SYMBOL AND MEANING IN MATHEMATICS

Transcription

1 ,,. SYMBOL AND MEANING IN MATHEMATICS ALICE M. DEAN Mathematics and Computer Science Department Skidmore College May 26,1995 There is perhaps no other field of study that uses symbols as plentifully and precisely as does mathematics. figure 1, for instance, contains a concise, grammatically correct sentence (except for the lack of a period at the end) that uses mathematical symbols to state one of the most fundamental definitions of calculus. This mathematical sentence illustrates the fact that mathematicians have, over thousands of years, developed a language with which to express their ideas. This language is in some sense more complex than a natural language (indeed it builds upon natural language) since the meanings of its symbols and words are highly precise; on the other hand, its words have little of the ambiguity and nuance of natural language and so in that sense it could be considered simpler. Of course, mathematics is far more than a language: it is the goal of mathematics to uncover and describe the fundamental patterns and symmetries underlying our universe. In this article I hope to show some of the ways in which the symbols of mathematics, like the symbols of any language, help us express our thoughts as well as shape the way we think. Figure 1. A mathematical sentence Mathematical symbols and language undergo a Darwinian sort of evolution. As new ideas are discovered, mathematicians introduce new symbols and language to describe them. The fittest of these survive through the ages to be used by subsequent generations. The great mathematician Paul Erdos speaks of a supreme being who has a book. In this book are all the great ideas and proofs of mathematics, expressed in their clearest and most elegant form. The highest compliment Erdos can pay to a mathematical proof is to call it the "book proof." By analogy, the "book symbols" might be those that manage to express our mathematical thoughts in the simplest and most beautiful manner possible. The symbols and language we use to express our ideas profoundly affect how we think about those ideas. From the most fundamental object, the number, to college topics like calculus and abstract algebra, to some of the most complex concepts of mathematics that are studied by a few experts,'the notation and definitions that describe mathematical ideas dictate, to large extent, how easily they will be understood and used. page 10/5

2 , An example that most non-mathematician readers might feel comfortable with is mathematical notation for numbers. Most readers feel at home with standard notation for whole numbers, also called integers: 0,1,2,3, -4, -15,197, -753, etc. Our notation uses a positional number system (i.e., 1'97 is different from 971) based on the use of the ten digits from 0 to 9. We call t:b.1s a base 10 representation;' if we used instead, say seven digits, it would be called base 7. Computers, for example, usually use base 2, base 8, or base 16. Of course there are other ways to represent numbers: most of you probably know how to use Roman numerals for e'xample. Mathematicians also like to represent numbers pictorially, usually as points on a line, like markings on a ruler: o Figure 2. A number line These representations of the whole numbers, or integers, give us a way to express thein, but they also help to shape the way we think about them. For instance, in the number line in Figure 2, the fact that each integer is one unit away from the next one is reflected by the little line segment between them. But what about the positions not marked as integers - for example, what about,the position halfway between 1 and 2? Again we have mathematical symbols to represent this notion. We usually would describe this number either as 1~ or 1.5. The first notation uses fraction notation, which expresses the idea of dividing the numbers, or the line segments that represent them, into parts; the second uses base 10 decimal notation. The fractional notation ~ means to divide the number 1 mto 2 parts; 1~ means the sum 1+ ~. The decimal notation 1.5 is related to fractional notation, but all numbers are divided into parts that are powers of 10: 1.5 =1 + UO' = (2 x 10) +3+ Ko + ~OO' etc. These three ways of representing numbers - as fractions, as decimals, and as points on a line form a foundation on which we can now build, introducing concepts that combine and otherwise use numbers, such as arithmetic, calculus, statistics, etc. But at all stages, our choice of representation either facilitates or impedes our ability to express these concepts, and it helps or hinders us in discovering new concepts as well. ' By continuing with this example of the three number representations, we can demonstrate how the symbols we use can actually inspire us to discover new ideas (and hence new symbols as well!). Let's think about the number represented by the fraction 3{1. To represent this number in. decimal notation, we could simply do long division: = 11) = Figure 3. From fractions to decimals page 2 0/5

3 This calculation reveals several things. First of all, the decimal representation for 7{1' unlike that for ~ = 0.5, is non-terminating: to be exact, it must go on forever! (Of course, we could think of all. decimal representations as non-terminating; for instance, 0.5 = ) It's also interesting to note that this decimal representation has the repeating pattern "18." This observation - which arose from comparing the two notations - motivates the following interesting question: Does every fraction of whole numbers have a repeating decimal expansion? If so, then a number like is interesting because it does not have a repeating pattern (it has an obvious pattern, but not one that repeats). Well, it turns out that the answer to the question, ''Does every fraction of whole numbers have a repeating decimal expansion?," is yes, and it's not even that hard to prove, although I won't prove. it here. Thus our use of decimal notation has led us to the discovery that not all numbers can be represented by fractions of whole numbers. This in turn leads us to new language: a number that can be represented as a fraction of whole numbers (or equivalently, with a repeating decimal expansion) is called rational (so ~, 7{1 ' , etc., are all rational numbers); otherwise it's called irrational (so is irrational). The existence of irrational numbers was discovered by the early Greeks some time around the fifth or sixth century BC, l and the discovery was monumental be~ause it upset many previously held assumptions about numbers. For an example, consider the number 7C, which represents the ratio of the circumference of a circle to its diameter. The number 7C, which figures in many physical applications, has been considered equal to 3 (by the ancient Chinese), 25%1 (by the ancient Egyptians), and 3JIg (by the ancient Babylonians). Even today there are many people who think that 7C equals 2~ or All of these are close to correct, but none is exactly right because it turns out that 7C is irrational (this is quite hard to prove). This is the reason thaut's convenient to invent a special symbol - the Greek letter 7C - to represent this number whose decimal expansion is non-terminating and non-repeating. 2 Actually, the number first discovered to be irrational by the early Greeks was not 7C but.j2, that number which when multiplied by itself (i.e., "squared") gives the answer 2 (thusm = 4 because 4 2 = 4 x4 = 16). The value of.j2 is somewhere between 1 and 2 because 1 2 = 1 is too small, while 2 2 =4 is too big. We can get closer and closer estimates of.j2 by squaring test values and seeing whether the result is bigger or smaller than 2. For instance, = 2.25 is too big so.j2 < 1.5. By continuing this process, we can get better and better approximations of.j2, and it turns out that.j2 is approximately How did the early Greeks prove that.j2 is 1 All historical data in this article comes from Great Moments in Mathematics Before 1650, by Howard Eves, Mathematical Association of America, Washington, DC, For those of you who "surf' the World Wide Web, here is a node where you can find some humorous and interesting information about 7C, including the first million digits of its decimal expansion: page 3 of5

4 irrational? Well first of all, they used the idea that a rational number is one that can be expressed as a fraction of two whole numbers. Second, they used a proof technique called proof by contradiction,3 in which you assume exactly the opposite of what you think is true, and then you show that if this were the case, then something you know to be false would also be true (this is the contradiction). This proof is not very hard, but it illustrates how mathematicians combine symbols and their meaning to draw conclusions, so let's take a look at it. Theorem..J2 is an irrational number. Proof by contradiction. Assume the assertion is false; in other words, assume that.j2 is a rational number. We will show that this leads to a contradiction. Our assumption that.j2 is rational means that we can find two whole nuinbers, which we'll call a and b, with.j2 = %. We may also assume that we have reduced this fraction to lowest terms; in other words, that a and b have no factors in common (so for example, we would reduce the fraction %1 to. 7j by canceling the common factor 3 from top and bottom). From our assumption that.j2 =%it follows that 2 =(%f =a%2. If we multiply both sides of the equation by b 2, we see that 2b 2 = a 2. Since 2b 2 is obviously an even number (i.e. a multiple of 2), so is a 2 But if a 2 is even, then so is a, since the square of an odd number is also odd. Now we know that a must be an even number. To say a number is even means it's a multiple of 2 (for example, the even number 34 is 2 times 17), so we can write a as 2 times something: say, as a =2c. Now let's use this way of writing a_in the equation 2b 2 =a 2 If we replace a by 2c, we get 2b 2 =(2C)2 =4c, and dividing both sides by 2 gives us the new equation b 2 =2c. From this last equation, b 2 = 2c, we see now that b 2 is an even number, so b must also be even. So, based on 'our original assumption that.j2 = %' we have shown that both a and b must be even, i.e., they are both multiples of2. However, this contradicts our other assumption that a and b had no factors in common, in other words, that the fraction %had been reduced to lowest terms. Since any fraction can be reduced to lowest terms, that means our assumption that J2 = %must have been false. Thus we have proven the theorem, because we have shown that.j2 cannot be written as a fraction of whole numbers and hence is an irrational number. I The little black bar on the line after the proof is a standard symbol used to indicate that a proof has been completed. Notice how carefully the definition of rational number was used to prove 3 Also called indirect proofor reductio ad absurdum. page 4 of5

5 OIl.'",.. that..fi could not be rational. We also used several basic properties of aritlnnetic (reducing fractions, clearing parentheses) that we were able to express clearly and succinctly through the use of well-chosen symbols. In particular, the use of the letters a, b, and c to represent undetermined numbers permitted 'us to use concise expressions to describe the relationships among various quantities. From this example, we can see how the careful use of symbols with very precise meanings not only lets us express mathematical ideas, but also leads to new ideas. This cycle of new concepts begetting new words and symbols, which in tum motivate other new concepts, is one that occurs throughout mathematics. Mathematicians will often describe a definition or theorem with words that others use to describe art or literature, e.g., beautiful, elegant, clever (or sometimes clumsy, cumbersome, or ugly). Mathematics can help us understand the underlying structure and logic of the physical world around us, sometimes revealing that two seemingly different phenomena have a great deal in common. For instance, automobile suspensions and electrical circuits behave' according to the saine mathematical rules, and chemical isomers and English grammar can be represented using the same mathematical notation. Ultimately the symbols of mathematics convey the meaning of nature. Exercises 4 1. In the paragraph where we discuss the number 1C, we gave five different rational approximations of 1C.' Find their decimal expansions to determine which one is closest to the actual value of 1C, whose decimal expansion begins , Consider the following two decimal expansions: 1= and Do you think these represent the. same number? 3. We have shown in this article the existence of irrational numbers. If x is an irrational number, show that there are rational numbers arbitrarily close to x. HINT: Think about the numbers you get by chopping off the decimal expansion of x after a fmite number of digits (e.g., chopping off the end of the decimal expansion of..fi to get 1.4, or 1.414, or , etc.). OPTIONAL: Show that ify is any rational number, there are irrational numbers arbitrarily close to y. 4. When I ran my word processor's grammar checker on this article, it repeatedly prompted me to replace the word "discover" with the word "invent," and vice versa. To what extent do you think mathematics is discovered, and to what extent is it invented? 4 After all, this is a mathematics article! page 50/5