One and the Same
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
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:
- 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,
- 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.
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.