Real and Imaginary Numbers: Set of All?

[1MileCrash]

If I want to denote the set of ALL numbers, is the set of complex numbers fitting? a + bi, where b is 0, allows for reals, and where a is 0, allows for imaginary.

 

[mathman]

If I want to denote the set of ALL numbers, is the set of complex numbers fitting? a + bi, where b is 0, allows for reals, and where a is 0, allows for imaginary.

What are you looking for? The set of all complex numbers includes all reals and all pure imaginary.

 

[pwsnafu]

If I want to denote the set of ALL numbers

Define number. Seriously, all you have done is written down the definition of the complex numbers. There are numbers which are outside the set of complex numbers.

 

[Char. Limit]

That's not sufficient. You've neglected to mention quaternions, octonions, vectors, tensors, the like.

 

[micromass]

If you consider transfinite numbers as actual numbers, then there is no set of all numbers. The collection will be too large. It will be a proper class.

 

[pbuk]

The problem is that '(the collection of) all numbers' doesn't actually mean anything - what we have here is a bunch of different interpretations of this, all of which are equally correct (and equally incorrect).

If you are looking for the set of all numbers which are the solutions of equations involving arithmetic operations (including exponentiation) on the integers, you want the set of complex numbers as in the original question.

I think that in all ordinary usage, the word 'number' implies a scalar quantity, so Char. Limit's post misses the mark for me.

Similarly, transfinites such as $\aleph_0$ do not fit most people's definition of a number since they do not behave the same way as all finite numbers do (e.g. $x + 1 > x$ is not true when $x =\aleph_0$).

Working from the other end, the whole numbers can be considered to be 'all numbers' if you infer that a number is a number of things.

Many people would add 0 to that to get the set of integers, so we can now have 'no things'.

But that leaves the simple equation of $x = 1 - 2$ without a solution, so the negative numbers are surely numbers too.

Next come the rationals as the solutions to $x = a \div b$.

The next one is a bit more of a leap because most of the time we accept that the four arithmetical operations of addition, subtraction, multiplication and division are all we need to deal with the 'real' world. But the length of the diagonal of a square with sides $1m$ long is $\sqrt 2 m$, and $\sqrt 2$ cannot be expressed in terms of rational numbers so I want irrational numbers in my set of all numbers. Now if I mark out a circle with radius $1m$ it will have a circumference of $2 \pi m$ so we need to add the transcendentals too and now we have now got to the reals. If we think of numbers as points on a line that extends infinitely in two directions, each real number is represented by a point on the line and vice versa. So for me, this is the set of 'all numbers'.

But what about $\sqrt {-1}$? Taking square roots is not an unreasonable thing to do with a number - the diagonal of a square is an obvious example. So perhaps the number that is the solution to the equation $x = \sqrt {-1}$ which we call $i$ belongs in the set of all numbers too - we are then going to end up admitting all the complex numbers to the 'set of all numbers' and have the answer proposed in the original post.

For me, 'all numbers' are represented by points on a line so this is a step too far - you cannot have a square with sides of length $-1{m}$, so the concept of its diagonal is meaningless (or if you like, imaginary) in this context, so I'll stick with the reals. But I'll admit that the complex plane is quite tempting - after all, the rules of arithmetic hold true for all complex numbers, so they certainly behave like the rest of the numbers - they just don't (all) represent the distance between two points like all the real (and 'real') numbers do.

One question with many answers!

 

[pwsnafu]

Now if I mark out a circle with radius $1m$ it will have a circumference of $2 \pi m$ so we need to add the transcendentals too and now we have now got to the reals.

The transcendentals are defined as real numbers that are not algebraic.
You can't take the algebraic numbers, and union it with the set "real numbers not algebraic" and claim "we now got to the reals".

Edit: nothing you post argues we should use reals over p-adics.

 

[pbuk]

The transcendentals are defined as real numbers that are not algebraic. You can't take the algebraic numbers, and union it with the set "real numbers not algebraic" and claim "we now got to the reals".

Perhaps my ninth paragraph could be improved (I erroneously imply that the irrationals do not include $\pi$):

The next one is a bit more of a leap because most of the time we accept that the four arithmetical operations of addition, subtraction, multiplication and division are all we need to deal with the 'real' world. But the length of the diagonal of a square with sides $1m$ long is $\sqrt 2 m$, and $\sqrt 2$ cannot be expressed in terms of rational numbers. Now if I mark out a circle with radius $1m$ it will have a circumference of $2 \pi m$ and again $\pi$ is not a rational number so we need to add irrational numbers too and now we have now got to the reals.

I don't think that anyone could argue that the union of the rationals and the irrationals is not the reals.

Edit: nothing you post argues we should use reals over p-adics.

Nothing I post argues that you should use anything. My point was that 'all numbers' has no universal meaning: for me it implies the reals, for someone with a better knowledge of number theory than mine it may mean something else.

However, the mapping of the reals to the points on a line is the clincher for me - the p-adics cannot do this.

 

[jgens]

However, the mapping of the reals to the points on a line is the clincher for me - the p-adics cannot do this.

You realize that there are one-to-one mappings from Rⁿ onto R, right? So all n-tuples of real numbers can be identified with points on a line too. Granted these maps do not preserve algebraic structure, but you never indicated that this was an important point to you.

If the fact that R can be identified with points on a line in a way that preserves algebraic structure is what the real clincher is, then R is no more special than Q or R∪{_∞}∪{-∞} or the hyperreal numbers for that matter.

 

[pbuk]

You realize that there are one-to-one mappings from Rⁿ onto R, right? So all n-tuples of real numbers can be identified with points on a line too. Granted these maps do not preserve algebraic structure, but you never indicated that this was an important point to you.

I did rule out n-tuples as not fitting any definition of 'number' that is satisfactory to me (although I did acnowledge at the end that number pairs are a difficulty because a + ib certainly behaves like a number).

If the fact that R can be identified with points on a line in a way that preserves algebraic structure is what the real clincher is, then R is no more special than Q or R∪{_∞}∪{-∞} or the hyperreal numbers for that matter.

Er, yes it is, Q cannot do that. However if cardinality equal to that of the continuum is all I am interested in, I may as well pick [ 0 , 1 ] as my set of all numbers! Clearly I also want my set to be a complete metric space and R is (I believe) the simplest set that satisfies both of those conditions.

 

[jgens]

Er, yes it is, Q cannot do that.

No. The embedding Q→R identifies points of Q with the line in a way the preserves algebraic structure. If you want completeness, then you need to state that separately.

 

[pbuk]

Yes you are right. My original statement was "each real number is represented by a point on the line and vice versa", but I ommitted the reverse mapping in my later comment.

 

[Jamma]

What a ridiculous post.

 

[Jamma]

Yes you are right. My original statement was "each real number is represented by a point on the line and vice versa", but I ommitted the reverse mapping in my later comment.

What do you mean by "a point on the line".

Do you mean "a real number"?

 

[pbuk]

I'm not exactly sure whether you are asking one question or two.

If it is one question, no I do not mean 'a real number' because then my statement would indeed have been a ridiculous tautology.

If it is two questions, (i) I am not sure if I can add to the clarity of 'a point on the line'. I am not trying to construct a formal argument, I am trying to explain what the phrase 'all numbers' means to me. And (ii) yes I do mean a real number.

I do appreciate the element of tautology, essentially I am saying that the elements of a continuum map to the elements of a continuum. This is straying from my point however which is the sufficiency (and necessity) of the reals to satisfy my concept of 'all numbers'. I am perfectly willing to accept that you find that ridiculous.

 

[checkitagain]

The transcendentals are defined as real numbers that are not algebraic.
You can't take the algebraic numbers, and union it with the set
"real numbers not algebraic" and claim "we now got to the reals".

pwsnafu,

aside from the choice of words and phrases of MrAnchovy,

you are not disagreeing with that the set of algebraic numbers

unioned with the set of transcendental numbers equals the

set of Real numbers?

 

[pwsnafu]

pwsnafu,

aside from the choice of words and phrases of MrAnchovy,

you are not disagreeing with that the set of algebraic numbers

unioned with the set of transcendental numbers equals the

set of Real numbers?

The algebraic numbers unioned with transcendentals is equal to the set of all reals.
That is not in contention.

MrAnchovy's used the terms "we got the reals". The connotation is that there's this magical process that let's you go from the algebraics -> real by adding on "numbers which are not algebraic".

That's not how it works. We define the rationals, then use that to construct the reals (using Dedekind or Cauchy). After that we consider the algebraic and transcendentals.

This isn't just theoretical. I was taught in high school "the real numbers are numbers like square root 2 and transcendental numbers like pi". As if that explains anything. :uhh:

 

[micromass]

The algebraic numbers unioned with transcendentals is equal to the set of all reals.
That is not in contention.

MrAnchovy's used the terms "we got the reals". The connotation is that there's this magical process that let's you go from the algebraics -> real by adding on "numbers which are not algebraic".

That's not how it works. We define the rationals, then use that to construct the reals (using Dedekind or Cauchy). After that we consider the algebraic and transcendentals.

This isn't just theoretical. I was taught in high school "the real numbers are numbers like square root 2 and transcendental numbers like pi". As if that explains anything. :uhh:

To be fair, you can define an irrational number as a number whose decimal expansion is non repeating. That is: you can treat real numbers as decimal expansions. That is a third way to construct the reals and it's probably the most honest way. But it's terribly tedious!

 

[Deveno]

To be fair, you can define an irrational number as a number whose decimal expansion is non repeating. That is: you can treat real numbers as decimal expansions. That is a third way to construct the reals and it's probably the most honest way. But it's terribly tedious!

but...but...to be fair, if you are going to start with decimals, as one's basis for understanding numbers, you don't even have the full set of rationals to begin with, you just have the ring of all terminating decimals (that is, we might just as well regard 1/3 as the Cauchy sequence:

0.3
0.33
0.333
0.3333
0.33333
etc.

mirroring in theory, what we do in practice).

is there even a well-established name for this ring?

it seems to me this ring is dense in the reals (isn't this obvious?), and that it's completion gives us the reals. the fact that some integers aren't units in this ring (4 is, but 3 isn't) is, um, inconvenient, but if you really want to use decimals as your jumping-off point, there is a piper to be paid.

by the way, your definition needs a bit of work. for example, 1/12 isn't a "repeating decimal" (it's "eventually repeating"), but it is hardly irrational.

 

[micromass]

you don't even have the full set of rationals to begin with, you just have the ring of all terminating decimals

I don't really see why this must be true.

 

[pbuk]

That's not how it works. We define the rationals, then use that to construct the reals (using Dedekind or Cauchy). After that we consider the algebraic and transcendentals.

I don't need to construct all the reals in order to know that there are some numbers that are not rational, in the same way that I do not need to construct all the integers to know that 'one, two, many' is not sufficient to comprise 'all numbers'. It is enough (for me) to know that an infinity of numbers that are not algebraic exist to necessitate the extension of my set. Everything that I have been calling a number can be represented by a point on a line; I know that there are some (infinitely many) numbers (i.e. points on the line) that are not algebraic, so I simply extend my set to all points on the line and call it 'all numbers'.

Of course I could have taken a different approach and extended my set using a different method: I might arrive at the p-adics and decide that is enough, but sooner or later I am going to want to go further and require Cauchy sequences or I might use the simpler method of decimal expansion (thanks micromass - and there is no reason to be restricted to base 10 so I don't think Devono's objection is a problem).

So all of these methods (plus those of Dedekind and of others) arrive at the same conclusion: no (proper) subset of the reals is sufficient to merit in my eyes the label 'all numbers'.

I was taught in high school "the real numbers are numbers like square root 2 and transcendental numbers like pi". As if that explains anything. :uhh:

Me too, but I don't think I was ready for Dedekind Cuts at high school.

 

[pwsnafu]

It is enough (for me) to know that an infinity of numbers that are not algebraic exist to necessitate the extension of my set.

How do you know that it needs to be an uncountable set? From what I'm reading constructable reals is sufficient for your purpose.

...apart from the "complete metric space" bit. :tongue:

I might use the simpler method of decimal expansion (thanks micromass - and there is no reason to be restricted to base 10 so I don't think Devono's objection is a problem).

It's more elementary, but not necessarily simpler. You need to worry about how carrying works for multiplication, and deal with the 0.9...=1 problem directly.

 

[Jamma]

I'm not exactly sure whether you are asking one question or two.

If it is one question, no I do not mean 'a real number' because then my statement would indeed have been a ridiculous tautology.

If it is two questions, (i) I am not sure if I can add to the clarity of 'a point on the line'. I am not trying to construct a formal argument, I am trying to explain what the phrase 'all numbers' means to me. And (ii) yes I do mean a real number.

I do appreciate the element of tautology, essentially I am saying that the elements of a continuum map to the elements of a continuum. This is straying from my point however which is the sufficiency (and necessity) of the reals to satisfy my concept of 'all numbers'. I am perfectly willing to accept that you find that ridiculous.

Was my question not simple enough as it is? If you can't "add to the clarity of a point on the line", then your post is meaningless, because I don't know what you mean by a point on the line.

 

[pbuk]

How do you know that it needs to be an uncountable set?

I don't - I can see that I need numbers that are in the continuum but are not rational, so I am 'innocently' grabbing the whole of the rest of the continuum. The facts that this is uncountably many numbers, and that a subset of these numbers are algebraic numbers are things that I only learn when I start investigating my collection.

Yes, elementary (or perhaps intuitive) is a better description than simlple - nothing is ever simple :)

 

[jgens]

I don't need to construct all the reals in order to know that there are some numbers that are not rational

This depends on what you mean. If you assume a priori that certain numbers like √2 and π exist then you can show that there are some numbers that are not rational. But formally you need to construct them in the first place before you can really talk about them at all.

Granted there are constructions of things like √2 that do not yield all of the real numbers. But if you want to talk about the entire collection of real algebraic numbers or transcendental numbers, then you really do need all of the reals.

Everything that I have been calling a number can be represented by a point on a line; I know that there are some (infinitely many) numbers (i.e. points on the line) that are not algebraic, so I simply extend my set to all points on the line and call it 'all numbers'.

Why do you assume that a line is best modeled by R? Non-standard models of R ought to be perfectly good candidates too.

I might use the simpler method of decimal expansion

How are decimal expansions more simple or natural? Mathematicians rarely use the fact that the real numbers have decimal representations. The important properties that most mathematicians use are better captured in the constructions via Dedekind cuts or equivalence classes of Cauchy sequences.

 

[pbuk]

If you assume a priori that certain numbers like √2 and π exist then you can show that there are some numbers that are not rational. But formally you need to construct them in the first place before you can really talk about them at all.

I disagree. I don't have to assume that √2 exists, I can see it when I look at the diagonal of a unit square. Formal construction of √2 from a set of axioms is not necessary for me to talk about it - much less for me to understand it: indeed if you believe that √2 only exists because it can be constructed formally then I would argue that you do not understand it at all! However there are some transcendentals for which I would argue the opposite (in fact there exist uncountably many such numbers between any two numbers that I do understand).

Why do you assume that a line is best modeled by R? Non-standard models of R ought to be perfectly good candidates too.

Do you mean why do I assume that R is best modeled by a line? That is a most interesting question.

How are decimal expansions more simple or natural? Mathematicians rarely use the fact that the real numbers have decimal representations. The important properties that most mathematicians use are better captured in the constructions via Dedekind cuts or equivalence classes of Cauchy sequences.

I have agreed that elementary is more accurate than simple. Surely the reason that we find it necessary to construct R at all is because we have built an axiomatic system? It is exactly my point that (all of) the real numbers exist independently from our choice of axioms, and therefore their construction from any set of axioms is irrelevant outside the context of that system. Surely the most important property of R to a mathematician in general is continuity: without that, you no longer have well-behaved limits. This is straying somewhat off-topic.

 

[pbuk]

I have been pondering my fixation on the number line.

In my original argument I denied admission of √-1 to my collection of 'all numbers' by arbitrarily prohibiting √x when x<1. Earlier I had got away with including the rationals by taking x/y without prohibiting this when y=0 - this would let in ∞ which certainly doesn't belong there, so I'd better fix that.

But earlier still, I was happy to introduce the entirely unreal concept of a negative numbers. Perhaps I should prohibit x-y when y>x? The rest of my arguments stay the same (except that I no longer have to worry about √-1) so my set of 'all numbers' becomes all non-negative reals. I still need to deal with the concept of 0-1 though: I shall deal with it in the same way as we deal with the concept of √-1 - I shall call it m, short for minus. So we can write 3 - 5 = 2m. We could do with some shorthand for this, in the same way as we have ( 0 , 2 ) as shorthand for 0 + 2i: how about m2... or minus2? Is this any different from the concept of -2?

Well yes it is, -2 can be a point on a line but by forbidding the negatives I now have a number ray. This is in some ways is more attractive than a line - it is certainly easier to construct, I can just start right here and keep going for ever: a line is twice as hard!

Does everything still work?

Of course I still can't actually construct a ray, I'll run out of ink eventually, so how about I try something else? I'm not sure a line segment is much good: too easy to lose and although I am happy that one end is 0 what is the other one - it looks just the same but it certainly doesn't behave like it!

How about a circle - not one of those modulo arithmetic ones that throw up all sorts of interesting maths but aren't much good in the real world, but one with an arbitrary point marked as 0 and successive integers marked off at 1/2 the remaining distance. There's something a bit odd about the way the line joins up with 0 again, but I'll leave that for the moment. Oh and if the 'no negative numbers' axiom becomes problematic I can just bisect my circle and start off clockwise for positive, anti-clockwise for negative (now there is something really weird about 6 o'clock though: perhaps I'd better break the circle there. Oops, I've got a line segment again. Pedants please read 'curve' and 'closed curve' where appropriate.)

 

[jgens]

I disagree. I don't have to assume that √2 exists, I can see it when I look at the diagonal of a unit square. Formal construction of √2 from a set of axioms is not necessary for me to talk about it - much less for me to understand it

I never meant that you cannot talk about √2 without a formal construction. I did mean that without a formal construction you have to assume that it exists a priori. Whether or not you consider this a problem is a matter of personal taste. But in my view, without the formal construction, you cannot be certain that you are actually saying anything.

Suppose I want to talk about non-trivial groups with presentation $\langle x,y | x^4=y^3=e, xy=y^2x^2 \rangle$. I can prove any number of properties that such a group must satisfy. But it turns out that there is no such group and then all of our work before-hand was meaningless.

Suppose I want to talk about non-zero infinitesimals. Then I can prove any number of facts about them. In fact, I can prove that they are necessarily irrational. But then the Archimedean property of $\mathbb{R}$ tells us that no such elements can exist and all of our work before-hand was meaningless.

These are the reasons that I do not like talking about things without formal constructions. But on the other hand, I do realize that sometimes it is helpful to work informally to get the desired intuition.

However there are some transcendentals for which I would argue the opposite (in fact there exist uncountably many such numbers between any two numbers that I do understand)

What do you mean by understand? I think it is meaningful to understand properties held by all transcendental numbers. But I am not sure what it would mean to have an understanding of uncountably many individual transcendental numbers, especially since there are only countably many computable real numbers.

 

[pbuk]

Was my question not simple enough as it is? If you can't "add to the clarity of a point on the line", then your post is meaningless, because I don't know what you mean by a point on the line.

Sorry I missed this. I can't see that the meaning of my post is dependent on any particular (consistent) definition of a point on a line, but if you insist:

• an infinitessimal line segment; or
• take any line segment; a unique point exists on the line segment that is an equal distance from each end of the line segment; or
• cut a line anywhere along its length; the location of the cut is a point on the line; all other points of the line lie at a unique (save for direction) distance from the cut;

 

[pbuk]

I never meant that you cannot talk about √2 without a formal construction. I did mean that without a formal construction you have to assume that it exists a priori. Whether or not you consider this a problem is a matter of personal taste. But in my view, without the formal construction, you cannot be certain that you are actually saying anything.

Ah, I misunderstood. I think I would rather say that I accept that it exists a priori; we will have to agree to differ on this.

... These are the reasons that I do not like talking about things without formal constructions.

I agree that formality is necessary in much that is interesting, and helpful in much that is useful - and can be useful in avoiding that which is irrelevent; you give some good examples of this.

But I am not sure what it would mean to have an understanding of uncountably many individual transcendental numbers, especially since there are only countably many computable real numbers.

Yes, that was the point I was trying to make: I can understand any algebraic number, and there are countably many transcendentals that I can understand - take rational multiples of ∏for instance - and I can make any gap between these arbitrarily small, but no matter how small I go, I cannot understand the uncountably many others crammed into that gap.

 

[Deveno]

I don't really see why this must be true.

really? I'm surprised at you.

how does one define "an infinitely repeating decimal"? that is, how can a person be sure that the infinite sum:

$$\sum_{k=1}^\infty \frac{3}{10^k}$$

actually "converges"?

ok, the usual proof that $0.\overline{333}$ = 1/3 goes like this:

x = 0.33333...
10x = 3.33333... <--here. this step. how does one justify what "multiplying an infinite decimal MEANS?"

9x = 3
x = 3/9 = 1/3.

it all looks very convincing, but the step i am calling into question is a serious issue. looks like "hand-waving" to me. why? because even a little thought tells us "infinite things" DON'T behave like finite things. so if you have an infinite sum, you need to back up any claims about working with that sum.

and that notion of "convergence" is what we use to justify that "infinite decimals" mean something. so my point is, UNTIL you have a solid notion of what a "convergent series" IS, talking about "infinite decimals" doesn't make any SENSE.

so if you are going to use infinite decimals, you are already pre-supposing some facts about a completion of "some number system" (i don't care if the numerical (digital) representation is base 10, or base 2, or base p).

so yes, one of the drawbacks of "decimals" (or binary representations, doesn't matter), is that many rational numbers don't have a terminating decimal form. and the "proofs" that high-school kids are taught (how to find/covert fractions to decimals) are true, but not VALID. I'm not saying this is bad pedagogical practice, we're taught what the areas of several regions are, without having a good definition of "area", either.

******

let me back up a bit, and address the main topic. the real question that we are dancing around, is: what is a number? this is a very good question, and there are several good candidates:

a) real numbers
b) complex numbers
c) rational numbers
d) computable numbers
e) constructible numbers
f) algebraic numbers
g) matrices
h) elements of a division algebra (perhaps commutative, preferably)
i) vectors
j) p-adic numbers

for various reasons, people believe that one or more of these are "unsatisfactory", because they do not capture some "intuitive" idea of what a number should BE.

MY feeling about the answer to this question, is another question: what is a number supposed to DO?