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 ?
In: 0
0.33333… *is* a precise value. We just don’t write all the 3’s because we don’t have infinite time but, mathmatically, that’s *exactly* 1/3.
And we can easily plot that on a numberline. Take a line of length 1, bend it around until it forms an equilateral triangle (angles exactly 60 degrees), mark the corners, unfold it. Those marks are at *exactly* 1/3.
If you mean “can we do this in real life” the answer is “no” but that has nothing to do with the math, that has to do with our physical universe being discrete(ish) at very small scales. Number lines are purely theoretical constructs in the first place, the fact that we run out of good measuring tools with a physical number line doesn’t change the math.
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