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, , 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 numbersare all numbers of the form where p and q are integers and q 0.Irrational numbersare all the numbers that cannot be expressed in the form of 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 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.

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 segments. 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. Read my post on why I think it is bad practice to teach a mathematical concept via its definition.

You are certainly right that the approaches you mention are poor IF the real numbers have not already been defined – either in terms of physical measurements (which is most suitable as motivation) or more abstractly in terms of completing the rationals by taking limits.

But they would be fine if that had actually been done.

An approach that I like is to start with a discussion of length measurement. First counting the basic units, then trying to match the left-over bit with successively smaller fractions (eg to get a decimal expansion) and then asking whether this will always terminate or repeat.

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.

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.

to measure an “irrational length”, you’ll need an “irrational ruler”. i will explain this further in my next post about this topic. thank you, too.

Not true. There is no such thing as an irrational ruler. In fact there is no such thing as an irrational number, only an incommensurable magnitude which is represented and used in the form of a rational approximation.

The following link explains how we got rational numbers:

Are real numbers well defined?

What is a limit?

There are many other interesting articles that include my new calculus.