Visualize \(3\overline{.765}\) on the number line, using successive magnification.

We know that, 3.765... lies between 3 and 4.

So, let us divide the part of the number line between and 3 and 4 into 10 equal parts, mark each point of division and look at the portion between 3.7 and 3.8 through a magnifying glass.

Now 3.765..... lies between 3.7 and 3.8 Figure (i). Now, we imagine to divide this again into ten equal parts. The first mark will represent 3.71, the next 3.72 and soon.

To see this clearly, we magnify this as shown in Figure (ii). Again 3.765.... lie between 3.76 and 3.77 Figure (ii).

So, let us focus on this portion of the number line Figure (iii) and imagine dividing it again into ten equal parts Figure (iii).

Here, we can visualize that 3.761 is the first mark and 3.765...... is the 5th mark in these subdivisions.




We call this process of visualization of representation of number on the number line through a magnifying glass as the process of successive magnification.

So, we get seen that it is possible by sufficient successive magnification of visualize the position (or representation) of a real number with a terminating decimal expansion on the number line.