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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The numbers we can “easily” plot on a number line are the constructible numbers, those that the ancient greeks worked with that can be made with a ruler and a compass. The constructible numbers are a tiny tiny subset of all the real numbers. Being able to be esily found on a numberline is not part of the definition of a Real number. All the Real numbers are there on the numberline but being able to easily find them has nothing to do with being Real.
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