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I've come to realize that the more I study some math subjects the more I question some results or ideas that seemed trivial or obvious to me. My question is about the real numbers and their geometric interpretation as a line.

I get that real numbers being complete and ordered are naturally visualized as a number line. Now, consider two positive numbers $a$ and $b$ and find them in the number line. We define the binary operation $a•b$ like this: "take a compass and open it from O to $b$, now draw a circle with such compass with $a$ as its center. The point on the right where the circle intersects the line is $a•b$".

This, of course, corresponds to $a+b$ but the reason is not that obvious to me. Intuitively we use addition when we "add" something so if I'm "adding" a line segment with length $a$ to another line segment with length $b$ the length of the resulting line must be $a+b$. But with a formal definition of addition (via Dedekind cuts for example) that property of addition doesn't seem that clear to me. I guess my question is: why must addition correspond to $a•b$? Why, if I have a line segment of length 3/2 and other one of length 5/3 can I be sure that if I arrange them so that one starts exactly when the other one ends, the length of the resulting line will be 19/6?

Please forgive me if this question is too obvious or trivial. Thank you!

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    $\begingroup$ Does the confusion disappear if you call those objects "intervals" instead of "line segments"? $\endgroup$ – AnoE Jun 26 '17 at 11:31
  • $\begingroup$ Consider an integer to be a one-dimensional vector. We know the effects of adding two vectors in higher order spaces, and this generalizes back down to 1-space. $\endgroup$ – Tyzoid Jun 26 '17 at 17:15
  • $\begingroup$ @Tyzoid But to prove the geometric interpretation of vector addition (like the triangular law) one uses the fact that a side which has two parts (one with length A and one with length B) has a total length of A+B. Wouldn't it be circular reasoning? $\endgroup$ – MathUser123 Jun 26 '17 at 19:33
  • $\begingroup$ @MathUser123 It was more illustrative than a rigorous proof. $\endgroup$ – Tyzoid Jun 26 '17 at 20:51
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In your example, if you multiply by $6$, you make it about integers. That is, $$ \frac32+\frac53=\frac16 \, (9+10)=\frac16\, (9\,\bullet\,10)=\frac32\,\bullet\,\frac53. $$ All you need to believe the above equalities is that the two operations agree on the integers (you can see the integer as "counting units" to see it as a length), and that dividing by an integer preserves scale.

Once you have the argument for the rationals, it extends to the reals by continuity.

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Let's completely forget all notions of number entirely.

It's not hard to show this operation (which I will call $\oplus$ rather than $\bullet$) has nice algebraic properties: it's commutative, associative, and it respects the ordering on line segments.

Alongside $\oplus$, we could pick some segment to be a unit, and use another geometric construction to define an operation $\otimes$.

These two operations on line segments have very good algebraic properties; without using any notion of number, we've still managed to construct a system in which we can do arithmetic, algebra, and analysis — in fact, once we add in a notion of direction, these operations satisfy the complete ordered field axioms.

What is the length of a line segment? The line segment itself. Addition is literally defined as "extending a length".


I believe this overall description is a relatively accurate description of actual history — e.g. that Greek geometers did algebra with line segments, considering the segments themselves as a notion of quantity; e.g. see Euclid's elements, book 2. The real number system came into being precisely as numeric representations of these geometric objects

In fact, AFAIK, for a long time mathematicians considered numbers quantifying length and numbers quantifying area as completely different kinds of numbers, rather than the point of view today where both kinds of quantities are described by the same number system, but possibly with units attached.

The perspective that treats real numbers as being more fundamental than geometric notions is, I think, mainly an artifact of how math is taught in modern times.

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    $\begingroup$ So you show that $\oplus$ and $\otimes$ satisfy the same axioms as $+$ and $\times$ (the latter two defined via Dedekind cuts or whatever), show that the axioms uniquely determine the operations, and conclude that $\oplus$ and $\otimes$ must equal $+$ and $\times$? $\endgroup$ – Akiva Weinberger Jun 26 '17 at 8:15
  • $\begingroup$ @AkivaWeinberger That's correct. These results are explained at an elementary level in the book by John Stillwell entitled "Four pillars of geometry", but they go back a long way. Of particular interest is way that the Pappus and Desargues theorems (from projective geometry) are used to prove the field axioms. $\endgroup$ – Lee Mosher Jun 26 '17 at 18:46
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There is an imagined (or abstract ) collection called the set of real numbers with operations and ordering so that it becomes what is technically called a real number system which is an ordered field with the least upper bound (or better called the order completeness property) .This is separate from reality ,it is a matter for set theory and its logic .Now the reason for the wide interest is that it has a natural interpretation in reality as the points of a straight line drawn say on a big piece of paper ,well of course you only show a piece of the line but ignore that. Your question is really how do you assign a real number to the points on the drawn line . Well first you just mark 0 somewhere . Then you put 1 somewhere ,say to the right of 0 Now do your compass construction with a=0 and b=1 ,The real number 2(by definition)= 1+1 is assigned to the point obtained with the compass ,Now repeat the construction to get the positions or 3=2+1 ,4=3+1 , 5 =4+1 etc .Science would just say use equal spacing . Now there is a lot of work with induction type arguments to get just the integers placed right and things like 5+3 = (5 + 2) + 1 =(5+1)+1)+ 1 to understand where to put a+b for just integers .Now you have to figure out where the fractions go and then one fills in the the rest of the reals by assuming the drawn line is what is called a continuum (between any 2 distinct points no matter how close there are points in between . This is not physically true if you look under a powerful microscope at your hand drawn line but is an idealization that is useful . Any way when you are done you should see empirically ,as clear as any part of physical science that is you do your compass construction on a and b that the point you get does correspond to the abstract real number a+b . Well what you are asking about is really not trivial . One last point .One can logically construct the real number system from the integers in set theory and each step of the construction can be interpreted on your drawn line . All this interpreting is empirical ,real science ,not just abstract mathematics (which is more precise ) ,it is accepted experimentally because it works. Hope this helps .

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    $\begingroup$ This is a good answer, however it could use proper spacing, mathjax, and especially paragraph splits to make it more readable and an even better answer. $\endgroup$ – Dando18 Jun 26 '17 at 15:47
  • $\begingroup$ Thank you very much! Do you know where I can find a rigorous construction and explanation of the number line and maybe proofs of it's properties? (for example, that every rational number is represented by just one point on the line or that distance between two points corresponds to subtraction etc.) $\endgroup$ – MathUser123 Jun 28 '17 at 4:10
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    $\begingroup$ What you are asking for is the construction of coordinate systems for the plane form in Euclidean Geometry from geometric axioms (ruler and compass constructions , A recent book is by John m Lee ,Axiomatic Geometry ,published by AMS should have it but I haven't seen it ;but note it is not trivial and might take 100 pages to get to it .. $\endgroup$ – StuartMN Jun 29 '17 at 8:42
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Based on the handout, section 7: $(A_1,B_1)+(A_2,B_2)=(A_1+A_2,B_1+B_2)=(A_3,B_3)$ is a Dedekind cut defining addition. Quoting from handout: "$A_3$ is the set of all rationals of the form $a_1 + a_2$ where $a_1 \in A_1$ and $a_2 \in A_2$". Similarly for $B_3$.

If the definition of $∙$ in terms of rationals being equivalent to $+$ is sufficiently well established for you by the other answers, then the quantities $A_1+A_2,B_1+B_2$ can be reformulated as $A_1∙A_2,B_1∙B_2$. Thereby, yielding a clear relationship between the definition of addition in terms of Dedekind cuts and the $∙$ operator. And extending $∙$ to the reals.

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  • $\begingroup$ Thank you! Now, suppose I don't have a definition for the reals. Could this property be obtained just from the axioms? And do you know where I can find a rigorous construction and explanation of the number line (geometric) and maybe proofs of its properties? $\endgroup$ – MathUser123 Jun 29 '17 at 2:47
  • $\begingroup$ @MathUser123 You're welcome. I'm not sure what you mean by "this property" - which property? Also which axioms? Right now I don't know where to find construction of real line. I usually google for things like this. Have you had any luck searching? $\endgroup$ – Χpẘ Jun 29 '17 at 16:41
  • $\begingroup$ I mean the property of addition being the same as extending a length (for the argument for the irrationals you used Dedekin's definition for addition). Could this be proven just with the axioms of the real numbers (that they are a complete ordered field)? I've also googled for a construction of the number line but all I can find is the construction for the reals. It's seems as most of the books just take the number line for granted and say "distance of a length segment from $a$ to $b$ is $ |b-a|$ without giving any reason for why this is true. $\endgroup$ – MathUser123 Jun 29 '17 at 18:54

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