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Representing the multiplication of two numbers on the real line

Consider the real line in the plane. Suppose you are given the location of the point associated to $0$ and two oter points $a$ and $b$ on the line, it's strightforward to provide a geometric construction(*) that allows you to identify the point associated to $a+b$. My question is: suppose we are given the locations of $0$ and $1$, given two points $a$ and $b$ can we provide a geometric construction which allows to identify the point $a \cdot b$?

(*) Compass and straightedge construction

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Marco Disce
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1 Answers1

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This can be done using similar triangles. Suppose we have a triangle with sides $1$, $a$, and $c$ (some length whose exact size does not matter). If we now get a similar triangle with sides $b$ times bigger, we now have a triangle whose sides are of length $b$, $a \cdot b$, and $b \cdot c$.

  1. Construct a triangle $ABC$ where $|AB|=1$ and $|AC|=a$. It does not matter what the length of $BC$ is.

  2. Extend the side $AB$ to get a line and find the point $D$ on this line (on the same side of $A$ as $B$) such that $|AD|=b$.

  3. Draw a line $L$ through $D$ parallel to $BC$.

  4. Extend the side $AC$ until it intersects the line $L$ at the point $E$. The line segment $AE$ has length $a \cdot b$. (This follows from the properties of similar triangles).

For further details see the lecture notes at this website.

Tom Cooney
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