Question

Visualise $4.\overline{26}$ on the number line, upto $4$ decimal places, that is upto $4.2626$.

Answer 1

We locate $4.\overline{26}$ on the number line, by the process of

successive magnification. This has been illustrated in Figure.

Step $1$: First we note that $4.\overline{26}$ lies between $4$ and $5$

Step $2$: Divide the portion between $4$ and $5$ into $10$ equal

parts and use a magnifying glass to visualise that $4.\overline{26}$ lies

between $4.2$ and $4.3$.

Step $3$: Divide the portion between $4.2$ and $4.3$ into $10$

equal parts and use a magnifying glass to visualise that $4.\overline{26}$ lies between $4.262$ and $4.263$.

Step $5$: Divide the portion between $4.262$ and $4.263$ into

$10$ equal parts and use a magnifying glass to visualise that $4.\overline{26}$ lies between $4.2625$ and $4.2627$.

We note that $4.\overline{26}$ is visualised closer to $4.263$ than to $4.262$.

The same procedure can be used to visualise a real number with a non-terminating and non-recurring decimal expansion on the number line to a required accuracy.

From the above discussions and visualisations we conclude again that every

real number is represented by a unique point on the number line. Further every point on the number line represents one and only one real number.