Teaching irrational numbers – break it to me gently

Numbers generally emerged from the practical need to express measurement. From counting numbers to whole numbers, to the set of integers, and to the rational numbers, we have always been able to use numbers to express measures. Up to the set of rational numbers, mathematics is practical, numbers are useful and easy to make sense of. But what about the irrational numbers? You can tell by the name how it shook the rational mind of the early Greeks.

Unlike rationals that emerged out of practical need, irrational numbers emerged out of theoretical need of mathematics for logical consistency. It could therefore be a little hard for students to make sense of and hard for teachers to teach. Surds, \pi, and e are not only difficult to work with, they are also difficult to understand conceptually.

It is not surprising that some textbooks, teaching guides, and lesson plans uses the following stunts to introduce irrational numbers:

After discussing how terminating decimal numbers and repeating decimal numbers are rational, you can then announce that the NON-repeating NON-terminating decimal numbers are exactly the IRRATIONAL NUMBERS.

What’s wrong with this? Nothing, except that it doesn’t make sense to students. It assumes that students understand the real number system and that the set of real numbers can be divided into two sets – rational and irrational. But, students have yet to learn these.

Some start with definitions:

Rational numbers are all numbers of the form  \frac{p}{q} where p and q are integers and q \neq 0. Irrational numbers are all the numbers that cannot be expressed in the form of \frac{p}{q} where p and q are integers.

How would we convince a student that there is indeed a number that cannot be expressed as a quotient of two integers or that there is a number that cannot be divided by another number not equal to zero? It’s not a very good idea but even if we tell them that \sqrt{2} is an irrational number, how do we show them that it fits the definition without resorting to indirect proof or proof of impossibility? What I am saying here is it is not pedagogically sound to start with definitions because definitions are already abstraction of the concept. I would say the same for all other mathematical concepts.

I think before introducing irrational numbers, students should be given tasks that raises the possibility of the existence of a number other than rational numbers. Another way is to let them realize that the set of rational numbers cannot represent the measures of all line segment, for example. Tasks that would help them get a sense of infinitude of numbers will also help. The idea is to prepare the garden well before planting.


6 Responses to Teaching irrational numbers – break it to me gently

  1. cani says:

    Excellent article.

    I believe this idea is bigger problem than just mathematical, it concerns the foundations of philosophy. Especially the arguement of rationalism Vs Empiricism. The whole story of Hippassus of the Pythagoras Sect, and the Greek model of rationalism and axiomaitisation of knowledge and Greek idea that science was not the way to knowledge. Yet Hippassus, destroyed this idea of rationalism, and hence the need for empiracal means of ascertaining knowledge.

    You can also say this problem is that we can not use the axiomatic appraoch for finding all of maths (a sub problem to one of Hillberts problems “the axiomatisation of physics”). Which was proved by Godel’s incompleteness theorem. So maybe we need a new way of doing maths, away from the rationalist Greek method.

    Can we truly understand the irrational, but more importantly can we truly understand the infinite.

  2. Erlina Ronda says:

    Of course, an irrational ruler does not exist, in physical form.
    Thanks for the links. I’m sure the reader will find it good read as I did.

  3. Great picture.

    What always seems strange is like you said irrational numbers aren’t used for measuring. But if we measure an area of a square to be a rational size 2 then the sides are the irrational length root 2. But we can also measure the length of the sides making them rational. I’ve never heard an explanation to this and was wondering if you knew where I’m making the mistake.

    Anyway interesting post,

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