Tag: rational numbers

Posted in Algebra

Teaching irrational numbers – break it to me gently

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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.


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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.

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.

Posted in Misconceptions, Number Sense

Technically, Fractions are Not Numbers

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It is misleading to put fractions alongside the sets of numbers – counting, whole, integers, rational, irrational and real. The diagram below which are in many Mathematics I (Year 7) textbooks is inviting misconceptions.

WRONG WAY

Fraction is a form for writing numbers just like the decimals, percents, and other notations that use exponents and radicals, etc.

The fraction form of numbers is used to describe quantities that is 1) part of a whole, 2) part of a set, 3) ratio, and 4) as an indicated operation. Yes, it can also represent all the rational numbers but it doesn’t make fractions another kind of number or as another way of describing the rational numbers. Decimals can represent both the rational and the irrational numbers (approximately) but it is not a separate set of numbers or used as another way of describing the real numbers! Note that I’m using the word number not in everyday sense but in mathematical sense. In Year 7, where learners are slowly introduced to the rigor of mathematics and to the real number system, I suggest you start calling the numbers in its proper name.

I prefer the Venn diagram to show the relationships among the different kinds of numbers like the one shown below:

The Real Number SYSTEM
The Real Numbers

The diagram shows that the set of real numbers is composed of the rational and the irrational numbers. The integers are part of the set of rational numbers just like the counting numbers are members of the set of whole numbers and the whole numbers are members of the set of integers. The properties of each of these set of numbers can be investigated. We do not investigate if fraction is closed or is commutative under a certain operation for example, but we do it for the rational numbers.

You may want to know why we invert the divisor when dividing fractions. Click the link.