Place Value System

< 6.3 Place Value in Daily Life | Topic Index | 6.5 Understanding Place Value (A) >

All number systems use a string of numerals (or symbols) to denote the value of a number. The Place Value System used internationally today uses the idea that a numeral (1 to 9) assumes a value in various powers of ten, depending on where it occurs in relation to the other numerals used to represent a number. It also uses Zero (0) in a special way which would be dealt with in chapter 6.8 "Zero & the Place Value System". It makes the representation of numbers of any magnitude easy to write with the same set of ten numerals 0 to 9.

The place value concept can be visually interpreted in the following manner. If 2 dozen pencils are made into bundles of ten, then we will have 2 bundles (of ten each) and 4 pencils. This is written as 24. Hence 2 represents 2 Bundles or Tens which is twenty and 4 represents 4 Units. So it is called Twenty Four. The place (position) of the rightmost numeral is called the Unit&rsquo;s place and the place (position) of the immediate numeral next to it is called Ten&rsquo;s place.

The 2 dozen pencils can also be represented with bundles made with numbers other than ten. Annexure 6.5A shows the various possibilities.

We use a system where two dozen pencils are bundled into 2 bundles and 4 sticks, with each bundle containing ten sticks. Why we adopted the idea of bundling in tens is possibly because all humans have ten fingers and bundling in tens seems a natural idea for humans! If insects were to invent a place value system, they may use bundles of six!

The quantity of bundles (2 in the above example) is written in the ten&rsquo;s place. Hence 2 here is not just 2 but 2 bundles of ten each, which is twenty. Hence ten&rsquo;s place has a value which is ten times the Unit&rsquo;s Place. With the same logic, the place just to the left of the Ten&rsquo;s Place will be the Hundredth&rsquo;s Place, Hundred being ten times Ten. (Visually still talking of numeral 2, it is now equal to 2 large bundles each of ten smaller bundles (with ten sticks). Any number written in this position would represent that many Hundreds. In a number written as 345, 3 would represent 3 Hundreds, 4 would represent 4 Tens and 5 would represent just 5.

We can visually represent the system as given below. It shows the value of numeral One (1) when it occurs in different places.

It is called Decimal System since there are ten numerals (1 to 9 &amp; 0) and the place values increase in powers of ten. Deci is a Latin name for Ten. It is also related to the idea that the bundles are made of ten sticks or a bigger bundle is made of ten smaller bundles. Deci is the adjective form of Ten. We will deal with Zero in a subsequent chapter.

Place Value Convention

By convention, in any number written using the decimal place value the right most numeral is assumed to be in the One&rsquo;s or Unit&rsquo;s place. So, in 4265 the numeral in the Unit&rsquo;s place is 5 &amp; in 42653, the numeral in the unit&rsquo;s place is 3. All other numerals of a number get their place values in relation to their position in relation to the unit place. Come to think of it, this is the inverse of the way we normally write language.

Reading Numbers

Reading numbers is slightly different from reading text. We read words from left to right. 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&rsquo;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 a sequence of 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

We have become so used to the place value system that we mistake the structure (456 where each numeral appears in a difference place) for its value (which can alternately be thought of as thirty-eight dozens). We start thinking of 456 as &ldquo;natural&rdquo; (everyday occurrence) and a notation like as &ldquo;artificial&rdquo; in the sense that it is &ldquo;mathematical&rdquo;&hellip; We need to remember that 456 is also &ldquo;artificial&rdquo;. It is only a code for indicating the actual magnitude; a code which demands an understanding of its structure (the decimal place value concept).

The Place Value System is a very sophisticated system which took human civilisations over 3000 years to fully develop &amp; adopt. We can justly be proud that it originated in India.

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 &amp; 0 and the next number is written by combining 1 &amp; 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 meanings 1 and 0. For an initial learner, it can be very abstract &amp; confusing. Hence it needs to be taught in a way that children can grasp the underlying idea.

This glaring difference between 1 digit &amp; 2-digit numbers does, however, not occur when we &ldquo;speak&rdquo; 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 a grasp of the place value system.

Now let us understand the number Zero and its critical role in the Place Value System.

< 6.3 Place Value in Daily Life | Topic Index | 6.5 Understanding Place Value (A) >