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.
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.
Real numbers on a number line can (in theory) be plotted with as much precision as your number line.
If your number line has interval precision, then pi can be plotted somewhere between 3 and 4.
If your number line has n’th decimal place precision, then pi can be calculated to the n’th decimal and plotted.
An infinitely precise number line cannot exist in real life. Just how we can’t calculate pi to infinite digits.
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.
You cannot plot ANY number with perfect precision on a number line. Each number is represented by only a single point on the line. A point is zero dimensional – it has no length, no width and no depth. We can mathematically determine where a point belongs on a number line, but we cannot actually plot such a point with perfect precision. This is true whether the number is rational or irrational.
Plotting a number on a number line (or any set of axes) is always an approximation. No matter how precise the tools we use to mark the spot, and no matter how fine the point of the actual marking device, the mark will always be infinitely larger than the actual point at issue. That is, there will always be an infinite number of points within the mark that are not the point that’s intended to be marked.
That said, for just about any human endeavors, we can plot points with “good enough” accuracy.
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.
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.
This applies to all numbers. How do you plot 1, for example> No matter where you put your pencil you will be a little bit off the mark, even if only a nanometer.
> Real numbers are those which can be represented on a number line. As per definition
That isn’t really the *definition* of real numbers – it’s just a way of thinking about them. Real numbers are usually defined in terms of sets or sequences of rational numbers (fractions) – you can google “Dedekind cut” or “Cauchy sequence” if you want the full details, but they’re maybe a bit beyond ELI5.
> but if we don’t know their exact precise value then how can we plot it?
On a real-life number line, we don’t really know the *exact* position of any number, since we don’t have any perfect measuring devices. We can use real numbers to model reality in approximate terms, but they don’t correspond exactly to reality. For example, we know that there are infinitely many real numbers between 0 and 1, but are there are infinitely many points between two distinct positions in space? Nobody really knows.
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