# Possible Maths FAQ

1. Sep 12, 2004

### matt grime

Is there support for a sticky thread answering commonly asked questions? I am specifically thinking of the 0/0, infinity/infinity, what is infinity, 0.999..=/=1 questions that come up far too often.

I am not suggesting this thread becomes it, nor that I should write it. If anyone wants to add a suggestion for a *general* topic please feel free, but please don't post disputing the fact that 0.999...=1 is true.

I'd be happy to write one, but don't think I'm the best person to do so.

Ideally it should simply explain why these are mathematical truths, not offer opinions on the "real world implication" of what it means to, say, divide infinity by infinity. I mean, the fact that 0.999 ..=1 cannot be in dispute, and the answer should simply explain what the mathematical ideas behind this are; it should not attempt to rebuff the crankish views on these subjects since they are often not mathematical. It should instead address people asking genuine mathematically based questions about why this is true, or that isn't a valid operation. I'd suggest that the final thread, if there were one, should be locked.

Any pros cons I've not mentioned and so on?

2. Sep 12, 2004

### marlon

Surely wanna help on this, matt

I suggest we also talk about some-integration rules and elementary vector-calculus since lot's of students are experiencing difficulties on these matters...

regards
marlon

3. Sep 12, 2004

### jcsd

I was thinking that a sticky on basic vector analysis might be useful in the homework section as over 50% of the questions seem to be about this topic.

A sticky on 0.999.. =/= 1 and other common misconceptions in maths would be a good idea as looking into my crystal ball I see the topic will come up again.

4. Sep 12, 2004

### matt grime

I don't think that counts as general maths, and would be better in the proper forum for that topic.
I can't say I bother reading most of the calc threads, what kinds of things crop up there frequently?

5. Sep 12, 2004

### Gokul43201

Staff Emeritus
Here's a hypothetical sequence of events :

(CON) Folks are not likely to read through the sticky before posting their query.

(PRO) They can simply be directed to the sticky once it is seen that they've posted a question that is addressed by the FAQ.

(CON) More often than not, they will be dissatisfied with the explanation given in the FAQ, and will pursue the matter further in their thread. And as often tends to happen, these threads will attract more cranks, all wanting to chip in their tuppence. Back to square 1 !

(PRO) A mentor need not feel pangs of guilt when locking down a thread that has been addressed by the FAQ.

Despite the cons, I think the idea is worth a shot. It is a small one-time investment, that may produce huge long-term returns.

6. Sep 12, 2004

### matt grime

I was worried about the possible effects it might have on spurring people to post their own (nonmathematical) theories on the subject. But if it is expressly written to explain the maths behind it, with the intention of helping those who want to learn the facts, then there can be no harm... right? Besides there will always be those who don't want to listen to the maths.

7. Sep 12, 2004

### arildno

I am quite certain that this will occur to some extent.
However, by having at their disposal a thread which goes through these examples in a satisfactory way, mentors/admins can more easily (sooner) choose to lock/delete threads on these topics as being irrelevant/not worth answering again.

On the whole, I think this is a good idea.

8. Sep 12, 2004

### Hurkyl

Staff Emeritus
IMHO It's certainly a good idea. I've been considering it myself for quite a while, now, but I've never found the time to sit down and draft just what I'd want it to say.

9. Sep 12, 2004

### matt grime

Why 0.999..=1

This is a fairly common topic, and the explanation is simple, but disguises some quite deep mathematics.
Rather than go too deeply into the mathematics straight away, let's start with a simple direct proof:

let 0.9[n] mean 0.9...9 with n nines.

0.9[n]<0.99....<=1

right? So

0<=1-0.999....<1-0.9[n]

for all choices of integer n. Now, 1-0.9[n]=10**(-n) (my six key is broken so I can't do powers), but we can pick n as big as we want and make that number as small as we care. Now, it follows from the definition of the real numbers that 1-0.999.. must thus be zero and therefore they are equal.

Of course, this may be the first time you've met the phrase "real numbers", and that might be part of the problem. No explanation would be complete without talking about them, since even if you know the name you might not know what it means.
At school you start using whole numbers, and fractions, and then you just start using infinitely long strings of decimal expansions without explaining what they are properly. This is necessary to make life easier but creates exactly the problem about not being sure why 0.99..=1.
We're all happy with fractions, and we're all happy that 1/2 = 2/4. Let's examine a little more what that means: 1/2 and 2/4 are just symbols, representations, of fractions. You're happy to say they're equal, and happy to manipulate them according to the rules you're given at school.
We also soon learn that fractions "aren't enough", that there are quantities that we want to describe that can't be written as fractions, such as pi and the square root of 2. At this point we start to think of these quantities as decimals, but really they aren't. Decimals are representations of these other quantities, which we call the REAL numbers (formal definition to follow), and just as fractions have may have different representations, so may real numbers.

<insert more explanation here>

back to me talking to the other posters here:

i'm very much unhappy with that, by the way, but I can't think of a way to start it. I mean, we all know that it's true simply because of the definition of the real numbers, but the chances are that the person who has the question has never heard the phrase "real number" and doesn't know anything about analysis. so, what should be done to improve the introduction, apart from a wholesale rewrite (i am not at all protective of this attempt, which was written off the cuff).

is it the right way to be explaining it, or should it just go for the full on: give the definition of a complete metric space.

if someone wants another view, here's a Field's Medal winner's view on a similar thing.

www.dpmms.cam.ac.uk/~wtg10/decimals.html

10. Sep 12, 2004

### Hurkyl

Staff Emeritus
Hrm, I've always considered starting with the definition of an ordered field to emphasize the algebraic properties we like in our number systems. Since even the crackpots presumably have some proficiency in arithmetic and elementary algebra, they will probably be willing to accept this part.

The problem with this approach is then one has to figure out how to motivate the completeness axiom.

But this isn't necessarily an advantage over your approach, since presumably the reader will have some proficiency with terminating decimals as well. You can avoid having to introduce the Archmedian axiom (or some form of completeness) if you talk about the "number" of digits in a decimal, which I think would need to be addressed in the FAQ anyways.

I've always preferred simply defining 0.999... = 1, but I don't think that would be satisfying to most people.

11. Sep 12, 2004

### matt grime

When a mathematician sees something like 0.999...=1, they are perfectly happy that it is true, perhaps even self evident, yet to a lot of people it seems just plain wrong. The difference is that a mathematician views this as a statement about "real numbers" and our representation of them as decimals. In order to understand why it is true it is necessrary to understand what the real numbers are. Different mathematicians have different ways of thinking about this, though they are all equivalent. For instance, if you were an algebraist you might think in terms of the real numbers as being an example of an "ordered field" with some other properties. An analyst might think in terms of sequences and "completeness".

12. Sep 12, 2004

### arildno

What I've noticed is quite typical of crackpots is how they assume that properties pertaining to finite quantities (in some sense) can unproblematically be transferred onto infinite quantities.
Hence, to make it clear through simple examples that these intiutively reasonable ideas simply don't hold, might possibly be of some advantage.

13. Sep 12, 2004

### Hurkyl

Staff Emeritus
Another thing that I've been unsuccessfully trying to put into words, and might not be useful at all in the FAQ, is how mathematics isn't exploring some mysterious "reality" about which we might be mistaken; it rigorously defines the reality that it explores.

14. Sep 12, 2004

### arildno

Agreed; IMO perhaps the most important insight to draw from this, is that unless a mathematical concept is rigourously defined, it is basically meaningless (without "reality").
Even if we need some basic concepts that cannot easily be derived from others (concepts in basic set theory, I would think), we should at the very least have clear relations between such concepts.

15. Sep 12, 2004

### Chrono

I agree. I'd defenitely find it useful and I can see myself looking to it as a great reference.

16. Sep 12, 2004

### Tide

$$1 = \frac {1}{3} + \frac {2}{3} = 0.333 \cdot \cdot \cdot + 0.666 \cdot \cdot \cdot = 0.999 \cdot \cdot \cdot$$

17. Sep 12, 2004

### HallsofIvy

That's assuming, of course, that the person you are responding to accepts that
1/3= 0.3333..., that 2/3= 0.6666..., and that you can add infinite decimals like that!

18. Sep 12, 2004

### Tide

Is there any other way to add decimals? There's nothing to "carry."

19. Sep 12, 2004

### mathwonk

another approach is simply to define what an infinite decimal means. i.e. an infinite decimal is by definition the smallest real number not smaller than any of its finite decimal approximations. Hence .99999..... = 1, the smallest number not smaller than any of the finite numbers .999....9, of any length.

But it seems simpler just to say that 1/3 = .3333....., so 1 = 3/3 =.99999.....

20. Sep 12, 2004

### phoenixthoth

Oh no this is turning into yet another one of those dreaded threads.

I think there should be an intuitive version and a rigorous version establishing the notion that 0.9...=1. The intuitive version was posted by mathwonk and the rigorous verison from Mr. Grime. And lastly, we can challenge anyone who thinks 0.9...!=1 to express the difference of the two numbers in decimal form which differs from 0! (Um.. I don't mean factorial)

We should make a list here in this thread of all the FAQs and then we can start a fully moderated thread where they get answered.

Some FAQs:
Is 0 a natural number?

What is 0^0?

What is 0! and why? (BTW why can't one figure out (-1)! using the same pattern?)

What's the deal with 0/0, "undefined", and "indeterminate"?

Why isn't oo/oo defined and why isn't oo-oo=0?

I don't know these are all I could think of quickly.

I suggest, again, that we collect some questions here and then someone can be either commissioned to answer them (like Mr. Grime or Hurkyl) and/or a fully moderated thread is started and/or Hurkyl collects PM'ed answers and posts the appropriate ones as a sticky.

Cheers

21. Sep 13, 2004

### shmoe

haha, at least this ones filled with good suggestions on how to show .99..=1 rather than someone adamantly refusing to believe it. I lean towards thinking of .99... as the value the infinite series .9+.09+...converges to (assuming they believe it converges). If they understand limits, then you can show them easily enough that there's no number between .99.. and 1. You could also use the old appeal to the formula for geometric series this way.

An obviously related topic is an explanation why there's no smallest real number or more generally no next real number (what comes right after pi?). This might be a good logical step before .99..=1.

Another topic could be an explanation of what the more popular "impossiblity results" actually mean, such as squaring the circle, non-existance of a "uintic formula", etc.

I can't think of any other topics off hand, but I think a faq is always a good idea. It at least gives a place to point at rather than repeat yourself every few weeks.

22. Sep 13, 2004

### matt grime

This should perhaps be the zeroeth answer. As most of the answers to the faqs I, and seemingly most other people here, have in mind are to explain what definitions we are using and how the result either follows from them, or doesn't make sense (infinity/infinity being inconsistent since people don't understand the definition of a muiltiplicative inverse)

23. Sep 13, 2004

### matt grime

I don't have time to do details right now, but how about then:

Mathematics is the study of which propositions may be deduced from axioms. The axioms are true, whether or not you agree with them is a philosophical question. In some cases we may take one of the axioms and negate it, and see what we can deduce then.
The mathematics can be used to model things in the real world, but if the model doesn't reflect observation it tells you you are using the wrong model, not that the model is internally inconsistent.

1. 0.99..=1

explain outline using 1/3 =0.3...
emphasize these are just representations of the real numbers, use the 1/2=2/4 example to show that they know this kind of thing happens
talk about the reals a little, but give links to places eg wolfram to explain what dedekind cuts, complete ordered fields, caucy sequences etc are.

2 1/0
having introduced reals with their algebraic properties explain that this isn't a real number.
mention tending ro infinity as a being a short hand for growing with out bound

3. infinity
cardinals (no need to mention cantor since that thankfully comes up very rarely)
perhaps continuum hypothesis mentioned as an example of something we can assum true or false as need be (though might need the axioms of ZF explaining so on second thoughts not a good idea).
point at infinity as a geometric construction perhaps.

4 infinity/infinity
again not in reals
extension to real numbers with conway's arithmetic?

24. Sep 13, 2004

### Hurkyl

Staff Emeritus
You're talking about the surreal numbers? The hyperreals are a good example too, though I think the simplest way might be to simply take the field R(w), and require that w is bigger than any real number.

25. Sep 14, 2004

### matt grime

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 infinte 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