Understanding Decimal Fractions 1

Originally fractions, i.e numbers less than 1, were represented as rational numbers of the type as described in the chapters on fractions. But the rules for arithmetic operations on rational numbers were quite different from and difficult compared to those applicable to whole numbers. This made mathematicians look for an easier way to write fractions. They found a way where fractions could be written like whole numbers and do the four operations easily in the same manner. This method extended the place value concept to numbers less than 1.

The concept of place value itself took a long time to get accepted. Extending it to numbers less than 1 took even longer. Though rational number representation of fractions was already known by 500 BC, at the time of Plato, the decimal representation using a decimal point, was accepted only by 1500 AD.

Research has proved that Italian merchant and mathematician Giovanni Bianchini had prepared astronomical tables in the 1440s, where he had used the decimal point in the form that we recognize today. This pushes back this invention by more than 150 years.

To understand the principle behind representing fractions using the place value concept, let us take the mixed fraction 267 373/1000. We have taken this fraction since it can be written as 267 + 3/10 + 7/100 + 3/1000.

1. The mixed fraction can now be rewritten as 2.102 + 6.101 + 7.100 + 3.10-1 + 7.10-2 + 3.10-3 with the dot representing multiplication sign.

2. In this form it looks like the extension of place value concept to negative powers of 10 i.e numbers less than 1. As we move from right to left, the power of 10 decreases by 1 at every step which is to say the place value gets less by a factor of 10 at each step.

3. If the whole number part can be written as 267, with the presence of powers of 10 assumed implicitly, why cannot the fractional part also written as 373 with the negative powers of 10 similarly assumed implicitly?

4. We have seen when we write numbers using the place value system, we are actually using a code. We are now just extending the code.

5. Hence the number could be written as 267373 provided there is some way to indicate that the place value of each of these numerals.

6. We have seen that in writing a whole number, the rightmost digit, by convention, is automatically assumed to be in the one’s place. But in decimal system, we cannot locate the one’s place easily. Hence another idea was used in decimal fractions.

7. A convention of placing a dot between the whole number and the fractional parts, like this 267.373, was used to locate the one’s place. The dot was placed just after the one’s place.

8. So, 267.373 became a short form (or code) to represent a number which was written in the rational form 267 + 3/10 + 7/100 + 3/1000.

9. This form of writing fractions is known as decimal fractions.

Once the place values related to the various numerals were identified, these decimal fractions can be operated arithmetically like whole numbers and the results interpreted similarly. We will see the operations in detail in subsequent chapters.

Decimal Point or Decimal Comma?

One math symbol which has still not been accepted by all countries is the “decimal separator”. Some countries, including India, use the “point” or the “period”. Many countries use the “comma”.

We will now see some other aspects of representing decimal fractions.