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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We do know the precise values of √2 and 1/3. They are, well, √2 and 1/3. We can also calculate their decimal representations as accurately as we need to as well. The fact that their decimal representations are infinite is more a consequence of how we write numbers down than anything else. For instance, if we used a based 3 number system instead of base 10, 1/3 would be written as exactly 0.1.
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