Fractions as Numbers 1

Fractions as Numbers

Fractions emerged originally as a relation between a part & a whole. Fractions are like the social concept of a "brother" which related 2 persons. The term "brother" has no "real" meaning unless the person to whom he is the "brother of" is also specified. Hence we cannot make sense of a fraction without referring to the whole.

In this sense it they different from counting numbers like 1, 2, 3 etc which children have become familiar with. Counting numbers can be thought of as independent entities.

Hence children find it difficult to think of them in the same way they think about counting numbers. This "change of perspective" is what makes fractions a universally "difficult" topic in schools. Children in early primary school are expected to understand this abstract idea, when their ability to do so has not fully developed.

Representing Fractions with Numbers - 1

For mathematicians, the challenge was to think of fractions as numbers and represent them with (counting) numbers. A solution possibly emerged because of a "change in perspective" about the number 1.

We have seen that the whole can almost refer to anything, from a grain of sand to the entire universe! When we write a fraction as 1/3, the "1" has this very general meaning. The exact meaning of "1" depends on the context. It is usually seen as a countable number.

What if the "1" here is thought of as a measuring number, indicating distance, weight, volume etc? Then any fraction can be assigned a meaning in that measuring unit. For example, "half" could represent half a meter or half a litre or half kg (of course the ancient humans may not have used these units). Thinking of "1" as a linear distance on the number line helps us plot fractions also on the number line. This enables us to think of fractions as numbers which can be plotted on a number line, compared, added & subtracted.

The idea of half can be represented by a whole being divided into two equal parts, each of the parts being half of the whole. So "half" can be represented using the numbers 1 & 2. Mathematicians used as representation where 1 & 2 were separated by a horizontal line, with 1 appearing above the 2 appearing below the line. This is like a code which we would learn more in a subsequent paragraph.

Representing Fractions with Numbers - 2

Mathematicians then generalised this idea. A whole which was to be divided into "y" equal parts out of which "x" parts are "of interest to us" would be represented as x/y. Obviously x is less than y and x/y is less in value than 1.

This was also in line with the division perspective of a fraction. That x/y is a fraction that you get when you divide x wholes equally among y persons. x/y also means x ÷ y.

Division was already being represented as ‘—‘. Hence the same notation was adopted for fractions.

Fraction Vocabulary

In a fraction written as x/y , the number above the line (in this case x), is called the Numerator. The number below (in this case y) is called the Denominator.

The term Numerator has most likely come from the verb "enumerate" meaning "to count". The Numerator tells us the count of the numbers which form the part.

The term Denominator has most likely come from the word "denomination" which indicates the total in a set. We say that the amount of money is in "denominations of hundreds."

(Provide a visual comparing 2/3 and 2÷3.

First picture is that of dividing a pizza into 3 parts & taking 2 parts out of it.

Second is that of keeping 2 pizzas one on top of another and dividing into 3 parts.

We can see that both the ways give you the same share of “one pizza”