In my last post I promised a follow up explaining why I feel that the concept of repeating decimals should not be introduced to students until at least Calculus. I’ve been delayed, but finally I’m ready to explain my thinking. My short explanation is that I find that my students often have a great difficulty with the concept of irrational numbers which hinders their understanding of trigonometric, inverse trigonometric, exponential, and logarithmic functions, and even just graphing polynomial functions. This difficulty is one that I fully expect–the concept of irrationals is rather abstract and filled with many subtleties that cannot be addressed–but I believe my job in developing student understanding of the concept would be easier if the students had never learned about repeating decimals. “Learning” repeating decimals pushes students to think about rationals as true numbers and irrationals as just symbols.
To see why this, let us first think about what we want students to understand about irrational numbers and why. A few weeks ago I overheard a conversation between a student and another math teacher in my office. The student was explaining that is not really a number. It’s just a concept like because you can “never get there.” In the spirit of Ben Blum-Smith’s post, I want to applaud this student. He senses a connection between those two concepts that is fairly deep as you move toward higher mathematics. That being said, the student is also missing something critical. Unlike , we want to be able to treat as a number in the sense that we should be able to perform arithmetic with it as a number, and it should follow all of the same rules we expect of rational numbers. Actually proving that this is true is well beyond the scope of K-12 mathematics, but intuitively this is an acceptable proposition if we think of real numbers as “points on a number line.” If we can get students to think about numbers as points on a number line I believe we have a nice balance of concreteness and abstraction.
One of the most critical ideas with fractions is that if we have two fractions with different denominators we can always find a common refinement of the number line that place both points on a number line marked by the same denominator. This makes it possible to describe the sum of two fractions (or difference of a smaller fraction from a larger fraction) using another fraction and indicates a procedure for finding the sum. Using the intuitive concept of area we can also see that the product of two fractions can be described using another fraction, and (using both ideas together) that we can describe the quotient of two positive fractions with another positive fraction. Because of all this we want students to understand that if a point on the number line can be expressed as a rational number, we want to do so. A second important realization is that any number on the number line can be approximated by a rational to within any positive length no matter how small. And if math were just a tool for science and applications rationals are all we need. There would be no need for irrational numbers. After all, in practical terms our measurements can only be so precise and if our computations are only approximations, that’s not a problem since we can control how good the approximations are. In fact, if we are willing to use approximations there is not even a need to consider all fractions. We can restrict our focus to decimal fractions, that is fractions where the denominator is a power of 10. As with fractions in general, since this denominator can be as large as we want, the approximations can be as good as we want. The quotient of a one decimals fraction by another (non-zero) is no longer necessarily exactly a third decimal fraction, but it can be approximated by one. So if we only care about practical computations, we only need to worry about approximations and can just use decimal fractions (aka terminating decimals). For this reason, a focus on decimals is perfectly appropriate in a science class.
I firmly believe, though, that mathematics is not just important for its applications to the sciences. Mathematics is also an important and beautiful subject in its own right as an endeavor in our search for truth. Furthermore, a better understanding of mathematics and its reliance on abstraction and logical thinking helps make students better problem solvers and critical thinkers. An approximation is useful, but it is not a substitute for precise numbers. One divided by three is approximately .33. That’s not exact, though, because one divided by three should have the property that when we multiply it by 3 we get 1, and .33 times 3 is only .99 which is too small. I think students need to understand that rational numbers are great and if that if we know how to express a point as a rational number, we generally want to do so. At some point (around middle school) students should discover the somewhat disturbing fact that there are points on the number line that we want to consider that are not exactly equal to a rational number. For example, if we draw a square with coordinates (0,0), (1,1), (0,2), and (-1,1) our intuitive sense of area should leads to say that this square is made of 4 triangles each of which is half a unit, so the square should have area 2. Therefore the length of the side of this square should be a number which multiplies by itself to make 2. We can show, however, that this number can not be written as a fraction. (See Kate Nowak’s lesson and comments). So here is a number which we will call that is not rational, but it it is stil a number in that it makes sense, for example, to add it to other numbers. In particular, if we take this length and add on to it a length of 1 we have a certain point on our number line. Does this point have a name? Not yet. It can’t be a fraction, or else we could write as a difference of two fractions and thus as a fraction itself. So we have to give a new name to this point. Unless somebody has a better name I propose we call it . It’s not that we CAN’T take the sum. Geometrically it makes perfect sense. It is a point on our number line, but unlike fractions where adding them always leads to another fraction with a nice ready made name, such is not always the case when doing arithmetic with irrational numbers. So that’s what I think students need to understand about irrational numbers and why. There are points on the number line that can’t be written as fractions, and we need to be aware of that because when we encounter such numbers arithmetic may look a little different (for example, the sum of two such numbers may have to be written simply as a sum), but we can still perform arithmetic and our usual rules of arithmetic will still apply.
That’s great, but what does this have to do with terminating vs. non-terminating decimals? Well, first of all, the most significant thing I want students to understand about rationals is that if you add, subtract, multiply, or divide (except by 0) two rational numbers you get another rational number. It may not be until we see irrational numbers, that we realize HOW nice this is, but is critical. This property is one that stems from thinking about rational numbers as fractions . When I first ask my students what a rational number is, though, over 90% of the time I get the response that it is a number which can be written as a terminating or repeating decimal (or some incorrect variation on this decimal notion). It is not at all clear, though, that if I add and that what I get should be another terminating or repeating decimal. It’s even less clear what happens when I multiply them. It’s also not clear what should be so special about a repeating decimal as opposed to one that doesn’t repeat. My students seem to think it’s because we can write it down (as opposed to which “goes on forever”) but I don’t buy that. Suppose I said that if instead of putting a line over a string of numbers in a decimal, I can instead choose to put a tilde above them. This will meant that I repeat the string over and over, but each time I insert some extra zeroes. The first time I repeat I insert one 0, the next time two 0’s, and so on. So, for example would mean the number I can write down that number to the same extent that I can write down , but the latter has the key property that it’s rational. If we want to consider the ability to write a number down as a decimal as an important property, then a number which can only be written as a repeating decimal really belongs on the same side of the line as irrational numbers as opposed to being lumped in with decimal fractions.
Not only does this thinking cause students to miss the significane of rational numbers, it causes undue hinderance to their acceptance of irrational numbers as numbers (which in turn makes it more difficult for them to work with them in Algebra II and Trigonometry). Students are bothered by as being an acceptable answer to a problem, or say that they can’t solve because they can’t figure out 2 raised to what number is 3. But if students can be reminded that they accepted fractions as numbers, this process of abstraction would not be a new one. When we were just dealing with integers we could add two integers and get another integer, or multiply two integers and get another integer. When it came to division, though, it was a different story. If we took 12 divided by 3, we were fine, we could write the quotient as 4. If we took 2 divided by 3, we were stuck. Once we introduced the concept of fractions, though, we were fine. The quotient is now , a number that uses a symbol which must make reference to two of our previous numbers together with an extra symbol (the fraction bar). Nevertheless, we are able to accept as a number, that has certain properties. In particular times 3 must be 2. If we are willing to accept as a number, it is not as far of a leap to accept or as numbers. When I try to draw on this past experience, though, I hit a bit of a wall. To my students is acceptable as a number only because we can write it as , but what does that even mean? Suppose students were instead convinced of the idea that 2 divided by 3 could be approximated by decimals to as close as we would like, but that if we want an exact expression for this quotient we must content ourselves to something like . Then perhaps they could take the same idea to arithmetic with irrational numbers. Yes, they can be approximated with decimals (and in many applied problems that would be particularly useful), but if we want exact expressions we often must be content with the use of several symbols in the same expression, and that makes it no less of a number.