Are the addition and multiplication of real numbers, as we know them, unique?

After recently concluding my Real Analysis course in this semester I got the following question bugging me:

Is the canonical operation of addition on real numbers unique?

Otherwise: Can we define another operation on Reals in a such way it has the same properties of usual addition and behaves exactly like that?

Or even: How I can reliable know if there is no two different ways of summing real numbers?

Naturally these dense questions led me to further investigations, like:
The following properties are sufficient to fully characterize the canonical addition on Reals?

1. Closure
2. Associativity
3. Commutativity
4. Identity being 0
5. Unique inverse
6. Multiplication distributes over

If so, property 6 raises the question: Is the canonical multiplication on Reals unique?
But then, if are them not unique, different additions are differently related with different multiplications?
And so on...

The motivation comes from the construction of real numbers.
From Peano's Axioms and the set-theoretic definition of Natural numbers to Dedekind and Cauchy's construction of Real numbers we haven't talked about uniqueness of operations in classes nor I could find relevant discussion about this topic on the internet and in ubiquitous Real Analysis reference books by authors as:

• Walter Rudin
• Robert G. Bartle
• Stephen Abbott
• William F. Trench

Not talking about the uniqueness of the operations, as we know them, in a first Real Analysis course seems rather common and not elementary matter.

Thus, introduced the subject and its context, would someone care to expand it eventually revealing the formal name of this field of study?

Solution 1:

The short answer is no: the operation defined by $a+_3 b=(a^3+b^3)^{1/3}$ also has all the properties 1 through 6 over the reals. This distributes over cannonical multiplication: for any $a,b,z\in \mathbb{R}$,

$z(a+_3b)=z(a^3+b^3)^{1/3}=(z^3)^{1/3}(a^3+b^3)^{1/3}=((za)^3+(zb)^3)^{1/3}=(za)+_3(zb).$

It might, however, be the case (and this is entirely speculation, not necessarily true) that only operations of the form $a+_f b = f^{-1}(f(a)+f(b))$ (where $f:\mathbb{R}\to \mathbb{R}$ is bijective and fixes the origin; or stated differently, $f$ is a permutation of the real numbers and $f(0)=0$) have all the properties 1 through 6. That would mean that addition is unique up to automorphism on the real numbers.

I would agree that this is not a trivial question.

Solution 2:

The proof goes in four steps:

• Addition on the natural numbers is uniquely determined by the successor operation (proved using induction)

• Addition on the integers is uniquely determined by addition on the natural numbers

• Addition on the rational numbers is uniquely determined by addition of integers

• Addition of real numbers is uniquely determined by addition of rationals, by continuity, using the fact that the reals are a complete ordered field containing the rationals as its prime subfield.

The same four steps can be used to show that multiplication on the reals is ultimately uniquely determined by the successor operation on the natural numbers.