Real numbers are those which can be represented on a number line. As per definition , we should be able to plot numbers like √2, 0.333333…. , π etc. on the number line, but if we don’t know their exact precise value then how can we plot it?
I have seen couple of answers on Google where people have used a right angled isosceles triangle with base and altitude of 1 , and with the help of a compass and ruler they plotted it , but still it isn’t the precise value, right?
Or for 0.333…. , they divided the length of 1 unit in 3 equal parts and marked the length of first part as 1/3=0.333…. ; 0.3333….. is not a precise value then how can it be accurately plotted on number line ?
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You cannot plot ANY number with perfect precision on a number line. Each number is represented by only a single point on the line. A point is zero dimensional – it has no length, no width and no depth. We can mathematically determine where a point belongs on a number line, but we cannot actually plot such a point with perfect precision. This is true whether the number is rational or irrational.
Plotting a number on a number line (or any set of axes) is always an approximation. No matter how precise the tools we use to mark the spot, and no matter how fine the point of the actual marking device, the mark will always be infinitely larger than the actual point at issue. That is, there will always be an infinite number of points within the mark that are not the point that’s intended to be marked.
That said, for just about any human endeavors, we can plot points with “good enough” accuracy.
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