The Rational Line has Irrational Holes

The Rational Line has Irrational Holes

Rational Line:

A number line in which we plot all the rational numbers is known as Rational Line.

We know that between any two rational numbers on the Rational Line, no matter how close, we can find infinitely many rational numbers.

The picture is very different for the Integer Line – we have an uncluttered number line with numbers at every unit interval. Thus going from the Integer Line to the Rational Line seems to fill up the line, in the sense that now every interval of the line (no matter how small) is populated with numbers.

Irrational Holes on Rational Line:

If we were traveling on a rational line, would we encounter holes, positions for which no corresponding rational numbers exist?

The answer is yes.

There do exist numbers that are not rational, which cannot be represented using integers and the four simple arithmetic operations. In other words, there are numbers that cannot be represented in the form of \(\frac{p}{q}\) where  \(p\) and \(q\) are integers. These numbers are called Irrational Numbers.

So, we can conclude that there will be gaps in the Rational Line and those gaps will correspond to the set of Irrational Numbers.

For example, there will be a hole at a length of \(\sqrt 2 \)  units from 0, since no rational number corresponds to \(\sqrt 2 \). In this sense, the Rational Line is discontinuous, even though it seems to be so densely packed with numbers. How many gaps do you think are there on the Rational Line? Since there are so many numbers on the Rational Line – since it is so densely populated with numbers – it might seem that there are very few gaps on the Rational Line –corresponding to the few “rogue” numbers which happen to be irrational. However, the actual answer to this question will totally surprise everyone who learns about it for the first time.

The truth is that there are so many irrational numbers that even if you put together the set of rational numbers n times, you would still not be able to match up to the size of the set of Irrational Numbers, no matter how large n is. In essence, what this means is that the set of Irrational Numbers is infinitely larger than the set of Rational Numbers (at the same time, note that both the sets have infinite elements). This is a very non-intuitive result but it is well-established. For now, you will have to accept it at face value, because the proof of this result requires advanced mathematics.

✍Note: The infinite set of irrational numbers is infinitely larger than the infinite set of rational numbers. surprise

Coming back to our geometrical number line picture, this result means that the Rational Line is riddled with holes everywhere, holes which correspond to irrational numbers. And even though the number line seems to be full of rational numbers in every interval of any length, the number of holes is still more (in fact, infinitely more).

✍Note: The rational line is discontinuous as there are infinitely many holes/gaps in between any two rational numbers that represent irrational numbers.

yesChallenge: If the set of rationals and irrationals put together, it is called the set of Real Numbers, and the corresponding number line is called the Real Line.

Is Real Line continuous?

⚡Tip: Go through Real numbers and Real Line

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