The question in the title is constantly discussed on web forums and at our department’s corridors; sometimes even by professional mathematicians and is in fact already quite seen and banal. In our department this question is often raised by the brilliant lecturer of the first-year calculus course as a kind of a test.
The appeal of this question to me is that it is not only a mathematical, but also a philosophical question. Should we derive the answer from our definition of the real numbers or should we cook up the definition of the real numbers according to our answer to the question? In the latter case, how are we supposed to derive the answer then? In the former case which part of the definition of the reals is crucial in that respect? Can that part be dropped or is it a fundamental part of the definition? Finally, what is the connection between sequences of digits and actual real numbers; what is the causes and consequences of these decimal representations?
In what follows I simulate an artifitial discussion:
Teacher: Is 0.999… equal to 1?
Student: I am not sure, but I think no.
Teacher: What you say is very interesting. Why do you think so?
Student: I can imagine that one minus an infinitely small number is less than one but greater than 0.999…. Thus there is a number in between and so the two cannot be equal.
Teacher: There are several problems with that. You can only subtract a real number from a real number. Do you think there are infinitely small real numbers?
Student: I’ve heard about a way to define such numbers. In that theory one can define infinitesimals and use them for defining limits for example.
Teacher: It is called the theory of hyperreal numbers, but not real numbers.
Vadim arrives on a spacecraft and joins the conversation.
Vadim: What’s wrong with hyperreal numbers?
Teacher: Well, suppose for a while, there is nothing wrong with them and suppose we asked our question in that theory instead. Denote an infinitesimal of your choice by [tex]d[/tex]. Why on Earth would [tex]1-d[/tex] be greater than 0.999…?
Student: Umm…
Vadim: Hint: you might want to use the definition of 0.999….
Student thinks hard.
Student: Aha! 0.999… equals to the limit of the increasing sequence 0.9, 0.99, 0.999, 0.9999, 0.99999 and so on!
Teacher: And you remember the definition of the limit from our calculus class: if the difference [tex]|a-b_n|[/tex] approaches 0 as [tex]n[/tex] increases, then we say that the sequence [tex](b_n)_{n=0}^\infty[/tex] approaches [tex]a[/tex].
Student: Yes… and so the sequence 0.9,0.99,0.999,0.9999,0.99999 and so on, seems to approach [tex]1[/tex]. I was wrong. Wasn’t I?
Vadim: You are hurrying too much. Remember that it is Teacher who tought you the definition of a limit. He might be tricking you!
Student: Are you (looks at Vadim) saying that you (looks at Teacher) made up the definition of a limit just in order to make 0.999…=1?
Everyone’s puzzled for 1.999… seconds.
Teacher: Certainly I didn’t make it up. Can you think of other definitions of a limit?
Vadim: The notion of a limit depends on the notion of a distance between two numbers [tex]x[/tex] and [tex]y[/tex], [tex]|x-y|[/tex], and the above `proof’ depends on the uniqueness of a limit.
Student: I see. Are you saying that 0.999… and [tex]1[/tex] can be distinct even though their distance is zero?
Vadim: No, I am saying stop measuring distances between real numbers by real numbers, because it sounds circular to me.
Student: Tarski’s definition of truth also sounds circular.
Vadim: That’s another matter.
Student: Since it is not clear whether 0-distance implies sameness, let us concentrate on sameness. 0.9 is not the same as [tex]1[/tex], because 0.99 is between them. 0.99 is not the same as [tex]1[/tex], because 0.999 is between them, 0.999 is not the same as [tex]1[/tex] because 0.9999 is between them.
Vadim: Go on.
Student: 0.999… is not the same as [tex]1[/tex] because… (hard thinking)… because 0.999… 9 is between them.
Vadim: Bingo.
Teacher prepairs to commit suicide.
Student: This looks like 0.000… 1 is the infinitely small number, I was talking about at the start.
Haha :D You rebel!
Sort of :)
I think it is important to question what is considered the truth once in a while. Even thought the truth holds. Maybe we have forgotten the logic behind the truth, or maybe we can find another set of axioms that turn out interesting.
You are very right about this. I have noticed that this is one of the (many ingenious) styles Terence Tao is up to! In his blog you might read expressions like “if R^2 and R^3 were homeomorphic then…..” and this kind of intuition just seems to give results.
1-d=.999…
The teacher can solve the problem by saying that the .999… equals the real number 1.
Students work with non real numbers all the time. It’s not like a calculus teacher would tell his students that the square root of -2 is only known to Jesus.