Teacher professional development and classroom resources across the curriculum
Teacher professional development and classroom resources across the curriculum
The pursuit of infinity begins with an examination of the idea of number, or quantity. Numbers originally were tools used to quantify groups of objects or to measure real things, such as the length of a pole or the weight of a piece of cheese. Measuring something requires some fundamental unit that can be used as a basis of comparison. Some things, such as rope or time, can be measured, or quantified, using a variety of units. In measuring a length of rope, for example, we might express the result as either "5 feet" or "60 inches." The length of time from one Monday to the next Monday is commonly called a "week," but we could just as easily—and correctly—call it seven "days," 168 "hours," 10,080 "minutes," or 604,800 "seconds." These number expressions all represent the same length of time and, thus, are interchangeable. Converting any one of these equivalent values into another simply requires multiplying or dividing by some whole number. For example, 10,080 minutes is 168 hours times 60. In fact, every one of the above measurements could be converted into seconds by multiplying by appropriate whole number values.
Now, we can take any two quantities and ask a similar question: can we find a common unit of measurement that fits a whole number of times into both? Take 8 and 6, for example. This is straightforward; both 8 and 6 are whole numbers, so we can use whole units to measure them both. What about and ? Here, each of the quantities uses different base units, namely halves and eighths, and comparing them would make little sense, because they are different things. We can find a common unit of comparison, however, by recognizing that is the same as .
So, we found a common denominator of 8, which implies that both of the fractions could be expressed as multiples of the same unit, "one eighth." In some sense, we have redefined our basic unit of measurement, or fundamental piece, to be of the original piece, thereby transforming the original fractions into easily-compared multiples of the same fundamental unit.
If two numbers can be expressed as whole-number multiples of some common unit, of whatever size, they are in some sense "co-measurable" in that we can measure both using the same ruler. The proper mathematical term for "comeasurable" is "commensurable." One way to think about this is that two lengths are commensurable if there is a basic unit of measure that fits into both of them a whole number of times. If we were to cut some length of rope into two pieces of lengths a and b, there would be some third length, c, such that a = mc and b = nc. In other words, these two numbers could be expressed as multiples of some common unit.
Using a little algebra, we can confirm that the ratio of magnitudes of our two commensurable quantities is equal to a ratio of whole numbers:
Canceling the common factor of c yields the equation:
Numbers that can be expressed as ratios of two whole numbers are called rational numbers. This idea of "creating" numbers that may not relate to any observable value in the real world is fairly modern. Although today we are comfortable speaking of a number such as as a concept that "exists" in its own right, the ancient Greeks generally took care to phrase things only in terms of geometric quantities—those that exist in the physical world. For instance, they might have spoken of two lengths of rope, one that could be described as 13 measures of a certain unit, and the other 25 units of that same measure, but they would not necessarily speak of the shorter length being of the other.
On the other hand, today we are comfortable saying, for example, that a length of string is of an inch long. In doing so, we are saying that it is commensurable with a piece of string one inch long, the fundamental unit of comparison being.
The modern and ancient views of rational numbers are intimately linked, but it is important to remember that the Greeks thought of commensurability in terms of whole units. The Pythagoreans, the followers of Pythagoras of Samos in 6th century BC Greece, held sacred the idea that the first principle underlying everything is "arithmos," the intrinsic properties of whole numbers and their ratios. It is certainly a tidy idea that whole numbers, or ratios of them, are all that is required to describe the world mathematically. It is thought that this belief had origins in both the study of figurate numbers and the recognition that strings or hammers of commensurable length sounded harmonious when played or struck together.
In any ratio of two whole numbers, expressed as a fraction, we can interpret the first (top) number to be the "counter," or numerator—that which indicates how many pieces—and the second (bottom) number to be the "namer," or denominator—that which indicates the size of each piece.
In modern arithmetic, we use a base-10 system to count, or evaluate, things. Large quantities are generally represented in terms of ones, tens, hundreds and the like, whereas small quantities are more easily represented in terms of tenths, hundredths, thousandths, and so on. Although the Greeks did not use a base-10, or decimal, number system, it is illuminating to see how rational numbers behave when expressed as decimals.
For example, we can interpret the number 423 as four 100s, two 10s and 3 units (or 1s), and the value 0.423 as four s, two s and three s. In such a decimal system it is necessary to think of all quantities in terms of units of tens, tenths and their powers. Thus, , for instance, must be interpreted as , to be written as 0.5.
The question of whether or not 0.5, or , represents the same quantity asdeserves a bit of thought, however, because it highlights a subtle difficulty with our understanding of rational quantities (and maybe with our understanding of "number" itself). To explain it, let's return to the Greek point of view of commensurability. Recall that two lengths, a and b, are commensurable if there exists a common unit of measure, u, such that each length can be generated by taking u a whole number of times: a = mu and b = nu. Applying algebra, it is easy to confirm that the ratio equals the ratio of whole numbers . What would happen, though, if we worked with a smaller unit, v, that fits five times into u (that is, u = 5v)? Then, we would have: a = 5mv and b = 5nv, and the ratio still would be equal to the ratio of whole numbers , or . This shows that a rational number is not simply a ratio of any two specific whole numbers, but rather, represents a collection of "equivalent" whole-number ratios. A consequence of all of this is that our modern notion of a rational number is, in itself, somewhat abstract and troublesome to comprehend. Putting philosophical woes aside for the present, we have at least seen that and are different representations of the same ratio.
Many find it useful to view rational quantities as answers to division problems. For example, sharing one apple equally between two students results in each student receiving half of an apple. Dividing two apples equally among three bins yields of an apple per bin. Thankfully, each equivalent representation of a rational number, interpreted as a division problem, yields the same physical result: dividing four apples among six bins, and 10 apples among 15 bins, and 200 apples among 300 bins, all yield the same result as dividing two apples among three bins.
We will use this division model to our advantage as we convert fractions into decimal representations. For example, to write as a decimal number, we can think of the process of dividing four things, such as apples, among seven bins. Our decimal representation is then the number of apples in each bin, with one whole apple being our fundamental unit. Because we are dividing only four apples equally into seven bins, we realize that each bin must receive less than one whole apple, so the value in the 1's place of our decimal-expansion number must be 0.
How do we actually envision this process, though? What would we actually do to the four apples to achieve the equal distribution into the bins? We could begin by cutting each apple up into ten equal pieces, which would give us 40 slices, each being a tenth the size of a whole apple. If we were then to apportion these slices equally into the seven bins, each bin would receive five slices (or fivetenths of an apple) with five slices left over. Note that the content of each bin after this initial distribution is represented by the decimal 0.5.
If we repeat this process, dividing the five leftover tenths each into ten equal pieces, we would have 50 slender slices, each being a hundredth of a whole apple. Apportioning these slices equally among the bins would mean that each bin receives seven slices (or of an apple), with one slice left over. The accumulated total in each bin can now be represented by the decimal 0.57.
If we take the one leftover slice and cut it into ten equal pieces, we will create slices that are each just a thousandth of a whole apple. With equal distribution, each bin receives just one of these slices, and three are left over. The total amount of whole apple now in each bin can be represented by the decimal 0.571.
We can continue this process of dividing each leftover slice into ten pieces, placing an equal number of slices into each of the seven bins, and then dividing the leftovers again, indefinitely. In this particular example, we would soon find that the number sequence repeats itself after six decimal places so that the decimal representation of is 0.571428571428…. Why must the decimal repeat? In our example, there are only six choices for non-terminating remainders (i.e., 1, 2, 3, 4, 5, and 6). Note that a remainder of zero would end the division process and create a terminating decimal. In the absence of termination, one of the remainder values must reoccur, thereby beginning a repeating sequence of numbers.
Note that this expansion never ceases, continuing for as long as we care to continue the division process. This is somewhat reminiscent of the potential infinity we talked about in the introduction to this unit. We can always determine another decimal place value, but after recognizing the repeating pattern, we don't need to.
There is nothing special about the ratio values 4 and 7 in this example. The logic of breaking up leftovers into ten equal pieces and distributing those pieces equally holds for whichever two numbers we choose. In this way, any rational number can be written as a repeating decimal. Even fractions that can be represented by "terminating" decimals, such as , can be thought of as repeating, if we recognize that = 0.5 = 0.5000000….
Conversely, any repeating decimal can be shown to be a ratio of whole numbers. Consider, for example, the decimal 0.4444…. If we let x = 0.4444…, we are saying that x consists of four s, four s, four s, and so on. Ten times this value (10x) would then be four units (1s), four s, four s, and so on—or, more concisely, 10x = 4.4444….
With the decimal values of both x and 10x established, we can construct this calculation:
Solving the resulting equation for x, we get:
Notice that this works because every 4 to the right of the decimal point in the number 4.4444… matches up with a 4 to the right of the decimal point in the number 0.4444…. When the two numbers are subtracted, all these 4s completely cancel out.
This method of converting a repeating decimal into a fraction also works for decimals that have longer repetition sequences, such as 0.325325325….
Let x = 0.325…
Then 1,000x = 325.325…
In the above example, both 0.325… and 325.325… exhibit an infinite decimal expansion, yet we can cancel all the digits to the right of the decimal point because it is plain to see that each decimal digit in 325.325…matches up with an equivalent decimal digit in 0.325…; leaving only the whole number 325 after subtracting the two quantities. This idea of establishing a one-to-one correspondence among the decimal digits provides a glimpse of how we might think mathematically about infinity that will be of supreme importance later on in this unit.
Early Greek mathematicians divided mathematics into the study of number, or multitude, and the study of geometry, or magnitude. The multitude concept presented numbers as collections of discrete units, rather like indivisible atoms. Magnitudes, on the other hand, are continuous and infinitely divisible. Because length is a magnitude, a line segment can be divided as many times as one likes. The Pythagoreans believed that magnitudes could always be measured using whole numbers, which would imply that lengths are not infinitely divisible. Other schools, such as the followers of Parmenides, known as the Eleatics, believed in the infinite divisibility of magnitudes.
Parmenides taught that true "being" is unity, static, and unchangeable. This is similar to the idea that "all is one," which implies that concepts such as multiplicity and motion are illusions. If everything is part of the same thing, then there are no "multiple" things and, consequently, no motion, which is the change in position of one thing relative to another. Pythagoreans believed in multitude and motion perhaps because these concepts are intuitive, part of collective common experience. A consequence of the Pythagorean notion of multiplicity is that magnitudes should be commensurable. To the Pythagoreans, the idea that between any two quantities in nature there exists a common unit of measure, a common denominator, may have been comforting. It perhaps suggested that the rational mind can always find a solid basis for comparison, and does not have to rely on guesswork to say definite things about reality.
It would be easy to dismiss the Eleatic view, if it were not for the arguments of one of Parmenides' most famous pupils, Zeno. As we shall see, Zeno argued against the Pythagorean notions of multiplicity and motion, using infinity to show contradictions in this view. Prior to Zeno, however, problems with the Pythagorean viewpoint arose from within their own ranks in the form of an independent thinker by the name of Hipassus of Metapontum. Hipassus showed that magnitudes are not always commensurable, an idea that upset his peers to such a degree that, as the legend goes, he was drowned for his heresy. In the next section, we shall examine the idea and consequences of incommensurable magnitudes.