Representing the multiplication of two numbers on the real line

There is a simple way to graphically represent positive numbers x and y multiplied using only a ruler and a compass: Just draw the rectangle with height y in top of it side x (or vice versa), like this

enter image description here

But is there a way to draw the number xy directly on the real line (i.e. not as an area on top of the real line) by using only some standard drawing means like using a compass, a ruler, a straightedge etc. (i.e. not multiplying x and y out and then putting the number xy at right spot), like indicated above ?

(I think this question actually asks if the multiplication is representable as a composition of (the mathematical translation of) operations of drawing circles using a straightedge etc.)

Answer

You can do so, but if you want to represent the result of the multiplication again as a length you have to choose a unit. The ancient Greeks didn’t come up with this idea; whence their products were always areas.

Responding to temo’s comment: When the point Og representing the number 0 has been chosen the geometric construction for the sum of two numbers is scale invariant, as a consequence of (λx)+(λy)=λ(x+y); but a similar identity for multiplication does not hold: (λx)(λy)λ(xy). You can test the effect in the figure by keeping x and y but choosing a new unit 1. The point xy will now be at a different location.

enter image description here

Attribution
Source : Link , Question Author : temo , Answer Author : Christian Blatter

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