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Say I have a two lines:

"_______"

"_________________"

Addition is clear:

"________________________"

Subtraction is clear:

"__________"

But multiplication and division seem to require numbers. What's going on?

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    $\begingroup$ Subtraction's not clear. What happens when you try to subtract a long line from a short line? $\endgroup$ – Ashwin Trisal Dec 19 '17 at 23:18
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    $\begingroup$ Before I can show you the "product line segment", you must present to me also the length of the $1$-segment. $\endgroup$ – Jeppe Stig Nielsen Dec 19 '17 at 23:23
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    $\begingroup$ You can define multiplication using a geometric construction as long as a unit length is defined. See math.stackexchange.com/questions/139340/… $\endgroup$ – Jair Taylor Dec 19 '17 at 23:36
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    $\begingroup$ I would just define the product of any two lines to be $0$, and the same for the quotient. Do you need something with specific properties? $\endgroup$ – Morgan Rodgers Dec 20 '17 at 0:22
  • $\begingroup$ @AshwinTrisal Nothing unclear here. When using lines you are almost by definition in the realm of natural numbers. And the substraction of a greater number from a smaller one just doesn’t have any solution in that context. $\endgroup$ – KPM Dec 20 '17 at 9:21
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Here is an Euclidean construction for the product. It does need a unit segment.

enter image description here

(Picture from this answer)

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    $\begingroup$ Very elegant. Now do tetrated Knuth up arrows for x and y. $\endgroup$ – Dale C Dec 20 '17 at 3:59
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    $\begingroup$ @TiwaAina: Euclidean geometry is based on axioms. Each axiom is like "given this and this, you can construct that, or something is true". It does not intrinsically permit you to construct something that depends on arbitrary repetition. For that you need to work outside of Euclidean geometry. For example to construct $x^n$ where $n$ is a natural, you could say "repeat lhf's construction $n$ times". That definition is just not within the axiomatic framework of Euclidean geometry. You can see from outside that you can construct such lengths, but you cannot refer to or reason about them inside. $\endgroup$ – user21820 Dec 20 '17 at 5:17
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    $\begingroup$ @TiwaAina: And a unit segment is required because the geometric construction is not scale invariant. $\endgroup$ – user21820 Dec 20 '17 at 5:18
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    $\begingroup$ @TiwaAina: More prosaically: Because in the degenerate case where $x = 1$, $xy = y$ and you need the corresponding segments to be congruent. But they must not be congruent in any other case, so you need to pick a unit distance to determine exactly when congruence occurs. $\endgroup$ – Kevin Dec 20 '17 at 5:29
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    $\begingroup$ @TiwaAina: In case it is not clear, in Euclidean geometry, given $1,x$ as lengths, you can construct by lhf's construction the lengths $x^2$ and $x^3$ and $x^4$ and so on, but each of these would take longer and longer to construct, proportional to the (natural) exponent. You cannot have a single uniform geometric construction that from lengths $x$ and $n$ (even if $n$ is guaranteed to be a multiple of $1$) will produce the length $x^n$. $\endgroup$ – user21820 Dec 20 '17 at 7:51
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Each line is constructed from individual segments of unit length (represented by underscores). For convenience, lets say that we want to multiply two shorter lines:

_____ ___

First, we insert spaces into the first line so that we can see the individual underscores.

_ _ _ _ _ ___

Then we replace each underscore in the first line by the second line.

___ ___ ___ ___ ___

Finally, we remove the spaces to obtain the result.

_______________

This process corresponds to rewriting multiplication in terms of addition.

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    $\begingroup$ Assuming _ = 1 and all numbers are natural... $\endgroup$ – Bart Michels Dec 20 '17 at 6:52
  • $\begingroup$ @barto That is how I interpreted the question since the OP used underscores to represent lines. The question leaves some room for interpretation. $\endgroup$ – Qudit Dec 21 '17 at 1:12
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The product of two positive real numbers $a$ and $b$ can be thought of as the area of the rectangle with side lengths $a$ and $b$.

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Take a look at Euclid's Elements, book VII, for one historical development of this. Qudit's answer above actually gives a nice graphical representation for the style of reasoning used by Euclid. For example Proposition 16, whose statement is "If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another". This is showing commutativity. What Euclid is calling numbers here are really dimensionless line segments, each measured by what he calls the "unit", which measures any "commensurable" magnitude (incommensurable = irrational). I won't try to reproduce the development, but to give a flavor of the dimensionless reasoning used, one phrase from the proof is, "for since A by multiplying B has made C, therefore B measures C according to the units in A". Other propositions develop other of the basic properties of multiplication. There are certainly flaws in Euclid from a modern view, but there is a lot of material that is well worth reading nevertheless and in this case some of it sounds like exactly what you are looking for.

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A reasonable arithmetic could be constructed where the product of $-$ and $|$ is $\square$.

In fact, my understanding of actual history is that lengths and areas were considered different kinds of quantities for over a millennium. Using the same kind of number to quantify both is a relatively modern idea that has only been around for a few hundred years.

More precisely, I think historically one never numerically talked about the "length" of a curve; instead the use of numbers was to refer to the proportion that one length makes with another. Similarly for area.

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  • $\begingroup$ Even modern physicists will maintain that length and area have different "dimensions" and different units of measurement. And it does not make much sense to ask whether 3 meters is more than 2.8 square meters, does it? However, for whole numbers, and possibly something similar to fractions, I believe multiplication of "pure" numbers to obtain a "pure number" product has been usual since before Euclid. And each of the pure numbers (multiplier, multiplicand, and product) could be translated into geometric quantities of the same sort afterwards. But for what use? $\endgroup$ – Jeppe Stig Nielsen Dec 20 '17 at 17:55
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Let's assume $x,y>0.$ If the unit length $1$ is available for plotting, make the points $A=(x,0),B=(0,y),I=(0,1).$ Then draw segment $IA,$ and from $B$ make a parallel to that, meeting the $x$ axis at $(p,0).$ Then $p=xy.$ As pointed out in a comment [J. Nielsen] this construction needs a unit length available.

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If we are assuming Euclidean geometry, Lebesgue measure, etc (that is, usual definitions about measures and its relations), the multiplication yields a rectangle with these two sides, and the division, the ratio between the segments (that can hardly be drawn, for it has no dimension).

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Multiplication can be described as a transformation, involving the stretching of a line (with numbers, you can call this line as the Number Line)
Over imaginary numbers, Multiplication is the transform involving stretching of a plane.

A great post and video explaining this, and more: http://www.3blue1brown.com/videos/2017/5/26/understanding-e-to-the-pi-i

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