Properties & Relations of Fractions

We understand anything new idea by comparing and contrasting it with something we have already understood. This is a part of the process of constructing knowledge.

While talking about fractions, we never deal with their properties and relations the way we do with whole numbers. If we compare and contrast the properties & relations of fractions with those of whole numbers, their understanding may become easier.

Properties that are shared - Fractions can be compared as well as Ordered

The only property that fractions share with whole numbers is that they can be compared and arranged in increasing or decreasing order. Every fraction is "less than" or "more than" some other fraction. Hence, they can be arranged in either an ascending or descending order.

From the idea of dividing a roti into several parts (denominator) and taking one part, will lead to the idea that when the numerator is same, a bigger denominator indicates a smaller fraction. The numerator can be any number as long as both the fractions have the same numerator.

Similarly, we can lead to the idea that if the denominators are same, the fraction with a bigger numerator would be the bigger fraction.

It is a good idea to start by comparing any fraction with familiar fractions like ½ or ¼ where the relation between the numerator & denominator are easy to see – in half, the numerator is always half of the Denominator and in Quarter the Numerator is a quarter of the denominator. Hence a fraction like 3/8 can be compared to half 4/8 and hence 3/8 is less than half.

When fractions with different numerators or denominators have to be compared, one strategy would be to find equivalent fractions (covered in the next paragraph) where either the numerator or denominator of both the fractions become same. In many cases, it would be enough if they are close enough.

Let us see the 2 strategies

1. 5/6 & ¾ = 10/12 & 9/12 (making denominators same).10/12 & hence 5/6 is bigger than ¾

2. ¾ & 4/7 = 12/16 & 12/21 (making numerators same). 12/16 is bigger since its denominator is smaller than 12/16 & hence ¾ is bigger than 4/7.

We can use this method where the computation to make the numerators same is simpler than that for denominators

0 as Denominator

What about a fraction whose denominator is 0?

Let us assume that 4/0 has a value "x". Then we have 4 =x.0 =0. Hence we get an absurd result. This will happen if we have any number instead of 4. Hence, 4/0 has no meaning in math. Hence, we can say that division by 0 is not "defined" in math.

Properties & Relations that are not shared

Ideas of Odd, Even, Composite, Prime, Factor of, Multiple of do not apply to fractions.