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

But is there a way to draw the number xy directly

onthe real line (i.e. not as an area ontopof 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 O∈g 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(x⋅y). 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.

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