Place Value System - 2

Reading Numbers

Reading numbers is slightly different from reading text. We interpret letters in a word from left to right. But in reading numbers, we quickly browse the entire number and work out the place value of each numeral. Then we read the number combining the numeral and its place value.

When we look at 4265, we scan it to quickly figure out that it has 4 numerals and hence the first numeral is in the thousand's place. So we read it as Four Thousand Two Hundred Sixty (which actually is Six Tens) Five.

Thus, a number of any magnitude requires only ten numerals from 0 to 9 to represent it. Hence there is no need to invent any more symbols or numerals.

Place Value is a Code Invented by Humans

We have become so used to the place value system that we mistake the structure (456 which is just a string of numerals 4, 5 & 6) for its value (which can alternately be thought of as thirty-eight dozens). We start thinking of 456 as "natural" (an everyday occurrence). We need to remember that 456 is an "artificial" human invention.

It is only a code representing the actual magnitude; a code which demands an understanding of its structure (the decimal place value concept). A person who has not learnt this code cannot understand the magnitude of a string of numerals.

It is a Sophisticated & Not-So-Easy-To-Understand Code

The Place Value System is a very sophisticated system which took human civilisations over 3000 years to fully develop & adopt. We can justly be proud that India invented the idea that zero can also be considered as a number. This idea removed any confusion in representing numbers and paved the way for universal adoption of the decimal place value system.

We should also realise that it is a very abstract concept which is difficult for children to grasp. We have unique numerals for numbers from One to Nine and also for Zero. But the number which comes after 9 (which is Ten) is written by combining 1 & 0 and the next number is written by combining 1 & 1.

We see the familiar numerals being written in a different way and also having a totally different meaning. The meaning of 10 is totally different from the individual values of 1 and 0. For an initial learner, it can be very abstract & confusing. Hence it needs to be taught in a way that children can grasp the underlying idea.

Spoken Word Representation Vs Numerical Representation

This glaring difference between 1 digit & 2-digit numbers does, however, not occur when we "speak" the number names. Ten, Eleven, Twelve sound quite different from numbers One, Two ..so on up to Nine.

Hence while teachers can teach oral counting up to twenty or thirty, they should postpone teaching children to write numbers greater than 9, until they have understood the place value system.

Standard Representation

There is another hidden convention in representing numbers in the place value system. This can be more easily understood with a daily life example.

Imagine we have to pay an amount of Rs 45 to a friend. We have several ways of paying this amount. We could pay as 3 ten rupee notes and 15 one rupee notes. Or we could even pay as 45 one rupee notes, in case our friend wants change.

However place value system insists that we have to imagine this transaction as having been done as 4 ten rupees & 5 one rupee notes. That is both the ten rupee & one rupee notes should be in quantities less than ten.

This convention requires that the number in any "place" should be less than or equal to 9. This may look obvious to us but the above example shows that there are other ways of representing 45. This convention is known as the "Standard Representation" of a number.

All other representations are called "Non-standard representations". You may wonder where do we use them in our computing algorithms.

The process of "carry over" during addition and "borrowing" during subtraction, temporarily create numbers in non-standard representations! If we are subtracting 28 from 43, then we temporarily rewrite 4(T) 3(U) as 3(T) 13(U) so that we can subtract 8 from 13.

The standard representation also has an important advantage. It ensures that all numbers have a unique representation. Non-standard representations are many ways of representing the same number.

Multiple Interpretations of Standard Representation

The standard representation itself allows automatic multiple non-standard interpretations of a number. Let us take number 346.

In the standard representation we can say that the number in the ten's place is 4. But we can also say that the number in the ten's place is 34! This is equivalent to paying an amount of Rs 346 as 34 ten rupee notes and 6 one rupee notes!

We can also say 3 is in the hundred's place and 46 is in the unit's place. This is equivalent to paying an amount of Rs 346 as 3 hundred rupee notes and 46 on rupee notes!

This is a demonstration of the flexibility and power of the place value system.