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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Basically, our eyes and the tools we use don’t have the precision to tell the difference.
Think about it. If I draw a line on a piece of paper then draw a dot on that line, the center of the dot can’t be EXACTLY any one value past a certain precision. And even if I have high precision, your eye won’t be able to tell the difference between “1.33333333333333” and “1.33333333333334” unless you use highly calibrated measuring tools and microscopes.
Likewise, on a computer screen, it’s hard to display things between pixels accurately, so there’s always a bit of fudge too.
TL;DR: Number lines aren’t 100% accurate. They’re “close enough”, where the definition of that word varies based on what tools are being used.
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