How to represent reciprocal of an irrational number on number line e.g $1/\sqrt{7}$?

Question

If we try to rationalize this reciprocal of irrational number like for $1/\sqrt{7}$ then we get $\sqrt{7}/7$ which poses a new problem of how to divide an irrational number into $7$ parts. Dividing an irrational number into $2$ or $4$ would be easy as we can geometrically bisect them. But what about bisecting in $3$ parts or $7$ or $6$ parts?

Answer 1

Construct a semicircle with radius $\frac{4}{7}$. Let $AC = \frac{1}{7}$ and $AB = 1$. Draw a perpendicular to $BC$ and call it $AD$. Then by $AA$ similarity (why?), $\Delta CAD \sim \Delta DAB$.

As a result, we have: $$\frac{AD}{AC} = \frac{AB}{AD}$$ $$AD^2 = AB \cdot AC = 1 \cdot \frac{1}{7}$$ $$AD = \sqrt{\frac{1}{7}}$$

This method generalises well to construct the square roots of any rational number.

Answer 2

Make a triangle $ABC$ with one side, say, $AB$ of length $1$ and another side, say, $AC$ of length $\sqrt7$. Find $D$ on $AC$ such that $CD$ has length $1$. Construct a line through $D$, parallel to $AB$. This line intersects $BC$ at a point we'll call $E$. Let $x$ be the length of $DE$. Then by similar triangles, $x/1=1/\sqrt7$, so you have constructed $1/\sqrt7$.

Answer 3

Division of a segment in equal parts is easily done by Thales.

By one endpoint, draw a straight line and tick $n$ points at equal distances, starting from that endpoint.

Join the last point to the other endpoint. Then by drawing parallels, you can obtain any $\frac kn$ fractions of the segment.