Mathematics for Teaching Technically, Fractions are Not Numbers

## Technically, Fractions are Not Numbers

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.

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 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.

## 3 thoughts on “Technically, Fractions are Not Numbers”

1. I agree that the word “fraction” more properly refers to an expression of the form “a/b” rather than the number a/b, but I think it would be less confusing if you said “fraction is not a number TYPE”, rather than implying that a “fraction is not a number”. Any particular fraction does *represent* a number, and to argue whether the fraction *is* the number is like arguing whether Erlina is person or just a name. Of course it would be great if kids did have this level of sophistication and precision in their use of language, but the fact that many teachers and textbook authors do not suggests that such a hope is overly optimistic.