OK, seeing as I've got an afternoon off, I'll write this. I think I'm going to keep editing this with people's suggestions, so if you post anything, I'll try and include it in this post rather than a new one. Or PM me and I'll put stuff in, cos right now this is mostly going to reflect my opinions, which isn't going to be a good thing necessarily. The 0.999..=1 is written with the observations of others so shouldn't be too opinionated.
Frequently posted topics, and some answers.
A good thing to bear in mind, and something that people often overlook, is how mathematical proofs work. We start from some axioms, or definitions, and deduce the answer. As long as the deductions are logical the proof is not in doubt, it is a consequence of the definitions. A lot of the answers here are to questions where the definitions aren't used.
1. 0.999...=1
To mathematicians this is clearly true, and it follows from the definitions of the objects involved. No answer to this would be complete without explaining what the Real Numbers are, but first let us give some indications of why this result is true and shouldn't cause concern.
a) If you accept 1/3=0.3... then 1=3/3=3*(0.333...)=0.999...
b) It's important bear in mind that decimals are just representations of numbers, and just like 1/2=2/4, sometimes some different representations may represent the same number
c) 0.999... is certainly greater than any of the finite decimals, 0.9,0.99,0.999 and so on. and it's aslo at most 1, since the gaps between the finitely long decimals and 1 gets as small as you like, then 0.999.. had better be eqaul to 1.
d) A bit like c) what numbers lie between 0.999... and 1?
e) 0.999... is the sum 0.9 + 0.09 + 0.009 +... it's a geometric series, we can work out its sum, and it's
\frac{9/10}{1-1/9}=1
These are useful arguments, and hopefully they're convincing. In order to rigorously prove it though we'd need to use the definitions. Firstly, this is a statement about the Real Numbers. So let's discuss what they are, since people start to use them without knowing how they are defined. There are several ways to do this, and it's important to remember that they are all equivalent.
We start with \mathbb{Z}, these are the integers, that is the positive and negative whole numbers and zero. Then there's \mathbb{Q} which are the rational numbers, that is the fractions with integer numerator and denominator. We know that there are quantities that cannot be expressed as fractions such as pi and the square root of 2, and adding in these quantities we get the real numbers, the elements in the real line. The actual definition of them mathematically is complicated, which is why it's not introduced at an early stage, indeed when mathematics was first being formalized it took a while for a good definition to be agreed upon.
When we use the real numbers we are using a model that satisfies certain axioms. To fully explain the axioms would take quite a while and go far beyond the scope of a FAQ section, instead we just indicate some of the important bits.
Here is one way of creating such a model algebraically:
www.dpmms.cam.ac.uk/~wtg10/decimals.html
and you'd need to know what field, ordered, complete mean.
We could do this analytically too, that is in terms of sequences.
http://mathworld.wolfram.com/CauchySequence.html
or Dedekind cuts
http://mathworld.wolfram.com/DedekindCut.html
which start from the rationals and again form a completion.
From all these we have the key propeties of the real numbers, which assure us that 0.999...=1
If you like, the real numbers are defined to be the set of numbers where 0.999... must equal 1. If it weren't so then you'd have several problems, principally that the limits of a sequence would not be unique, and that the arithmetic operations wouldn't have the desired properties (think of the 1=3/3=3*(0.333...) argument).
We perhaps also need to think about what 0.999... means too.
It is the limit of the sequence 0.9,0.99,0.999,0.9999, etc, or if we are purely thinking in terms of describing things in decimal expansions, it is the smallest real number greater than all decimals obtained by terminating the expansion after a finite number of steps. Again, these are definitions, and from all these definitions it must follow that 0.999... =1.
Common objections:
a) But you're adding up an infinite number of numbers (0.9+0.09+0.009...) and you never get to the end of the addition, so it can't be 1.
Ans: we aren't actually performing any infinite set of addition, we are taking a limit of a sequence of finite sums that is *defined* to be the infinite sum.
b) The difference between 0.999... and 1 is 0.0...1, an infinite string of zeroes and then a 1.
Ans, that isn't a valid decimal expansion, you're contradicting the definition of a decimal representation. (see also the section on infinity).
c) (Variation on b) The result of 1-0.999... is the smallest non zero positive real number.
Ans. by the definition of the real numbers, there isn't a smallest non-zero positive real.
d) They are different decimals so they must be different real numbers.
Ans. decimal expansions are not the same thing as real numbers, they are a convenient way of representing the real numbers.
<Anything to be added or removed?>
2. What is infinity?
There are several answers to this, but there is the common theme that it is to do with something that is not a finite (real) number. Despite the fact that mathematics needs rigorous definitions and proofs, often the practitioners of it wil abuse notation, which can cause confusion to people meeting something for the first time.
2.1 "As n tends to infinity <foo> happens..."
This just means for all n sufficiently large, that is greater than some integer, <foo> happens. There isn't some infinity *in* the integers. Often from this usage people think of infinity as being a number bigger than all the other numbers. This may be a useful idea but shouldn't be taken too literally, since it can and does lead people to say things about the last digit in 0.999... being the "infinitieth", there is no last digit, there is no position in the expansion that corresponds the the infinitieth place.
2.2 "As x tends to zero 1/x tends to infinity"
This just means that for any real number N you may pick some x (close to zero) such that 1/x>N. It does not mean 1/0 *is* infinity. Even usually reliable sources can define infinity (oo) to be 1/0, which isn't a good thing. We'll explain how to interpret this later in the 1/0 bit).
So, each of those statements involving the word infinity is actually about finite things.
2.3 "There are an infinity of integers"
This introduces us to the idea of
cardinal numbers. If S is a set with a finite number of elements then its cardinality (card(S) or #(S)) is the number of elements in it. If a set is not finite then we say its cardinality is infinite. Some people say "there is an infinity of...", which for what it's worth sounds ugly, and is to be discouraged <<note personal opinion>>.
Two finite sets have the same cardinality if and only if there is a bijection between them. Cantor defined the idea of infinite
cardinal numbers, often called transfinite numbers, by generalizing this property. Two sets have the same cardinality if there is a bijection between them. This is a definition. Sometimes people use the phrase infinite numbers to mean infinite
cardinal numbers, ie transfinite numbers. (see infinity/infinity)
So that's some of the uses of the word infinity. There's another important one, and in this case we actually end up with infinity as a point in the *extended* complex plane. And this we pick up in the next part.
3. "What's 1/0, why isn't it defined?"
So, the reals and complexes are fields right?
http://mathworld.wolfram.com/Field.html
And the symbol 1/b is the multiplicative inverse of b, which is only defined for non-zero real/complex numbers. If we want to define 1/0 to be another real number (or complex number) then it turns out we're going to have to break some of the other rules, thus we're inconsistent, and we can't do it, since it would have to be true that
1=0*(1/0) definition of multiplicative inverse
0*(1/0)=0 since in a field 0*k=0 for all k. Hence 1=0, but that contradicts the definition of a field.
So any attempt to make sense of 1/0 means we have to go outside the field of Real or Complex numbers., which is why we don't define it in the real or complexes: it is inconsistent with their definition.
Now there is a well known and useful construction to extend the complex numbers widely used in complex analysis where we often have functions that have poles.
Firstly, it is a convenient geometric construction, and as I don't have the ability to post an inline image, here's a link:
http://www.clowder.net/hop/Riemann/Riemann.html
<<note, has anyone got a better link than this?>>
This is written, as a set, \mathbb{C}\cup\{\infty\}
arithmetic on the complex numbers is the usual one plus the following rules:
\frac{z}{0}=\infty \forall z\neq 0
\frac{z}{\infty}=0 \forall z\neq \infty
Notice that we still can't define 0/0 and \frac{\infty}{\infty} or \infty*0, but more on that later.
Note this does not translate into meaning that "infinity" IS 1/0 in the sense that
\sum_0^{\infty}
does not mean
\sum_0^{1/0}
the use of infinity in that sum is not the use of infinity in the complex plane. The use depends on the context, this shouldn't be strange, often the meaning of a mathematical symbol is context dependent (i as a vector i as the square root of minus 1).
I#'ve hit the 10,000 char barrier
