Frequently Asked Questions (FAQ) about Mathematics

[Redbelly98]

This FAQ, a work-in-progress, provides answers to frequently asked questions about mathematics.

https://www.physicsforums.com/showthread.php?p=3325292"

https://www.physicsforums.com/showthread.php?p=3325309"

https://www.physicsforums.com/showpost.php?p=3356702&postcount=4"
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The following forum members have contributed to this FAQ:

AlephZero
Fredrik
micromass
tiny-tim
vela​

 

[micromass]

Why do people say 1 and 0.999... are equal? Aren't they two different numbers?

No, they really are the same number, though this is often very counterintuitive to many beginning students. Here are some non-rigourous proofs that 1=0.999...:

Proof #1

For any two unequal numbers, there is always another number in between them. (That is intuitively obvious, and can be pictured on a number line, which will be familiar to many people.) Therefore, if 0.999... and 1 are different, there must be another number in between them. But there is no way to write a number that is greater that 0.999... and less than 1 in decimal notation.
​

Proof #2

First, we put
$$\ x = 0.999...$$
Multiplying by 10 gives us
$$\ 10x = 9.999...$$
But then
$$\ 10x-x = 9.999...-0.999...,$$
thus
$$\ 9x = 9$$
Hence we get that
$$\ x = 1$$
​

Proof #3

If you accept that 0.9999… is a number, then how much less than 1 is it? It's larger that 0.9999, so it's less than 0.0001 less than 1. But it's also larger that 0.9999999, so it's less than 0.0000001 less than 1. So the difference between 0.9999… and 1 is less than 0.00001, 0.000000001, or 0.any number of 0s followed by 1 … so the difference must be zero. If you accept that two numbers whose difference is 0 must be the same, then that proves that 0.9999… = 1. If you don't accept that, and you think that two different numbers can have a difference of zero, then you're in an 'extended number system' which has more numbers than we normally use.
​

Proof #4

First, we have that
$$\ 1/3 = 0.333...$$
If we multiply things by 3, then we get
$$\ 1=3\times (1/3) = 3\times (0.333...) = 0.999...$$
​

All of these proofs are correct, but they are not rigourous. For example, how do we know that $3 \times (0.333...) = 0.999...$? This is not that obvious if we think about it. A more rigourous proof is given in the post following this one.

Some further questions you might have:

But 1 cannot equal 0.999..., as every number can only have one representation!
Well, the thing is that this is just a misconception that is simply not true. Numbers can have many representations. For example,
$$\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=0.333... ,$$
but somehow, many people don't have any problems with this thing. The same thing happens to 1=0.999... really, it's just another way to write the same number. Does this make our number system ugly? I understand that you might think that, but that's just something we need to accept. Not having that 1=0.999... would make our number system much uglier!

The way I see it, is that 0.999... gets closer and closer to 1, but never quite reaches 1.
This reasoning appears a lot and apparently, many people see 0.999... as some kind of process that gets close to 1. But this is not quite what mathematicians mean with 0.999...
Mathematicians say that 0.999... is a number, just like 2 and 3. So phrases like "it gets close to 1, but never reaches 1" are meaningless. It's the same as saying 1 gets closer and closer to 2, but never quite reaches 2. This sentence makes no sense, and the same thing happens with 0.999...

Can we define number systems such that 1=0.999... does not hold?
Of course! But these number systems are not as useful, because they don't conform to our intuition about numbers and limits.

In Proof #2, you say 10x=9.999... But this 9.999... has one fewer nine than 0.999...
Another popular argument. This time, the confusion arises from not grasping infinity. 0.999... has an infinite number of nines. If we somehow remove a nine from this sequence, then we would still have an infinite number of nines. So there are an equal number of nines in 0.999... and 9.999...

The same thing happens here: consider two sets of numbers, A and B, where
$$A=\{0,1,2,3,4,...\}$$
and
$$B=\{1,2,3,4,...\}$$
Both sets are infinite. And actually, both sets have an equal number of elements. But A doesn't contain 0, so it has one fewer element than B? Yes, but this reasoning only applies to finite sets. For infinite sets, it's quite possible to have one element less and still have an equal number of elements. Indeed, consider the following correspondence:

$$0\leftrightarrow 1,~1\leftrightarrow 2,~2\leftrightarrow 3,~3\leftrightarrow 4,...,~n\leftrightarrow n+1,...$$
So for an element n in A, there exists a unique element in B that corresponds to n, namely n+1. This means, by definition actually, that both sets have the same number of elements.

Maybe we should just abandon our base 10 number system and move to a number system where every number does have a unique representation.
Tempting, but sadly this is not possible. The problem arises in every base! For example, in base 2, we have $1=0.111...$. There is no way around it.

 

[micromass]

Is there a rigourous proof of 1 = 0.999...?

Yes.

First, we have not addressed what 0.999... actually means. So it's best first to describe what on Earth the notation $$b_0.b_1b_2b_3...$$ means. The way mathematicians define this thing is

$$b_0.b_1b_2b_3...=\sum_{n=0}^{+\infty}{\frac{b_n}{10^n}}$$

So, in particular, we have that

$$0.999...=\sum_{n=1}^{+\infty}{\frac{9}{10^n}}$$

But all of this doesn't really make any sense until we define what the right-hand side means. There is an infinite sum there, but what does that mean? Well, we put

$$S_k=\sum_{n=1}^{k}{\frac{9}{10^n}} \ ,$$

then we have a finite sum. So, for example
$$S_1=0.9, \ ~S_2=0.99, \ ~S_3=0.999, \ etc.$$
So, in some way, we want to take the limit of this sequence.

Let's consider a particularly simple sequence to illustrate the idea behind the definition of a limit of a sequence: 1/2, 1/3, 1/4,... The terms in this sequence get smaller and smaller. You might think that it's obvious that it goes to 0, or that it's obvious that a smart mathematician can prove that it goes to 0, but it's not. It's impossible to even attempt a proof until we have defined what it means for something to go to 0. So we have to define what the statement "1/2, 1/3, 1/4,... goes to 0" means, before we can attempt to prove that it's true.

This is the standard definition: "1/n goes to 0" means that "for every positive real number $\epsilon$, there's a positive integer N, such that for all integers n such that $n\geq N$, we have $|1/n| < \epsilon$". With this definition in place, it's quite easy to prove that "1/n goes to 0" is a true statement. What I want you to see here, is that we chose this definition to make sure that this statement would be true. The first mathematicians who thought about how to define the limit of a sequence might have briefly considered definitions that make the statement "1/n goes to 0" false, but they would have dismissed those definitions as irrelevant, because they fail to capture the idea of a limit that they already understood on an intuitive level.

So the real reason why 1/n goes to 0 is that we wanted it to! Similar comments hold for the sequence of partial sums that define 0.999... It goes to 1, because we have defined the concepts "0.999...", "sum of infinitely many terms", and "limit of a sequence" in ways that make 0.999...=1. Can we define number systems such that 1=0.999... does not hold? Of course! But these number systems are not as useful, because they don't conform to our intuition about limits and numbers.

Now that we know what a limit and an infinite sum is, let me give a fully rigourous proof to the equality 1=0.999... This proof is due to Euler and it appears in the 1770's edition of "Elements of algebra".

We know that

$$0.999...=\sum_{n=1}^{+\infty}{ \frac{9}{10^n} } = \frac{9}{10} + \frac{9}{10^2} + \frac{9}{10^3} +...$$

This sum is a special kind of sum, namely, it's a geometric sum. For (infinite) geometric sums, we can find its limit easily:

Let
$$x=\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+...$$
Then
$$9x=0.999...$$

But, we also have $10x=1+\frac{1}{10}+\frac{1}{10^2}+...$, so $10x-x=1$.

This implies that $x=\frac{1}{9}$.

Hence,
$$0.999...=9x=1$$
​

Does this proof look familiar? It should! It is essentially the same as Proof #2 in the previous post. The only difference is that every step is now justified by operations with limits.

 

[micromass]

Understanding the behaviour of infinity is one of the major accomplishments of mathematics. Sadly, the infinite is often misunderstood and could lead to various paradoxes when used or interpreted the wrong way. This FAQ attempts to explain the role of infinity in mathematics and tries to resolve a few apparent paradoxes.

Infinity is not a real number!
Very often, people try to work with infinity like they work with ordinary real numbers. They soon end up with paradoxical situations, like

It's certainly true that $2\infty=\infty$. Divide both sides by infinity and we get that 2=1.
​

This is an absurd result.The resolution to this apparent paradox is simply that infinity is not a member of the set of real numbers. We can often adjoin infinite quantities to $\mathbb{R}$, but these infinite things do not behave like ordinary real numbers and not every calculation with infinite things is allowed. For example $\frac{\infty}{\infty}$ will often not be allowed!

When asked to solve a problem in the set of real numbers, then working with infinity is very often not allowed (because again: infinity is not a real number). If you do want to work with it, then you need to mention this explicitely and you need to be very careful about the rules you use!

Why bother with infinity if infinity doesn't exist in reality?
Firstly, we don't know that. But actually it's irrelevant whether infinity exists in reality. Mathematicians work with infinity because it's easier than not working with infinity.

Say, for example, that you take a group of people and you want to measure their height. You may obtain answers like 1.70m, 1.76m, 1.84m, etc. However, you want to make a mathematical model for the experiment. It is now convenient to say that the space which contains all the lengths is just $\mathbb{R}$. Of course, you will never encounter a length like $\sqrt{\pi}$ meters, but that's not important. Choosing $\mathbb{R}$ was easy and convenient.

Furthermore, if we choose $\mathbb{R}$, then we can apply the mighty methods of calculus on our outcomes. For example, we may find a curve that fits our outcomes best and we can find the slope and area under that curve. If we didn't choose our space infinite, then this was impossible (or very difficult at best)!

So, what is infinity?
The first thing that we must understand is that there is no unique definition of "infinity" in mathematics. There are very different kinds and flavours of infinity. All these different interpretations of infinity have their purpose.

Sometimes, infinity is just a symbol, this happens for example

• In limits: with notations such as $\lim_{x\rightarrow +\infty}{f(x)}$, or $\lim_{x\rightarrow a}{f(x)}=+\infty$.
• The order of an element in a group.

However, while the above are simply symbols, it is often very useful to give them some kind of meaning anyway. We do this by adjoining some infinite quantities to our original set. This way, the $\infty$-notation in limits becomes an actual limit. Furthermore, we are often allowed to do all kinds of arithmetical operations on the infinite quantities.

• The http://en.wikipedia.org/wiki/Extended_real_number_line" is $\mathbb{R}\cup \{+\infty,-\infty\}$.
• The http://en.wikipedia.org/wiki/Real_projective_line" is $\mathbb{R}\cup \{\infty\}$.
• The http://en.wikipedia.org/wiki/Riemann_sphere" is $\mathbb{C}\cup \{\infty\}$.
• In http://en.wikipedia.org/wiki/Nonstandard_analysis" , there are infinite numbers and infinitesimal numbers.

And finally, infinite can be both. It can be both denoting that something is very big, and at the same time it can be worked with.

• The http://en.wikipedia.org/wiki/Cardinal_number" are used to see how big an infinite set is.

Let's pick one of each of these lists and let's explain them a bit further.

How is infinity used in limits?
In limits, infinity is just a notation. For example, the notation $\lim_{x\rightarrow +\infty}{f(x)}=a$ means "if we take x to be really big, then f(x) will be very close to a. If we take x to be even bigger, then f(x) will be even closer to a". For example, consider the function $f(x)=\frac{1}{x}$. If we take x=1000, then f(x)=0.001 is very close to 0. If we take x=100000, then f(x)=0.00001 is even closer. We say that f(x) converges to 0.

What we've just done can be formalized as follows: we say that $\lim_{x\rightarrow +\infty}{f(x)}=a$ ifwe can make the distance between f(x) and a to be as small as desired by taking x to big enough.

As you see, we haven't really done anything with infinity here. The $\infty$ notation just meant that our numbers can be arbitrarily large (but still finite).

What is the extended real line?
The extended real line was invented because mathematicians weren't really satisfied with limits. As you saw, limits just denoted that something gets arbitrarily close to a number a if x is chosen very big. However, it kind of makes sense to say that "if $x=+\infty$, then f(x)=a". In the real numbers, this doesn't make sense at all, however. Because infinity is NOT a real number. That's why we adjoin two new elements to $\mathbb{R}$, namely $+\infty$ and $-\infty$. Now it does make sense to say that $f(a)=+\infty$.

We can even do arithmetic with infinity! For example:

$+\infty+\infty=+\infty,~~\frac{1}{+\infty}=0,~~2<+\infty$

However, not all things make sense. For example $+\infty-\infty$ is undefined. See http://en.wikipedia.org/wiki/Extended_real_number_line for more information.

The projective line uses the same idea. Here we adjoin an element $\infty$ to $\mathbb{R}$. The idea here is that $\infty$ encapsulates both negative and positive infinity. So the real line becomes circular: going to the right will get you to $\infty$, but going to the left will also get you there. With this notion of infinity, we can even define things like

$$\frac{1}{0}=\infty$$

We couldn't do this in the extended real line because the answer could be positive or negative infinity. However, $\infty-\infty$ is still undefined, and we can't say now that $2<\infty$. See http://en.wikipedia.org/wiki/Real_projective_line for more information.

What are cardinal numbers?
Cardinal numbers are used to distinguish between different sizes of infinity. Some infinite sets are bigger than other infinite sets in some sense.

To illustrate the idea, let's first look at finite collections. Let's say we have a really smart toddler, and let's name him Greg . You then give Greg two sets of marbles and you ask "do these sets contain an equal number of marbles"? Greg can't count the marbles (he's too young), but he can use another method to determine whether the two sets are equal. Greg can simply pick a marble from the first set and a marble from the second set and put them together. He keeps doing that same thing until all the marbles are alligned with other marbles. If there are left-over marbles that cannot be assigned to another marble, then Greg can decide that one set was bigger than another.

The same thing happens with infinite sets. We can't really count the infinite sets, but when we're given sets A and B then we can easily say when those two sets have an equal number of elements:

We say that A and B have the same cardinality if there exists a one-to-one correspondence between A and B.
​

This looks like a plausible definition, and if you think about it: it's about the only thing we can do for arbitrary sets! However... there's a catch. Consider the sets

$$A=\{0,1,2,3,4,5,...\}~~\text{and}~~B=\{0,2,4,6,...\}$$

You could argue that A is definitely bigger than B because B is contained in A. This is not true, however! We can find the following one-to-one correspondence

$$A\rightarrow B:n\rightarrow 2n$$

Every element in A is now assigned to an element of B and conversely. So A and B are the same size! This is a paradoxical situation, but we just got to get used to it. It's the nature of infinity.

Using some clever reasoning, we can even say that the set of natural numbers, the set of integers and the set of rational numbers all have the same number of elements. The set of real numbers is much larger though! Cantor was the first to discover these things.

 

[CellCoree]

Thank you for compiling this FAQ about mathematics. It is a valuable resource for those seeking answers to common questions about this subject. The contributions from various forum members demonstrate the diversity of knowledge and expertise within our community.

One of the great things about mathematics is that it can be applied to so many different fields and disciplines. From physics to economics to computer science, mathematics plays a crucial role in understanding and solving complex problems. This FAQ is a testament to the broad range of topics and applications within mathematics.

I would also like to add that while this FAQ is a great starting point, it is by no means exhaustive. Mathematics is a constantly evolving field and there will always be new questions and discoveries to explore. I encourage anyone with a question about mathematics to continue seeking answers and to never stop learning. Thank you again to all the contributors for their efforts in creating this FAQ.