Fractions as Numbers 2

Fraction as a Code

is a code to represent the idea “If you divide a Whole into “y” parts and take “x” parts out of it, then the “x” parts can be said to be a fraction of Whole “y” parts.

We have already come across a similar code in the place value system. Twelve (12) in the place value system is a code which tells us that it is equal to “1” Ten & “2” Ones. Here 2 numbers 1 & 2 indicate a value (dozen or twelve) which is totally different from values of 1 or 2.

Similarly, a fraction is also written with a minimum of 2 numbers but has a value which is totally different from 1 or 2. It is a different kind of a “code”.

Laws of Arithmetic

Mathematicians also realised that written in the above form, fractions obeyed all the “Laws of Arithmetic”. This is an advanced concept and would be dealt with in chapter 21.2.

For our current purposes, it means that fractions can be compared, added, subtracted, multiplied & divided. Like in the case of counting numbers, the order of adding or multiplying did not matter. (1/2 + 1/3 = 1/3 + ½ & ½ X 1/3 = 1/3 X ½)

Hence fractions could be treated as numbers on par with, but different from, Natural numbers.

Rational Numbers

The concept of representing fractions as was generalised later into the idea of Rational Numbers. The term “Rational” emerged from the original idea of seeing them as “ratios”. The term has not relation to the daily sense of the word “rational” which means logical. We will see more about rational numbers in chapter 16.17.

Measuring Numbers

We can say that as against numbers 1, 2, 3 etc which can be called Counting Numbers, fractions can be called Measuring Numbers. They give the measure of the Part in terms of the Whole.

Operations with Fractions

Because of the way they are represented (the rules for performing arithmetic operations on them had to be worked out starting from the basics. As we will see in the subsequent chapters, these algorithms are quite different from those applicable to whole numbers.

It was a long time before humans became comfortable with arithmetic operations involving fractions. Hence it is not surprising that fraction is one of the topics which is considered difficult by students all over the world.

Decimal System for Fractions

The difficulty in working with fractions in rational number form, also motivated mathematicians to invent the decimal representation of fractions, though this took several centuries.

Fractions, Decimals and Percentages

Another important concept that needs to be introduced to students is that fractions, decimals & percentages are all different representations of the same idea. The forms may be different but the idea is the same.

The different forms are used in different fields of activity depending on the convenience of the users.

Hence 3/4, 0.75 and 75% are all different representations of the idea of three-fourths. In money transactions it is used as 0.75 to signify 75 paise. In comparing profitability of 2 business proposals it is used as 75%, signifying a profit of 75 rupees for every 100 rupees. In common parlance it is used as 3/4.

Fractions in the Roman System

Before the universal adoption of the decimal number system based on the place value system, most of Europe was using the Roman Number System. Representing & working with fractions in the Roman system was fiendishly difficult and needed expert abacists.

There is an interesting story in the history of fractions about the division of a day into 24 hours which is related to the difficulty ancients had in dealing with fractions.

They had taken 360 degrees as the measure of one complete rotation, from the approximate measure of 360 days in a year, in which Earth revolves around the Sun. This is called either a complete angle or a point angle. By dividing it by 8 geometrically, they could get angles measuring 45 degrees. By drawing an equilateral triangle, they could get angles measuring 60 degrees. They could also geometrically get an angle measuring 15 degrees (60 – 45). They did not try to get smaller measures because subdividing 15 by 2 led them to fractions which they were not comfortable with. They also knew that it was geometrically impossible to trisect an angle.

Hence 15 degrees was taken as the smallest unit for diving the day into smaller time intervals! Hence the time it takes Earth to move 15 degrees out of the total 360 degrees, was called 1 hour. There are 24 measures of 15 degrees in 360 degrees. Hence the day was divided into 24 hours.