | | H2, unit disc, d-line, d-segment, d-circle, boundary |
| 335 | | |
| 336 | | Background Knowledge: |
| 337 | | Students should be familiar with Cabri software. The hyperbolic menu should be downloaded from the internet. The lab should be completed after the students have studied the axioms and theorems in absolute and Euclidean geometry. Specifically, students should be familiar with the linear pair axiom, vertical pair theorem, properties of circles, properties of triangles, isosceles triangle theorem and circumcenter of triangles. |
| 338 | | |
| 339 | | Learning Objectives |
| 340 | | 1. To become familiar with the concept of distance and angle measure in hyperbolic space to test axioms and theorems which are true in Euclidean space. |
| 341 | | 2. Understand the existence of perpendicular and parallel lines in hyperbolic space. |
| 342 | | 3. Determine if isosceles triangle theorem holds in hyperbolic space. |
| 343 | | 4. Use understanding of the nature of hyperbolic space to determine if hyperbolic triangles have circumcenters. |
| 344 | | |
| 345 | | Materials |
| 346 | | Cabri |
| 347 | | Access to lab via the internet |
| 348 | | |
| 349 | | Assessment |
| 350 | | A lab report with answers to the questions and constructions illustrating completion of the lab |
| 351 | | |
| 352 | | |
| 353 | | Circmcenter of d-Triangles |
| 354 | | |
| 355 | | |
| 356 | | |
| 357 | | Lab Goal: To determine if the hyperbolic triangles have circumcenters. |
| 358 | | |
| 359 | | Activity: |
| 360 | | Part I. Comparing Euclidean and Hyperbolic Space (open Hyperbol.men) |
| 361 | | Euclidean Space (this section should confirm what you already know) |
| 362 | | |
| 363 | | 1. Create two intersecting lines AB, and CD. (Use line tool) |
| 364 | | Label the point of intersection P. (intersection point tool ) |
| 365 | | |
| 366 | | 2. Create segment AP, PB, CP, and PD. (Use segment tool) |
| 367 | | Hide the lines. (Use hide/show tool) |
| 368 | | |
| 369 | | 3. Measure <APD and <APC. Use the calculator function to add the angles. Record the value. What theorem or axiom have you just illustrated? Do you predict that this will hold in hyperbolic geometry? Why? (use the angle measurement tool) |
| 370 | | |
| 371 | | |
| 372 | | |
| 373 | | |
| 374 | | |
| 375 | | 4. Measure <CPB. Record the value. Compare to <APD. What theorem or axiom have you just demonstrated? Do you predict that this will hold in hyperbolic geometry? Why? |
| 376 | | |
| 377 | | |
| 378 | | |
| 379 | | |
| 380 | | |
| 381 | | |
| 382 | | |
| 383 | | |
| 384 | | |
| 385 | | Hyperbolic Space – The hyperbolic menus appears on the last four buttons on the toolbar. These will be referred to as: |
| 386 | | 12. Figure Menu |
| 387 | | 13. Construction Menu |
| 388 | | 14. Reflection Menu |
| 389 | | 15. Measurement Menu |
| 390 | | |
| 391 | | |
| 392 | | 5. Create your hyperbolic plane. (On the Measurement menu – Button 15, create a unit disc by choosing a center point and a point on the x-axis which will represent 1 unit. All constructions made here have the properties of H2. Create a d-line (on the Figure Menu) by choosing two points A and B and then choosing the unit circle. (Label A and B using the Euclidean label tool.) |
| 393 | | |
| 394 | | |
| 395 | | 6. Create a d-segment AB on the d-line (Figure Menu) by choosing A and B and then the unit circle. This segment is also called an “arc”. Create d-line CD and d-segment CD such that AB and CD intersect. Label the intersection P. |
| 396 | | |
| 397 | | 7. Measure the non-E distance of AB and CD (Using the Measurement menu, select two points, the axis and then the unit circle). What values do you get? |
| 398 | | |
| 399 | | |
| 400 | | |
| 401 | | |
| 402 | | 8. Measure < APD and <APC using “angle” on the Measurement menu (On the hyperbolic menu). Add them together. Record the value. |
| 403 | | |
| 404 | | |
| 405 | | |
| 406 | | |
| 407 | | |
| 408 | | 9. What does this suggest? |
| 409 | | |
| 410 | | |
| 411 | | |
| 412 | | |
| 413 | | 10. Measure < DPB and compare to < APC. What do you notice about the measurements? |
| 414 | | |
| 415 | | |
| 416 | | |
| 417 | | 11. Move segment AB. What do you notice about the distance and angle measurements? What does this demonstrate? |
| 418 | | |
| 419 | | |
| 420 | | |
| 421 | | |
| 422 | | 12. Move Point B outside the circle? What happens? Why? |
| 423 | | |
| 424 | | |
| 425 | | |
| 426 | | |
| 427 | | |
| 428 | | Hyperbolic Inquiry Lesson |
| 429 | | |
| 430 | | Do d-Triangles have circumcenters? |
| 431 | | |
| 432 | | INTRODUCTION |
| 433 | | More than two thousand years ago Euclid of Alexandria collected, compiled, and composed the thirteen volumes of geometry known as the Elements. This magnum opus would become the quintessential model of the way in which mathematics is structured, namely, the axiomatic method. Euclid began by defining his terms and then laying forth his postulates and common notions, both of which can be viewed as the assumptions he would work from as his did his geometry. He then set to work in a proposition-proof format wherein each result was proved using only that which came before it. Now, it should be noted that Euclid, though his work was masterful, was not without error. He failed to recognize as we do now that it is logically futile to define all terms and so there must be undefined terms; it has also been uncovered that right from his first proof he made assumptions about things like betweenness and continuity that were not listed in his postulates and common notions. Nevertheless, Euclid’s Elements was a logical and mathematical tour de force that was the standard-bearer of mathematical reasoning and certainty, the standard-bearer, that is, until it all came crashing down. |
| 434 | | The crash occurred when two mathematicians—János Bolyai of Hungary and Nikolai Lobachevsky of Russia—independently discovered that Euclid’s famous fifth postulate was independent of the others, leading to a consistent non-Euclidean geometry. So it was that mathematics’ surest foundation was shaken. To get a better perspective on this historic event let us take a moment to consider Euclid’s postulates, giving particular attention to his famous fifth. |
| 435 | | Euclid’s postulates, as recorded in Book I of the Elements, are as follows: |
| 436 | | 1. A unique line segment exists between any two distinct points. |
| 437 | | 2. A line segment can be uniquely extended in a straight manner. |
| 438 | | 3. A circle exists given any center and radius. |
| 439 | | 4. A right angle is equal to any other right angle. |
| 440 | | 5. If a line falling on two other lines makes the interior angles on the same side less than two right angles, then the two lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. |
| 441 | | Many mathematicians felt (and it is hard to blame them!) that the fifth was too long and complicated to be a postulate and believed that it could be derived from the first four, all of which were intuitively clear and acceptable. The many attempts to prove the fifth postulate, however, were unsuccessful. Then in the early 19th Century Bolyai and Lobachevsky published their discoveries of hyperbolic geometry, the former’s work based on replacing the fifth postulate with a parameter and the latter’s based on the postulate’s negation. No longer was Euclidean geometry the sole study of shape and space. Eventually it would be proved with the introduction of hyperbolic models (embedded in Euclidean space) by Klein, Poincaré and Beltrami that the consistencies of hyperbolic geometry and Euclidean geometry were logically equivalent. Alas, the proof attempts of Euclid V were doomed from the start! |
| 442 | | A close inspection of the fifth postulate reveals that two negations exist. One negation is the statement that there exist two lines such that a transversal forms angles on one side less than two right angles but, when produced indefinitely, the two lines do not meet on either side; but to say that the two lines, if produced indefinitely, also meet on the other side is another negation. The first negation leads to hyperbolic geometry, which will be the environment of the explorations to come. The second negation, on the other hand, leads to spherical geometry which is itself an intriguing world in which to do geometry but, unfortunately, does not satisfy Euclid’s first postulate (there is more than one line segment between two distinct points) and will not be discussed in the remainder of this paper, except for a few comparative comments in passing. |
| 443 | | As indicated above, what follows is a collection of explorations in the world of hyperbolic geometry. The sections are written in an active voice, much like Euclid’s own Elements (e.g., he would write “let AC be drawn through B” rather than “let AC be the segment containing B”). As the reader, you should envision the paper as documentation of a student’s investigative excursion into this non-Euclidean landscape, complete with false starts and modifications. |
| 444 | | Euclid’s first four postulates will be cited as axioms, as will a few of Hilbert’s additional axioms, and there will be conjecturing and proving that takes place. All the while, though, the geometry will appeal to intuition and be grounded on the models. So that is where we begin. |
| 445 | | EXPLORATION 1: FINDING A MODEL |
| 446 | | Euclidean geometry is the study of size, shape, distances, and so forth, in an ambient space that is in some-sense flat. The most common manifestation of this is the doing of geometry on a piece of paper on a desk. The ground on which we walk, run, and generally live is also perceived to be flat. From such experiences it is natural to assume several things because they seem to be intuitively true. First, between any two points we can find a unique line. Second, if we have a segment of a line then we can extend it in a straight manner. Third, we can construct a unique circle so long as we know the center and the radius. And fourth, a right angle is a right angle is a right angle. These assumptions, or axioms, are based on the familiar “flat” geometry, but also hold on other surfaces such as surfaces with constant positive curvature (e.g., a sphere) and surfaces with constant negative curvature (e.g., a hyperboloid). Let us see what happens if we delve into the latter case, known as hyperbolic geometry. |
| 447 | | Our first order of business is to make sure that we understand what the axioms are saying in a negatively curved environment. We will take words like “between,” “on”, “point,” “line” and “congruent” to be undefined terms. This does not mean we are without guidance with regard to their meaning because intuition plays an important role. For instance, we can think of two figures as being congruent if we can rigidly move one precisely onto the other, and a line can be conceptualized as the path marking the shortest distance between its points. |
| 448 | | What is the shortest path between two points A and B in hyperbolic geometry? Using our model, we can stretch a string tautly along the surface of the hyperboloid. Based on investigations of this sort we see that a “straight line” on our model is the intersection of the hyperboloid with a plane through the central point. The result of such an intersection can be a hyperbola (of which we would only use half), an ellipse or a circle. In the latter two cases we run into a problem because two points can determine more than one line. Specifically, if A and B are antipodal points of a circle or ellipse, such as the one shown in figure 2, then either arc of the circle or ellipse is a line segment between the two points. This is a clear violation of Axiom 1. |
| 449 | | Perhaps we will not be able to proceed in a way similar to the explorations of spherical geometry. Perhaps it is not easy to find a negatively curved surface on which to physically conduct hyperbolic business. Hence we must return and contemplate what it is we are trying to accomplish. |
| 450 | | We have four axioms in hand and want to explore a geometry in which the ambient space is not necessarily “flat.” Another way to think about this is that the lines in the geometry are not necessarily “straight.” These two ideas are related because a perceived curvature of lines could really be just a symptom of the curvature of the underlying space, but rather than try to identify that space we can just accept the fact that lines appear to be curved. Of course line segments would still be the shortest path between two points because it could be the case that what looks like a straight path actually rises or dips through the ambient curvature, making it longer than it seems. So how can we model this geometry containing “curved” lines? |
| 451 | | Let A and B be two distinct points. We want to define a line l through A and B, but it has to be unique to satisfy Axiom 1. Thus it cannot be simply any curve containing the points because there are many of those. A third point C that was non-collinear with A and B would determine a unique circle, and we could define l to be the minor arc between A and B of that circle. Assuming C is fixed, for points D and E that are collinear with C we could define the line segment between them to be the normal straight line segment. However, as soon as we fix C there are points, say F and G, which lie diametrically opposed to each other with respect to their circle formed with C. In this case there is not a unique line segment and Axiom 1 is violated. (Axiom 2 also fails—the “lines” are compact.) |
| 452 | | Again, let A and B be distinct points. Instead of fixing a point we can fix a line l below A and B. If m is the perpendicular bisector of the Euclidean segment AB, then m either intersects l at a point C or is parallel to l (again, in the Euclidean sense). In the first case, Axiom 3 gives us a unique circle Γ through A and B with C as the center. In the second case, we have a ray n emanating perpendicularly from l and containing A and B. In either case, we have a way to define line segments for all points lying in the half plane above line l. |
| 453 | | |
| 454 | | Let us quickly check the four axioms. Per the paragraph above, we know that a unique line segment exists for any two points above line l because we can choose the arc of the circle that lies above l (or else we have a case of the vertical ray which also presents a unique line segment). If we use the open half-plane above l then any line segment has an open neighborhood around it, and thus we can extend the line segment to include a bit more of the hemisphere. (This suggests, however, that distances grow exponentially as you get nearer to l.) We can define a hyperbolic circle as the set of all points a fixed distance away from a fixed center, which satisfies the third axiom by design. Finally, we can define hyperbolic angle measures to be the same as the Euclidean angle measures between the tangent lines of the intersecting arcs; ergo, the fourth axiom in Euclidean geometry implies the fourth axiom in our model. |
| 455 | | Thankfully, we seem to have found a workable model for the geometry that we wish to investigate (indeed, in finding the model we have already been investigating quite intensely). A summary seems appropriate. |
| 456 | | • Rather than construct an explicit surface on which to do hyperbolic geometry, we have changed our visual image of “line” and relegated the ambient curvature to the background. |
| 457 | | • The set of points for our hyperbolic plane model is the open upper half-plane as determined by a line l. |
| 458 | | • The line segment between two points is either the arc of the circle with center on l containing the two points, or is the segment of the ray perpendicular to l containing the two points. |
| 459 | | • As you move closer and closer to l the underlying space curves more and more, that is to say, the hyperbolic distances do not match the Euclidean distances present in our model. |
| 460 | | |
| 461 | | |
| 462 | | EXPLORATION 2: PARALLEL LINES |
| 463 | | With a model of hyperbolic geometry at our disposal we can now examine the nature of lines and line segments in this new world. From past experience we know that parallel lines in Euclidean geometry are everywhere equidistant in a certain sense, and in spherical geometry parallel lines do not exist. One illuminating way to formulate this distinction is by choosing a line m and a point P not on m. The question is: how many lines parallel to m contain P? The Euclidean answer is one, and the spherical answer is zero. Let us seek the hyperbolic answer. |
| 464 | | To proceed, it is necessary to make explicit what we mean by “parallel.” |
| 465 | | Definition. Two lines are parallel if they have no points in common. |
| 466 | | Furthermore, it is important to note that in Euclidean geometry two distinct circles can meet in 0 points, 1 point, or 2 points, and the single point situation occurs if and only if the circles meet tangentially at that point. We will use this because the hyperbolic lines of our model can also be thought of as circles in the traditional Euclidean sense. |
| 467 | | Now, let m be a line in the hyperbolic plane and let P be a point not on m. Label the boundary points of m as A and B. We can construct the Euclidean line segment PA and then bisect it perpendicularly. If this perpendicular bisector intersects line l then we can use this intersection point as the center of a circle and construct the hyperbolic line n that passes through P and A (though A is not actually in the hyperbolic plane, this is important!). The Euclidean circles m and n meet at the point A, and there they are both orthogonal to line l which means that they meet tangentially. This means that A is the only point at which they meet. But A is technically off the hyperbolic plane, so necessarily m and n do not meet in the hyperbolic plane. Thus, by definition, they are parallel hyperbolic lines. If the perpendicular bisector of PA does not intersect line l then we can construct the ray from A to P. This is a hyperbolic line that meets m only at the point A, and so is also parallel to m in hyperbolic geometry. |
| 468 | | The paragraph above has proven the following result in hyperbolic geometry. |
| 469 | | Proposition 1. If m is a line and P is a point not on m, then there exists a |
| 470 | | line through P parallel to m. |
| 471 | | So we see that hyperbolic geometry is inherently different than spherical geometry. Moreover, it is inherently different than Euclidean geometry because we can repeat the argument above using the point B in place of A, and this will give us another line through P parallel to m! |
| 472 | | Proposition 1 (updated). There exist at least two lines through P parallel to m. |
| 473 | | A bit more examination uncovers infinitely many parallels to m through P (see figure 8). However, we are seeing that the difference between parallels in hyperbolic geometry and Euclidean geometry is more than just a matter of multitude, there is a qualitative difference as well. In hyperbolic geometry we have some parallel lines (like m and n in figure 7) that diverge in one direction but converge in the other, and we have other parallel lines that diverge in both directions. |
| 474 | | Definition. Parallel lines are ultraparallel if they diverge in both directions, and are asymptotically parallel if they converge in one direction. |
| 475 | | |
| 476 | | |
| 477 | | |
| 478 | | |
| 479 | | |
| 480 | | |
| 481 | | |
| 482 | | The asymptotically parallel lines (of which there are two, based on our proof of Proposition 1) seem to be the bounds of a region that contains m, and any hyperbolic line through P contained in that region will necessarily intersect m. Conversely, any hyperbolic line through P outside of that region will be ultraparallel. |
| 483 | | We have defined parallel as non-intersecting. There is another notion, however, related to parallelism that is worth consideration—the parallel transport. |
| 484 | | Definition. Two lines are parallel transports of one another if there exists a transversal that creates equal corresponding angles. |
| 485 | | In Euclidean geometry two lines are parallel transports if and only if they are parallel. Does such a result hold in hyperbolic geometry? |
| 486 | | Proposition 2. If two lines are ultraparallel, then they are parallel transports. |
| 487 | | Let m and n be ultraparallels. Our task is to find a third line p that creates equal angles in corresponding positions with regard to m and n. Recall that the hyperbolic angles in our model are conformal to the Euclidean angles. |
| 488 | | Intuitively, if we think of a very small (in the sense of Euclidean circles) transversal, this will create an angle with respect to m that is nearly zero and a corresponding angle with respect to n that is nearly two right angles (see figure 9). Now, we let the radius of the transversal circle (i.e., the hyperbolic line) grow until it is nearly the largest transversal possible. In this case, the angle in the same position as before is nearly two right angles with respect to m and is nearly zero with respect to n. They have switched the inequality! Since this process of growth was continuous, by the intermediate value theorem, there exists some transversal p that creates equal corresponding angles. Thus m and n are parallel transports along p. |
| 489 | | It is important to note that the argument for Proposition 2 fails for asymptotically parallel lines. |
| 490 | | |
| 491 | | |
| 492 | | web posting, November, 20062 Exponential, Logarithmic and Hyperbolic Functions |
| 493 | | Definition 3: |
| 494 | | If , the logarithm to the base a of x is . |
| 495 | | Definition 4: |
| 496 | | The number e is defined by |
| 497 | | . |
| 498 | | Note: |
| 499 | | . |
| 500 | | |
| 501 | | Theorem 5: |
| 502 | | is differentiable for and |
| 503 | | . |
| 504 | | [justifications of theorem 5:] |
| 505 | | |
| 506 | | Theorem 6: |
| 507 | | is differentiable for all x and |
| 508 | | . |
| 509 | | [justifications of theorem 6:] |
| 510 | | |
| 511 | | Note: |
| 512 | | . |
| 513 | | Definition 5 (The hyperbolic cosine and sine functions): |
| 514 | | The hyperbolic cosine function is defined as |
| 515 | | , |
| 516 | | while the hyperbolic sine function is defined as |
| 517 | | . |
| 518 | | |
| 519 | | |
| 520 | | Theorem 7: |
| 521 | | . |
| 522 | | |
| 523 | | Note: |
| 524 | | . |
| 525 | | Note: |
| 526 | | |
| 527 | | since |
| 528 | | |
| 529 | | |
| 530 | | Definition 6 (other hyperbolic functions): |
| 531 | | The hyperbolic cosine function is defined as |
| 532 | | |
| 533 | | |
| 534 | | Hyperbolic Functions |
| 535 | | Hyperbolic cosine of x: |
| 536 | | |
| 537 | | (vi) Sector Area |
| 538 | | |
| 539 | | |
| 540 | | |
| 541 | | |
| 542 | | |
| 543 | | Example For the cardioid , the total area enclosed by the curve will be given by |
| 544 | | |
| 545 | | |
| 546 | | Example For one of the leaves of the four leaved rose |
| 547 | | |
| 548 | | |
| 549 | | (vii) Intersections of curves expressed in polar form. |
| 550 | | Example Find the points of intersection of and . Find also the area enclosed between the two graphs, outside the cardioid. |
| 551 | | |
| 552 | | For intersections |
| 553 | | |
| 554 | | Points of intersection are in the form |
| 555 | | and |
| 556 | | The sketch below helps. |
| 557 | | |
| 558 | | |
| 559 | | |
| 560 | | |
| 561 | | |
| 562 | | |
| 563 | | Shaded area = area of sector of circle |
| 564 | | |
| 565 | | |
| 566 | | |
| 567 | | |
| 568 | | |
| 569 | | |
| 570 | | Hyperbolic sine of x: |
| 571 | | |
| 572 | | Hyperbolic tangent: |
| 573 | | Hyperbolic cotangent: |
| 574 | | Hyperbolic secant: |
| 575 | | Hyperbolic cosecant: |
| 576 | | |
| 577 | | Identities |
| 578 | | |
| 579 | | |
| 580 | | Derivatives |
| 581 | | |
| 582 | | |
| 583 | | |
| 584 | | Integrals |
| 585 | | |
| 586 | | |
| 587 | | |
| 588 | | Useful Identities |
| 589 | | |
| 590 | | |
| 591 | | |
| 592 | | Derivatives of Inverse Logarithm Formulas for Evaluating |
| 593 | | Hyperbolic Functions Inverse Hyperbolic Functions |
| 594 | | |
| 595 | | |
| 596 | | |
| 597 | | Integrals of Inverse Hyperbolic Functions |
| 598 | | |
| 599 | | |
| 600 | | Lesson Summary: |
| 601 | | Students will measure distances and angles in Euclidean and Hyperbolic space on intersecting line segments, circles and triangles to discover the character of hyperbolic space. Students will use this knowledge to construct a triangle and determine whether triangle in hyperbolic space have circumcenters. |
| 602 | | |
| 603 | | Key words: |
| 604 | | H2, unit disc, d-line, d-segment, d-circle, boundary |
| 605 | | |
| 606 | | Background Knowledge: |
| 607 | | Students should be familiar with Cabri software. The hyperbolic menu should be downloaded from the internet. The lab should be completed after the students have studied the axioms and theorems in absolute and Euclidean geometry. Specifically, students should be familiar with the linear pair axiom, vertical pair theorem, properties of circles, properties of triangles, isosceles triangle theorem and circumcenter of triangles. |
| 608 | | |
| 609 | | Learning Objectives |
| 610 | | 1. To become familiar with the concept of distance and angle measure in hyperbolic space to test axioms and theorems which are true in Euclidean space. |
| 611 | | 2. Understand the existence of perpendicular and parallel lines in hyperbolic space. |
| 612 | | 3. Determine if isosceles triangle theorem holds in hyperbolic space. |
| 613 | | 4. Use understanding of the nature of hyperbolic space to determine if hyperbolic triangles have circumcenters. |
| 614 | | |
| 615 | | Materials |
| 616 | | Cabri |
| 617 | | Access to lab via the internet |
| 618 | | |
| 619 | | Assessment |
| 620 | | A lab report with answers to the questions and constructions illustrating completion of the lab |
| 621 | | |
| 622 | | |
| 623 | | Circmcenter of d-Triangles |
| 624 | | |
| 625 | | |
| 626 | | |
| 627 | | Lab Goal: To determine if the hyperbolic triangles have circumcenters. |
| 628 | | |
| 629 | | Activity: |
| 630 | | Part I. Comparing Euclidean and Hyperbolic Space (open Hyperbol.men) |
| 631 | | Euclidean Space (this section should confirm what you already know) |
| 632 | | |
| 633 | | 1. Create two intersecting lines AB, and CD. (Use line tool) |
| 634 | | Label the point of intersection P. (intersection point tool ) |
| 635 | | |
| 636 | | 2. Create segment AP, PB, CP, and PD. (Use segment tool) |
| 637 | | Hide the lines. (Use hide/show tool) |
| 638 | | |
| 639 | | 3. Measure <APD and <APC. Use the calculator function to add the angles. Record the value. What theorem or axiom have you just illustrated? Do you predict that this will hold in hyperbolic geometry? Why? (use the angle measurement tool) |
| 640 | | |
| 641 | | |
| 642 | | |
| 643 | | |
| 644 | | |
| 645 | | 4. Measure <CPB. Record the value. Compare to <APD. What theorem or axiom have you just demonstrated? Do you predict that this will hold in hyperbolic geometry? Why? |
| 646 | | |
| 647 | | |
| 648 | | |
| 649 | | |
| 650 | | |
| 651 | | |
| 652 | | |
| 653 | | |
| 654 | | |
| 655 | | Hyperbolic Space – The hyperbolic menus appears on the last four buttons on the toolbar. These will be referred to as: |
| 656 | | 12. Figure Menu |
| 657 | | 13. Construction Menu |
| 658 | | 14. Reflection Menu |
| 659 | | 15. Measurement Menu |
| 660 | | |
| 661 | | |
| 662 | | 5. Create your hyperbolic plane. (On the Measurement menu – Button 15, create a unit disc by choosing a center point and a point on the x-axis which will represent 1 unit. All constructions made here have the properties of H2. Create a d-line (on the Figure Menu) by choosing two points A and B and then choosing the unit circle. (Label A and B using the Euclidean label tool.) |
| 663 | | |
| 664 | | |
| 665 | | 6. Create a d-segment AB on the d-line (Figure Menu) by choosing A and B and then the unit circle. This segment is also called an “arc”. Create d-line CD and d-segment CD such that AB and CD intersect. Label the intersection P. |
| 666 | | |
| 667 | | 7. Measure the non-E distance of AB and CD (Using the Measurement menu, select two points, the axis and then the unit circle). What values do you get? |
| 668 | | |
| 669 | | |
| 670 | | |
| 671 | | |
| 672 | | 8. Measure < APD and <APC using “angle” on the Measurement menu (On the hyperbolic menu). Add them together. Record the value. |
| 673 | | |
| 674 | | |
| 675 | | |
| 676 | | |
| 677 | | |
| 678 | | 9. What does this suggest? |
| 679 | | |
| 680 | | |
| 681 | | |
| 682 | | |
| 683 | | 10. Measure < DPB and compare to < APC. What do you notice about the measurements? |
| 684 | | |
| 685 | | |
| 686 | | |
| 687 | | 11. Move segment AB. What do you notice about the distance and angle measurements? What does this demonstrate? |
| 688 | | |
| 689 | | |
| 690 | | |
| 691 | | |
| 692 | | 12. Move Point B outside the circle? What happens? Why? |
| 693 | | |
| 694 | | |
| 695 | | |
| 696 | | |
| 697 | | |
| 698 | | Hyperbolic Inquiry Lesson |
| 699 | | |
| 700 | | Do d-Triangles have circumcenters? |
| 701 | | |
| 702 | | INTRODUCTION |
| 703 | | More than two thousand years ago Euclid of Alexandria collected, compiled, and composed the thirteen volumes of geometry known as the Elements. This magnum opus would become the quintessential model of the way in which mathematics is structured, namely, the axiomatic method. Euclid began by defining his terms and then laying forth his postulates and common notions, both of which can be viewed as the assumptions he would work from as his did his geometry. He then set to work in a proposition-proof format wherein each result was proved using only that which came before it. Now, it should be noted that Euclid, though his work was masterful, was not without error. He failed to recognize as we do now that it is logically futile to define all terms and so there must be undefined terms; it has also been uncovered that right from his first proof he made assumptions about things like betweenness and continuity that were not listed in his postulates and common notions. Nevertheless, Euclid’s Elements was a logical and mathematical tour de force that was the standard-bearer of mathematical reasoning and certainty, the standard-bearer, that is, until it all came crashing down. |
| 704 | | The crash occurred when two mathematicians—János Bolyai of Hungary and Nikolai Lobachevsky of Russia—independently discovered that Euclid’s famous fifth postulate was independent of the others, leading to a consistent non-Euclidean geometry. So it was that mathematics’ surest foundation was shaken. To get a better perspective on this historic event let us take a moment to consider Euclid’s postulates, giving particular attention to his famous fifth. |
| 705 | | Euclid’s postulates, as recorded in Book I of the Elements, are as follows: |
| 706 | | 1. A unique line segment exists between any two distinct points. |
| 707 | | 2. A line segment can be uniquely extended in a straight manner. |
| 708 | | 3. A circle exists given any center and radius. |
| 709 | | 4. A right angle is equal to any other right angle. |
| 710 | | 5. If a line falling on two other lines makes the interior angles on the same side less than two right angles, then the two lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. |
| 711 | | Many mathematicians felt (and it is hard to blame them!) that the fifth was too long and complicated to be a postulate and believed that it could be derived from the first four, all of which were intuitively clear and acceptable. The many attempts to prove the fifth postulate, however, were unsuccessful. Then in the early 19th Century Bolyai and Lobachevsky published their discoveries of hyperbolic geometry, the former’s work based on replacing the fifth postulate with a parameter and the latter’s based on the postulate’s negation. No longer was Euclidean geometry the sole study of shape and space. Eventually it would be proved with the introduction of hyperbolic models (embedded in Euclidean space) by Klein, Poincaré and Beltrami that the consistencies of hyperbolic geometry and Euclidean geometry were logically equivalent. Alas, the proof attempts of Euclid V were doomed from the start! |
| 712 | | A close inspection of the fifth postulate reveals that two negations exist. One negation is the statement that there exist two lines such that a transversal forms angles on one side less than two right angles but, when produced indefinitely, the two lines do not meet on either side; but to say that the two lines, if produced indefinitely, also meet on the other side is another negation. The first negation leads to hyperbolic geometry, which will be the environment of the explorations to come. The second negation, on the other hand, leads to spherical geometry which is itself an intriguing world in which to do geometry but, unfortunately, does not satisfy Euclid’s first postulate (there is more than one line segment between two distinct points) and will not be discussed in the remainder of this paper, except for a few comparative comments in passing. |
| 713 | | As indicated above, what follows is a collection of explorations in the world of hyperbolic geometry. The sections are written in an active voice, much like Euclid’s own Elements (e.g., he would write “let AC be drawn through B” rather than “let AC be the segment containing B”). As the reader, you should envision the paper as documentation of a student’s investigative excursion into this non-Euclidean landscape, complete with false starts and modifications. |
| 714 | | Euclid’s first four postulates will be cited as axioms, as will a few of Hilbert’s additional axioms, and there will be conjecturing and proving that takes place. All the while, though, the geometry will appeal to intuition and be grounded on the models. So that is where we begin. |
| 715 | | EXPLORATION 1: FINDING A MODEL |
| 716 | | Euclidean geometry is the study of size, shape, distances, and so forth, in an ambient space that is in some-sense flat. The most common manifestation of this is the doing of geometry on a piece of paper on a desk. The ground on which we walk, run, and generally live is also perceived to be flat. From such experiences it is natural to assume several things because they seem to be intuitively true. First, between any two points we can find a unique line. Second, if we have a segment of a line then we can extend it in a straight manner. Third, we can construct a unique circle so long as we know the center and the radius. And fourth, a right angle is a right angle is a right angle. These assumptions, or axioms, are based on the familiar “flat” geometry, but also hold on other surfaces such as surfaces with constant positive curvature (e.g., a sphere) and surfaces with constant negative curvature (e.g., a hyperboloid). Let us see what happens if we delve into the latter case, known as hyperbolic geometry. |
| 717 | | Our first order of business is to make sure that we understand what the axioms are saying in a negatively curved environment. We will take words like “between,” “on”, “point,” “line” and “congruent” to be undefined terms. This does not mean we are without guidance with regard to their meaning because intuition plays an important role. For instance, we can think of two figures as being congruent if we can rigidly move one precisely onto the other, and a line can be conceptualized as the path marking the shortest distance between its points. |
| 718 | | What is the shortest path between two points A and B in hyperbolic geometry? Using our model, we can stretch a string tautly along the surface of the hyperboloid. Based on investigations of this sort we see that a “straight line” on our model is the intersection of the hyperboloid with a plane through the central point. The result of such an intersection can be a hyperbola (of which we would only use half), an ellipse or a circle. In the latter two cases we run into a problem because two points can determine more than one line. Specifically, if A and B are antipodal points of a circle or ellipse, such as the one shown in figure 2, then either arc of the circle or ellipse is a line segment between the two points. This is a clear violation of Axiom 1. |
| 719 | | Perhaps we will not be able to proceed in a way similar to the explorations of spherical geometry. Perhaps it is not easy to find a negatively curved surface on which to physically conduct hyperbolic business. Hence we must return and contemplate what it is we are trying to accomplish. |
| 720 | | We have four axioms in hand and want to explore a geometry in which the ambient space is not necessarily “flat.” Another way to think about this is that the lines in the geometry are not necessarily “straight.” These two ideas are related because a perceived curvature of lines could really be just a symptom of the curvature of the underlying space, but rather than try to identify that space we can just accept the fact that lines appear to be curved. Of course line segments would still be the shortest path between two points because it could be the case that what looks like a straight path actually rises or dips through the ambient curvature, making it longer than it seems. So how can we model this geometry containing “curved” lines? |
| 721 | | Let A and B be two distinct points. We want to define a line l through A and B, but it has to be unique to satisfy Axiom 1. Thus it cannot be simply any curve containing the points because there are many of those. A third point C that was non-collinear with A and B would determine a unique circle, and we could define l to be the minor arc between A and B of that circle. Assuming C is fixed, for points D and E that are collinear with C we could define the line segment between them to be the normal straight line segment. However, as soon as we fix C there are points, say F and G, which lie diametrically opposed to each other with respect to their circle formed with C. In this case there is not a unique line segment and Axiom 1 is violated. (Axiom 2 also fails—the “lines” are compact.) |
| 722 | | Again, let A and B be distinct points. Instead of fixing a point we can fix a line l below A and B. If m is the perpendicular bisector of the Euclidean segment AB, then m either intersects l at a point C or is parallel to l (again, in the Euclidean sense). In the first case, Axiom 3 gives us a unique circle Γ through A and B with C as the center. In the second case, we have a ray n emanating perpendicularly from l and containing A and B. In either case, we have a way to define line segments for all points lying in the half plane above line l. |
| 723 | | |
| 724 | | Let us quickly check the four axioms. Per the paragraph above, we know that a unique line segment exists for any two points above line l because we can choose the arc of the circle that lies above l (or else we have a case of the vertical ray which also presents a unique line segment). If we use the open half-plane above l then any line segment has an open neighborhood around it, and thus we can extend the line segment to include a bit more of the hemisphere. (This suggests, however, that distances grow exponentially as you get nearer to l.) We can define a hyperbolic circle as the set of all points a fixed distance away from a fixed center, which satisfies the third axiom by design. Finally, we can define hyperbolic angle measures to be the same as the Euclidean angle measures between the tangent lines of the intersecting arcs; ergo, the fourth axiom in Euclidean geometry implies the fourth axiom in our model. |
| 725 | | Thankfully, we seem to have found a workable model for the geometry that we wish to investigate (indeed, in finding the model we have already been investigating quite intensely). A summary seems appropriate. |
| 726 | | • Rather than construct an explicit surface on which to do hyperbolic geometry, we have changed our visual image of “line” and relegated the ambient curvature to the background. |
| 727 | | • The set of points for our hyperbolic plane model is the open upper half-plane as determined by a line l. |
| 728 | | • The line segment between two points is either the arc of the circle with center on l containing the two points, or is the segment of the ray perpendicular to l containing the two points. |
| 729 | | • As you move closer and closer to l the underlying space curves more and more, that is to say, the hyperbolic distances do not match the Euclidean distances present in our model. |
| 730 | | |
| 731 | | |
| 732 | | EXPLORATION 2: PARALLEL LINES |
| 733 | | With a model of hyperbolic geometry at our disposal we can now examine the nature of lines and line segments in this new world. From past experience we know that parallel lines in Euclidean geometry are everywhere equidistant in a certain sense, and in spherical geometry parallel lines do not exist. One illuminating way to formulate this distinction is by choosing a line m and a point P not on m. The question is: how many lines parallel to m contain P? The Euclidean answer is one, and the spherical answer is zero. Let us seek the hyperbolic answer. |
| 734 | | To proceed, it is necessary to make explicit what we mean by “parallel.” |
| 735 | | Definition. Two lines are parallel if they have no points in common. |
| 736 | | Furthermore, it is important to note that in Euclidean geometry two distinct circles can meet in 0 points, 1 point, or 2 points, and the single point situation occurs if and only if the circles meet tangentially at that point. We will use this because the hyperbolic lines of our model can also be thought of as circles in the traditional Euclidean sense. |
| 737 | | Now, let m be a line in the hyperbolic plane and let P be a point not on m. Label the boundary points of m as A and B. We can construct the Euclidean line segment PA and then bisect it perpendicularly. If this perpendicular bisector intersects line l then we can use this intersection point as the center of a circle and construct the hyperbolic line n that passes through P and A (though A is not actually in the hyperbolic plane, this is important!). The Euclidean circles m and n meet at the point A, and there they are both orthogonal to line l which means that they meet tangentially. This means that A is the only point at which they meet. But A is technically off the hyperbolic plane, so necessarily m and n do not meet in the hyperbolic plane. Thus, by definition, they are parallel hyperbolic lines. If the perpendicular bisector of PA does not intersect line l then we can construct the ray from A to P. This is a hyperbolic line that meets m only at the point A, and so is also parallel to m in hyperbolic geometry. |
| 738 | | The paragraph above has proven the following result in hyperbolic geometry. |
| 739 | | Proposition 1. If m is a line and P is a point not on m, then there exists a |
| 740 | | line through P parallel to m. |
| 741 | | So we see that hyperbolic geometry is inherently different than spherical geometry. Moreover, it is inherently different than Euclidean geometry because we can repeat the argument above using the point B in place of A, and this will give us another line through P parallel to m! |
| 742 | | Proposition 1 (updated). There exist at least two lines through P parallel to m. |
| 743 | | A bit more examination uncovers infinitely many parallels to m through P (see figure 8). However, we are seeing that the difference between parallels in hyperbolic geometry and Euclidean geometry is more than just a matter of multitude, there is a qualitative difference as well. In hyperbolic geometry we have some parallel lines (like m and n in figure 7) that diverge in one direction but converge in the other, and we have other parallel lines that diverge in both directions. |
| 744 | | Definition. Parallel lines are ultraparallel if they diverge in both directions, and are asymptotically parallel if they converge in one direction. |
| 745 | | |
| 746 | | |
| 747 | | |
| 748 | | |
| 749 | | |
| 750 | | |
| 751 | | |
| 752 | | The asymptotically parallel lines (of which there are two, based on our proof of Proposition 1) seem to be the bounds of a region that contains m, and any hyperbolic line through P contained in that region will necessarily intersect m. Conversely, any hyperbolic line through P outside of that region will be ultraparallel. |
| 753 | | We have defined parallel as non-intersecting. There is another notion, however, related to parallelism that is worth consideration—the parallel transport. |
| 754 | | Definition. Two lines are parallel transports of one another if there exists a transversal that creates equal corresponding angles. |
| 755 | | In Euclidean geometry two lines are parallel transports if and only if they are parallel. Does such a result hold in hyperbolic geometry? |
| 756 | | Proposition 2. If two lines are ultraparallel, then they are parallel transports. |
| 757 | | Let m and n be ultraparallels. Our task is to find a third line p that creates equal angles in corresponding positions with regard to m and n. Recall that the hyperbolic angles in our model are conformal to the Euclidean angles. |
| 758 | | Intuitively, if we think of a very small (in the sense of Euclidean circles) transversal, this will create an angle with respect to m that is nearly zero and a corresponding angle with respect to n that is nearly two right angles (see figure 9). Now, we let the radius of the transversal circle (i.e., the hyperbolic line) grow until it is nearly the largest transversal possible. In this case, the angle in the same position as before is nearly two right angles with respect to m and is nearly zero with respect to n. They have switched the inequality! Since this process of growth was continuous, by the intermediate value theorem, there exists some transversal p that creates equal corresponding angles. Thus m and n are parallel transports along p. |
| 759 | | It is important to note that the argument for Proposition 2 fails for asymptotically parallel lines.Kline, M. (1982). Mathematics: The Loss of Certainty. Oxford: Oxford University Press. |
| 760 | | |
| 761 | | 3.4 Exterior Angle Inequality |
| 762 | | 3.5 The Inequality Theorems |
| 763 | | 3.6 Additional Congruence Criteria |
| 764 | | 3.7 Quadrilaterals |
| 765 | | 3.8 Circles |
| 766 | | |
| 767 | | |
| 768 | | Homework 5 |
| 769 | | |
| 770 | | |
