I Algebraic property of real numbers

[Mr X]

TL;DR Summary

Why is the algebraic property considered one of the the properties of real numbers when it's defined by addition and multiplication? Are those two linked to real numbers in a way no other binary operators are, and in a way it's not linked to any other system? If so, why?

The properties of real numbers are listed as order, completeness and algebraic properties. I can understand order and completeness, atleast i get why it's there but why algebraic?
If we consider real numbers as the number line or an infinite set, where does addition and multiplication come in? Or does it always assume by default that whenever we say real numbers it is a field? And even if it so, why addition and multiplication? Is there any part they play in real numbers that cannot be done by any other binary operators, and do they only play such a part in real numbers only? If no, then why are they there, and if yes, what is it?

 

[jedishrfu]

That's a deep collection of questions. I think your best bet is to watch a set of real analysis lectures which should answer most if not all of your questions:



Operatorwise addition and multiplication are close to our hearts because of their practical use in everyday tasks. Mathematicians have taken them to develop the subject of real analysis.

While there are undoubtedly other possible operator choices these two seem the most practical and intuitive to use. One example is modulo arithmetic where the same operators are constrained to work with a smaller finite set of numbers.

 

[Mr X]

I think your best bet is to watch a set of real analysis lectures which should answer most if not all of your question

I've seen a few but most of them simply go to addition and multiplication but never explains why we choose those specific operators. Not even a statement saying that it's for convenience. I'll go through the playlist you've attached as well as soon as I can.

While there are undoubtedly other possible operator choices these two seem the most practical and intuitive to use. One example is modulo arithmetic where the same operators are constrained to work with a smaller finite set of numbers

Understood. Thankyou

Ill get back on the other questions after I watch through then.

 

[PeroK]

I've seen a few but most of them simply go to addition and multiplication but never explains why we choose those specific operators.

Addition and multiplication (of whole numbers) is where mathematics starts. What use would mathematics be without those?

 

[Mr X]

Addition and multiplication (of whole numbers) is where mathematics starts. What use would mathematics be without those?

It's not about mathematics, it's about properties of real numbers. Not even the real number field, but just real numbers. Ordering and completeness are innate properties of the real numbers, but what about algebraic? They're operators used on real numbers, not a part of real numbers itself. Well that's what I'm thinking anyways.

 

[PeroK]

It's not about mathematics, it's about properties of real numbers.

The properties of real numbers is mathematics.

 

[Mr X]

The properties of real numbers is mathematics.

That's.. not quite how that question works. By that logic we should add algebraic properties in properties of triangles, as well as for circles. I'm not sure how to rephrase the question better than I've done, so I'll leave it be here.

 

[pasmith]

We usually think of the real numbers not just as a set of objects, but as a complete ordered field. In particular, this means that the basic arithmetic operations work on them. It is inherent in the definitions of 0 and 1 that they are respectively the additive and multiplicative identities.

The real numbers are mathematical constructs; they do not have any physical existence. Any statement about them is part of mathematics.

 

[PeroK]

Ordering and completeness are innate properties of the real numbers, but what about algebraic? They're operators used on real numbers, not a part of real numbers itself. Well that's what I'm thinking anyways.

Imagining that ordering and completeness, are "innate" properties and are more fundamental than addition is misguided. The usual ordering on the real numbers requires addition as part of its definition. See here, for example:

https://math.stackexchange.com/questions/1357771/foundation-of-ordering-of-real-numbers

Completeness requires at least an ordering.

Not to mention that the negative whole numbers are defined as the additive inverse of the positive whole numbers - with zero being the additive identity. And, the rationals are defined using the notion of multiplication. For example, the number $\dfrac 1 2$ is defined as the multiplicative inverse of $2$. With $1$ being the multiplicative identity.

 

[WWGD]

Well, there are tons of additional structure to the Reals as well as $\mathbb R^n; n\geq 1$, i.e., Euclidean Space.They are a (Complete) Metric space, an inner-product space and a Hilbert Space.

 

[fresh_42]

If we consider real numbers as the number line or an infinite set, where does addition and multiplication come in?

By the requirement that we want to define a field extension of the rational numbers, and field means addition and multiplication with their inverse operations.

 

[jedishrfu]

One could study the history of numbers and learn that we either discover or invent rules called operators for working with them.

As an example:

- counting numbers were used to describe how many of some item we have so we can say:

I have X apples.

- addition was implicitly recognized when people would say:

I have 3 apples here and you gave me 4 apples so I have 7 apples

- multiplication appears when people realize we can organize items in rows and columns like

5 apples per row and 7 rows hence I have 35 apples

- zero was recognized when people needed a good way to describe having no apples

- subtraction when people selling apples would remember they sold 2 apples leaving 5 in their possession
...

- fractional numbers came from the measurement of quantities like selling pizza where one could say "I have half of a pizza" left versus "I have 3 slices left", which loses the notion of whether I have half a pizza or a quarter of a pizza or something in between. (This is not the greatest analogy, but when you're hungry, pizza appears)

- fractional numbers were combined with counting numbers to make rational numbers
...

- then there are real numbers, complex numbers, vectors...


Addition and multiplication operations (rules) are redefined to work with the number system chosen so we have rules for adding and multiplying complex numbers, vectors and ...

It may sound silly, but we learn a lot of this in elementary school, and it becomes so implicit in our being that we never stop to think that there was a time when people first learned to count.

 

[fresh_42]

Here is an article about numbers of all kinds:
https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/
and how they relate to each other.

 

[Mr X]

Well, there are tons of additional structure to the Reals as well as $\mathbb R^n; n\geq 1$, i.e., Euclidean Space.They are a (Complete) Metric space, an inner-product space and a Hilbert Space.

I don't quite understand this, but I'll look into it, thankyou

 

[Mr X]

By the requirement that we want to define a field extension of the rational numbers, and field means addition and multiplication with their inverse operations.

I'm postive a field requires any two binary operators, but I don't know about the construction of real numbers so if you're talking about that specific case, then understood.

 

[Mr X]

I think I've got the gist of things, real life was as well as abstract maths vise. Thankyou all for helping out. I'll look into the materials provided and continue by myself later on.

 

[mathwonk]

Briefly:
If you think of the real numbers as points on a line (ordering) without gaps (completeness), then addition is the result of translating segments and joining them together to form a new segment. This translation process is necessary for the comparison of sizes of different segments. Multiplication of two segments is naturally the process of forming a rectangle with those segments as sides. Then the product of two segments is, initially, an area.
In order to think of points on the real line as numbers, including 0 and 1, one chooses an origin point 0, and a second point 1, forming together a unit segment.
Then multiplication of two segments can be considered to yield another segment rather than an area, by forming another rectangle with the same area, but having one side a unit segment.
In this way, addition and multiplication of real numbers (segments) yields real numbers (i.e. segments). This is all in Euclid, whose geometry is the origin of our system of real numbers. For a beautiful modern reference one may very profitably consult Hartshorne's Geometry: Euclid and Beyond, and in particular the section on (Hilbert's) "segment arithmetic". This harks back of course to Hilbert's Foundations of Geometry, about 1899.

 

[fresh_42]

I'm postive a field requires any two binary operators, but I don't know about the construction of real numbers so if you're talking about that specific case, then understood.

There are, in principle, two ways to construct real numbers: a set-theoretic approach via so-called Dedekind cuts and a topological one via Cauchy sequences. I learned the first one, but meanwhile prefer the latter. Both approaches basically add all missing numbers between $\mathbb{Q}$ and the number line $\mathbb{R}$, and both have the problem of implementing what you call algebraic structures. It is a bit easier with Dedekind cuts, and a bit more technical with Cauchy sequences. Hence, in a way, you are right. The algebraic structures don't come naturally. However, it has to be done since we don't want to lose the structures we already have for rational numbers. We can add lengths geometrically, and the area of a rectangle gained from multiplication is obviously real, so there has to be a way to define addition and multiplication.

 

[Erland]

If you can't add and multiply real numbers, then what use do you have for them?

 

[lavinia]

If you can't add and multiply real numbers, then what use do you have for them?

The real line without addition and multiplication has structure as a continuum. The continuum generalizes to Euclidean space and then further to manifolds.

My sense is that extending the rational points on the line using equivalence classes of Cauchy sequences was motivated by the quest for finding the mathematical structure of the continuum.

 

[Erland]

The real line without addition and multiplication has structure as a continuum. The continuum generalizes to Euclidean space and then further to manifolds.

My sense is that extending the rational points on the line using equivalence classes of Cauchy sequences was motivated by the quest for finding the mathematical structure of the continuum.

Yes but what is the continuum used for? You need addition to put together and calculate distances and multiplication to form and calculate areas.

 

[PeroK]

The real line without addition and multiplication still has structure as a continuum. The continuum generalizes to Euclidean space and then further to manifolds.

My sense is that extending the rational points on the line using equivalence classes of Cauchy sequences was motivated by the quest for finding the mathematical structure of the continuum. Formally, this involves completing a metric space. This is a topological procedure and is not unique to the rational points on the line using the Euclidean metric. Any metric space has a completion.

The usual definition of the Euclidean metric in $\mathbb R^3$ for points $\mathbf x = (x_1, x_2, x_3)$ and $\mathbf y = (x_1, y_2, y_3)$ is:
$$d(\mathbf x, \mathbf y) = \sqrt {(x_1 - y_1)^2 + (x_2-y_2)^2 + (x_3 - y_3)^2}$$Now, there may be ways to develop topological spaces without requiring algebraic properties. But, the idea that the Euclidean metric is intrinsic to $\mathbb R^n$, whereas, addition and multiplication of real numbers is not, doesn't stand up to scrutiny.

 

[lavinia]

The usual definition of the Euclidean metric in $\mathbb R^3$ for points $\mathbf x = (x_1, x_2, x_3)$ and $\mathbf y = (x_1, y_2, y_3)$ is:
$$d(\mathbf x, \mathbf y) = \sqrt {(x_1 - y_1)^2 + (x_2-y_2)^2 + (x_3 - y_3)^2}$$Now, there may be ways to develop topological spaces without requiring algebraic properties. But, the idea that the Euclidean metric is intrinsic to $\mathbb R^n$, whereas, addition and multiplication of real numbers is not, doesn't stand up to scrutiny.

Not sure what you mean by intrinsic. There are many possible metrics on the continuum. Are they all intrinsic?

 

[lavinia]

Yes but what is the continuum used for? You need addition to put together and calculate distances and multiplication to form and calculate areas.

Yes but it is also used in the theory of manifolds and in many other mathematical contexts. Generally, when one has a topological space, measurement is an added structure.

 

[fresh_42]

Note: This statement contains rhetorical exaggerations and should be read as such.

The real numbers are a standard example for really many objects, from vector spaces, over algebras and fields, to manifolds and topological groups. Depending on your favorite subject, you need different properties. Topologists do not need addition and multiplication, algebraists do not need the metric, only an evaluation, and number theorists do not need real numbers at all. It is pointless to talk about the intersections because the real numbers are in practically all of them, and all make use of all properties simply because they are there.

However, I stick with my opinion that they are created by topological means, and the algebraic structure has to be implemented afterwards. This implementation took Hewitt, Stromberg about five pages, starting with the Cauchy approach. One aspect that supports this point of view is the ancient Greeks. Although they found the entire Euclidean geometry with its lengths of continuums, they weren't able to work out the concept of real numbers. They had $\pi$ and square roots, but that was it.

 

[PeroK]

Not sure what you mean by intrinsic. There are many possible metrics on the continuum. Are they all intrinsic?

I thought this was the point of thread? That completeness and order were "innate" properties of real numbers; whereas, addition and multiplication are not.

I thought we were discussing this:

It's not about mathematics, it's about properties of real numbers. Not even the real number field, but just real numbers. Ordering and completeness are innate properties of the real numbers, but what about algebraic? They're operators used on real numbers, not a part of real numbers itself. Well that's what I'm thinking anyways.

 

[bhobba]

Some nice answers here.

Specifically, the Wikipedia article (link below) explicitly answers your original question. See, for example, the definition of addition and multiplication in Cauchy's definition (or construction if you prefer) using Cauchy sequences of rationals.

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

It's the type of thing that when you see what is going on, you go - oh Jesus - whack your head - and say, since this is a family forum, some rude expletive.

Most students don't worry about such things. I will outline something genuinely profound because you are what my professors called a 'thinking' student.

You have probably heard of Gödel's Incompleteness Theorems. Just for the heck of it, I will outline the proof. First, the holding problem.

The program HALT (P, I) returns true if program P, which accepts one input, eventually terminates using input I, but false if it continues to loop. Consider the program TROUBLE

TROUBLE (X)
If HALT(X, X)
loop forever

What happens when TROUBLE is run with the program TROUBLE as input? Suppose HALT returns true (ie TROUBLE according to HALT eventually stops). TROUBLE then loops forever. Suppose HALT returns false; then it stops. Both are contradictions, hence HALT does not exist.

This leads directly to the incompleteness theorem. Suppose we have a mathematical system that obeys rules that a computer program can check to determine whether a statement written in the system's rules is true or false. The system is called 'complete' if any statement written in the system's rules is true or false. Write a program that checks all possible statements and determines whether they are true or false. Among those statements are whether programs will halt or not. If the system is complete, you have solved the halting problem. So, no system can be complete if it is strong enough to allow the statement and proof about whether programs will halt or loop. It can be shown that arithmetic is such a system (after all, computer instructions are nothing but binary numbers). This is just an outline to get across the gist of Gödel.

Believe it or not, despite the reals containing the integers, Tarski's construction evades Gödel. The same with Euclidean geometry. Now that is something profound you may wish to delve deeper into.

Thanks
Bill

 

[Mr X]

I've read through the further replies, but I'll need some time to fully understand the ideas as well as go through atleast bare minimum of the materials given, so I'll post another reply afterwards.
Thanks again.

 

[jedishrfu]

Remember some questions are just unknowable. We can speculate on an answer but may never have the actual reasoning behind why something is the way it is.

 

[Mr X]

Believe it or not, despite the reals containing the integers, Tarski's construction evades Gödel. So does Euclidean geometry. Now that is something profound you may wish to delve deeper into.

Yes thankyou, it seems really interesting. I'll definitely look more into it.

I've went through them all, and explored the best I can with the time I have at my level. This entire discussion was, least to say, fascinating. My question is now answered and I've got some ideas on which direction to further explore as well, so thankyou to all those who helped.

 

[lavinia]

I thought it might be relevant to describe the topology of the real line without any use of the Euclidean metric or any other metric for that matter and without any algebraic structure.

A little web research revealed several ways to characterize the topology of the real line or equivalently any open interval on the real line. The reference is this post on Mathoverflow

https://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line

The easiest one for me to understand was

- The topological real line is charactrized as a connected locally connected separable regular topological space in which the complement of any point is two disjoint connected sets. The quoted references for this description are

"The topological characterization of an open linear interval", Proc. London Math. Soc.(2) 41 (1936), 191-198
"On the topological characterization of the real line", Department of Mathematics, Carnegie-Mellon University, Report #69-36, 1969

A relevant point made in the post was that the first reference did not say regular space but instead said metric space. Regular is less restrictive than metric space which shows that a metric is not required to define the topology of the real line.