One and the Same

Some years ago when my daughter was in early middle school (or thereabouts) she learned about fractions and repeating decimals.  She complained that the teacher said that the repeating decimal 0.9999… was the same as one.  She wasn’t convinced.  She said, “it couldn’t be because no matter how far you go there’s always a difference.”  I smiled and said I would prove it to her.  Here’s how my proof goes.


Since we are trying to figure out what 0.9999… is, let’s give it a name so we can talk about it.  Let’s call it “A”.  So we start with the definition

          A = 0.9999…  .

Now let’s multiply A by 10.  This just shifts the decimal, giving the result

          10A = 9.9999… .

Since the decimal goes on forever, we haven’t changed the part beyond the decimal at all.  The right side is therefore what we started with +9, or:

          10A = 9 + A.

Since the equals sign means that the things on the two sides are different ways to represent the same thing, if we subtract the same thing from both sides, we will still get two things that are equal.  Subtracting A from both sides gives

            10A – A = 9 + A – A
or
            9A = 9.

This is easily solved by dividing both sides by 9 to give A = 1 as her teacher claimed.

She was mollified, but not satisfied.  Eventually she accepted it (I won’t say “figured it out” because of what I say below) and became a master of such things as calculus, differential equations, and how to calculate pi.  But the incident stuck in my mind and now I’m the one not entirely satisfied.

This little exchange illustrates something that I have found increasingly interesting in the years since it happened: the relationship between mathematics and the way we construct it, and the way we apply it to our world.  Math by itself is a self-contained structure with principles and rules so rigid that they can be carried out by a computer – sometimes.  (The places where this fails are particularly interesting.)  But when you think about how it actually works, there are often some hidden psychological elements.  (Please don’t tell my mathematician friends that I think math rests on psychology.  I might get thrown out of the club.)  The critical issue is: When do we decide two things are “the same”?

The questions of identity – when two things can be thought of as “the same” – is a tricky one in everyday life as well as in math.  In Star Trek, since he didn’t have the budget to show a spaceship landing and taking off, Gene Rodenberry created the “transporter” – a device that dissolved any object (including people) into their molecules, carefully taking note of where each one was, beaming the information down to another location and reassembling the object out of different molecules in that place in exactly (one hopes) the same configuration as the original.

[In one episode, the beaming is assisted by “Heisenberg compensators.”  Legend has it that at a press conference at a Star Trek convention, someone asked Rodenberry, “How do those Heisenberg compensators work?”  He responded, “Very well.”]

The crotchety medical officer, Bones McCoy, was very nervous about this – as well he might be!  Suppose what really happens is that pulling you apart into your molecules simply kills you and assembling those molecules elsewhere simply (!) recreates a clone of you with exactly your memories who thinks he is you.  We know from quantum physics that all atoms of a particular kind are truly identical (that is a story for another entry), so it shouldn’t matter.  Since he has all your memories, he looks to everyone else as if he is you – and they don’t care that the “real” you has been destroyed. Is your consciousness continuous from the destroyed you to the recreated one?  Or has the second consciousness been created anew?

Of course, this technology doesn’t exist (yet), so we don’t have to worry about it.  Except of course we do.  When we go to sleep, or become unconscious due to injury, when we wake up are we the same person?  When you meet an old flame years later, they have had a range of experiences that have changed them.  Are the still “the same person you knew”?

These issues go quite deep into our whole interpretation of our lives.  When do we consider two different things “the same”?  And things are always different.  The laptop computer on which I am composing this is continually picking up and emitting individual atoms and molecules – gas, dust, etc.  Its transistors and chips are continually changing as the carry electric currents, heat up, and cool down.  The magnetic storage on the hard drive may deteriorate as it is used.  In a few years, it “won’t be the machine it used to be” – but what will it be?

The basic principle seems to be, when it’s useful to treat something as a single thing over time, let’s just go ahead and do it. “A difference that makes no difference is no difference.”

Let’s return now to the issue of identity in math.  With my daughter’s example of the repeating 9’s, we have to say, If we are going to use this in a system where we manipulate symbols in the systems of arithmetic and algebra, then we have to consider “1” and “0.9999…” as being two different representations of the same thing.  Similar things happen with fractions.  Math is a study of relationships and equality.  Two principles that are critical to it are:

1. We are interested in knowing when two different ways of representing something are the same so the definition of equality (or identity) is crucial and, 
2.  We assume that if we do the same thing to two different representations of the same thing we still get two representations of the same thing.

These rules don’t always hold.  For example, in arithmetic, you can easily generate nonsense by dividing by 0.  Funny things can happen whenever infinities or things that in principle take an infinite number of steps are involved.  (Think of Zeno’s paradox, for example.)  There is a lot of fun math that studies how to handle these situations – math like calculus, the theory of complex variables, and functional analysis.

We are usually comfortable when we do arithmetic and algebra as long as no infinities are involved, but the issues of identity are still deep, especially when we are trying to apply math to the physical world.

An example of this is fractions.  Our arithmetical system needs fractions to be complete and consistent.  If we start with integers, it’s easy to figure out what adding means – just “counting on”.  Five year olds usually get this.  To be able to go backwards and solve equations with adding (like, what do you have to add to 5 to get 12, that is, 5 + x = 12), we have to invent the opposite of addition – subtraction.  This then leads us to construct negative numbers.  We can then add anything to anything and subtract anything from anything.  Repeated addition leads to multiplication.  To be able to go backwards and solve equations with multiplication (like, what do you have to multiply by 5 to get 30, that is, 5x = 30), we have to invent the opposite of multiplication – division.  This then leads us to construct fractions.  We can then multiply anything by anything and divide anything by anything (except 0) and solve all kinds of equations.

Our rules now lead us to identify certain fractions as “the same”.  The fraction rules say that when you multiply fractions you multiply the tops and put the result on top and multiply the bottoms and put the result on the bottom.  This accords well with our everyday sense that if I take ½ of something and divide it in two parts again, I will get ¼ of the original thing; that is, (½)x(½) = (1x1)/(2x2) = ¼.  This also tells us that

          2/4 = (1x2)/(2x2)  = (1/2)x(2/2).

But since 2/2 = 1, we must identify 2/4 and 1/2 as the same number.
This is all very well and good, since it increases the number of ways we can represent the same thing and that leads to more things we can do with it.  But it creates some problems with mapping our numbers into the real world.

As a young physicist, I was strongly tempted by the idea that the mathematics I was learning in physics was the real world – that the world was somehow number.  As I have aged, I have become increasingly convinced that this is the wrong way to look at it.  A better way now seems to me to be the following:

Mathematics is the abstract study of relationships and how things can be looked at in different ways without changing some essential essence (often quantitative – but not always).  We look for patterns in the physical world that match these relationships and rules.  When we find them, the tools developed for abstract relationships in math can be carried over and will tell us things about physical relationships that cannot be easily seen directly.

But we do have to carefully watch for cases where there is a shift in the “essential essence” of what we care about.  If we are talking about figuring out how many bricks are needed to build a building, multiplying and dividing and adding and subtracting will work just fine.  If we are talking about dividing a pizza into slices, the fact that 1000/1000 = 1 does not really tell me that if I ask for a pizza, that it is OK to take the pizza, cut it up into 1000 pieces, and give me all the pieces.  When applying math to the real world in any way, we have to be careful about whether the “essential element of reality” that is correctly represented by the math is what we care about, or whether other essential elements have been missed.

83333

The other day while driving to the office, the odometer of my car turned to 83,333. Pretty typical. My car is 8 years old and I commute nearly 20 miles each way to work, so I’m putting on about 10,000 miles per year. But I liked that number. It reminded me of something. What? Was it the cosine of some angle? I’m a physicist and I like to work with numbers, so I do run into cosines all the time. That didn’t seem to ring true. Was it some simple fraction? That probably was it. I decided to see if I could figure it out. Here’s my approach.

Consider the infinitely repeating decimal 0.8333333…. . Is it some simple fraction? Well I know that 0.333333…. is just 1/3 and this looks almost like it – at least the repeating part. Can I manipulate this to make it look like that? First there this “0.8” in front that doesn’t belong. So let me write my decimal at 0.8 + something else. Subtracting off the 0.8 I get 0.033333…. So my decimal = 0.8 + 0.03333…. The 0.8 is just 8/10 which equals 4/5 by cancelling the common factors of 2 at the top and bottom. The other part is almost the 1/3 but it has a 0 in front after the decimal. I know that you put extra zeros after the decimal by dividing by 10. So 0.03333…. = (0.3333….)/10 or (1/3)/10 = 1/30. So my fraction is 4/5 + 1/30. I can combine these together by putting them over a common denominator. If I multiply the first fraction by 1 = 6/6 (you can always multiply something by one without changing its value – but you can change the way it looks to help you) I get 4/5 = (4x6)/(5x6) = 24/30. I can now add this to the 1/30 to get 25/30. This is 25/30 = (5x5)/(5x6) so I can cancel the common factor of 5 to get 5/6. A nice simple fraction as I had expected (and should have remembered). Then I see that the complement of my decimal – it’s difference from one – is just 0.166666…. (You have to make the two of them add up to 0.99999….. which actually is just equal to 1 – but that’s a discussion for another entry.) If I had seen that I would have gotten it right away. I work a lot with fractions and I know that 1/6 is about 0.16 and is a repeating decimal.
I know, I know! DWDM (Driving while doing math) – or maybe it should be “DWA” (Driving while academic) – is probably as dangerous as talking on a cell phone while driving. Not a good idea! But the exercise illustrates some lovely principles about number and education.

First, one of the things I have been trying to do with my work recently is to teach non-physicists (biologists, actually) to think a bit “in the way that a physicist does.” They often tell me this when I ask what I should teach. But what does this mean? I think my example, although it’s just about number and not about physics itself, tells me something. As a physicist I see numbers as real things, not just abstract relations. They’re things with structures and connections and properties. I can work with them, turning them twisting them, multiplying by 1 in various forms (5/5 or 6/6 for example) or adding 0 in various forms to change the way they look and give insight into what they are. Being able to see them in different ways gives them a solidity and a reality that I suspect most of my students don’t feel. Seeing that I can do the problem in the two ways discussed – manipulating it into a fraction by taking it apart and looking at the complement and recognizing a familiar number – gives me the comfort that arithmetic and all the complicated stuff about fractions and decimals isn’t just something I have to remember. It’s something that makes sense. It’s reliable and consistent. I can look at things in a variety of ways and confirm my answer and catch my mistakes, though, luckily in this case, I didn’t make any. I often do, especially when doing math in my head, but having the multiple perspectives on number (and equations) usually helps me nail down the correct form pretty quickly.

I showed this analysis to one of my students who came in to my office for help on this week’s homework. She was flabbergasted. She said she had never seen anything like that. How sad! This is the sort of thing I would love to see taught to every 5th grader. We seem to be making a useful transition in our teaching of arithmetic – from doing tedious rote math (adding long columns of 4 and 5 digit numbers) to using calculators to eliminate the tedium. But learning to use a calculator shouldn’t just get rid of the tedium but also give students the sense that numbers are somehow magic and can’t really be thought about.

[If you can find it, I recommend Isaac Asimov’s 1939 short story, “The Weapon Too Dreadful to Use”, which describes a world in which number had been automated by computers so much that people didn’t realize you could figure things out. Asimov isn’t usually predictive, but this one hits the nail on the head.]

Calculators can in fact be used to help students develop this “sense of number” very effectively, though they often are used in exactly the opposite way. For some references to this, check out the literature review in my paper with my student Tom Bing, “Symbolic manipulators affect mathematical mindsets,” (Am. J. Phys. 76, 418-424 (2008)). Interesting books on the sense of number, how people develop it and build it into math include:

Stanislas Dehaene, The Number Sense – a very readable book about the neuroscience and psychology of basic math.

George Lakoff and Rafael Nunez, Where Mathematics Comes From: How the embodied mind brings mathematics into being – this one is quite a bit more technical and takes one deep into sophisticated math.