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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Units are arbitrary. Consider, you wouldn’t balk at me having a line that is 6 units long and then dividing it into three segments each 2 units long apiece.
But all I have to do is then say that the whole line is 1 unit. That automatically makes the segments 1/3 long each. Exactly. You can’t say they were all exactly 2 units long but then refuse to accept them being 1/3 units long now. They didn’t change their actual length simply by me using different units to represent that length.
So either no number can be exactly represented or they all can.
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