Types of Fractions

Mixed Fractions or Mixed Numbers

Fractions could occur either on their own or along with whole numbers. When a fraction and a whole number occur together it is called a mixed fraction or mixed number.

We are very familiar with quantities like 1 ½ or 3 ¼ in terms of weights or money.

Convention in Writing Mixed Fractions

A mixed fraction is written simply as 1 ¾ though it actually means 1 + ¾. By convention, the "+" sign is not written because the whole number part looks quite different from the fraction part. This also saves space & time.

Such decisions are called Conventions rather than concepts. It is just a convenient way which is adopted by one and all. There is no other logic behind it.

Improper Fractions

While addition & subtractions with mixed fractions can be handled fairly easily, multiplying and dividing with them can pose several difficulties.

Imagine multiplying 3 ½ by 4 ⅓. This has to be as ( 3 + ½)(4 + ⅓) using the distributive law of multiplication.

Dividing 3 ½ by 4 ⅓ can pose even more difficulties.

Mathematicians found a simple solution to the above problems by converting mixed fractions into what are called "improper" fractions.

3 ½ of a pizza can also be thought of as 7 slices ordered of a "particular" pizza which is divided into 2 pieces.

Hence 3 ½ can be written as 7/2.

Since it is in the "rational number" format it can be operated upon like a "normal" fraction.

The general idea of a fraction was that it is less than 1 or a whole.

Since the "idea" of a fraction was now being applied to entities which were more than a whole, such fractions were known as "improper" fractions as opposed to a "proper" fraction which was less than a whole.

We normally do not use "improper" fractions in our daily transactions. Instead we prefer to use their mixed fraction format.

Hence "improper" fractions are mathematical objects created basically to make operations with mixed fractions easier.

Teachers should clarify that the term improper is just a mathematical term without any value connotations. There is nothing "improper" about an improper fraction!

Unit Fractions

In the early days of visualizing fractions & operations with them, mathematicians found it easy to imagine fractions as representing "a single part out of a whole which has been divided into several parts" In practice this works out to be fractions where the numerator is always 1, regardless of the value of the denominators. Examples are ½, ⅓, &frac17; etc.

They would try and express any other fraction, say "3 parts out of a whole divided into 4 equal parts" as a sum of fractions where the numerator was always 1. In the above case, it was represented as ¼ + ¼ + ¼ . Such fractions whose numerator was always 1, were called Unit Fractions

Like & Unlike Fractions

It was also realized that fractions could compared, added or subtracted only if they were parts of the same whole, divided into the same number of parts. I.e their denominators had to be same. Such fractions are called Like Fractions. 3/7, 5/7 & 1/7 are Like Fractions.

All other fractions which do not have the same denominator are called Unlike Fractions.

Standard & Non-Standard Representations

Fractions are rational numbers. Hence their numerator & denominator should always be whole numbers, except that the denominator should not be 0.

However for operational purposes, fractions can, in the intermediate stages, may be represented in non-standard formats.

Imagine that we are comparing 3/7 & 5/8 using number sense techniques.

We can think of 3/7 as less than 3.5/7 and 5/8 being more than 4/8. Both 3.5/7 & 4/8 can be thought of as equal to half! Hence 3/7 is less than half while 5/8 is more than half. Hence 5/8 > 3/7.

3.5/7 can be thought of as a non-standard representation of half, used for mentally comparing two fractions.

Irrational Numbers

There are some fractions which are called Irrational Numbers. In brief, these are "parts of a whole" which cannot be represented with a numerator and denominator which are both whole numbers.

These would be dealt with in chapter 17.10.