Place Value System

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

Decimal Place Value System

All number systems use a string of numerals (or symbols) to denote the value of a number.

The Decimal 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.

Laplace on the Decimal Place Value System

The importance and enormity of the invention of the Decimal Place Value System can be gauged by this quote from French mathematician Pierre Simon Laplace.

“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.”

Visualization of a 2-digit number in the Place Value System

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's( or One's) place and the place (position) of the immediate numeral next on its left is called Ten's place.

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!

Structure of the Place Value System

We used 24 to represent two dozens.

The numeral 4 is written at the rightmost end of the number string, which represents the Unit's place. Since 4 is in the "unit's place" its value is just 4.

The numeral 2, which represents 2 bundles (of ten sticks each) is written just to its left. That place is the "ten's place". Since 2 is in the ten's place, its value is twenty ( ten times 2)

Extending the same logic, the place just to the left of the Ten's Place will be the Hundred's Place, Hundred being ten times Ten. 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 structure as given below. It shows the value of numeral Two (2) when it occurs in different places.

Please note that the way Come to think of it, this is the inverse of the way we normally write language.

Why the name Decimal?

It is called Decimal System since the quantities are grouped in ten or powers of ten. The system has ten numerals (1 to 9 & 0) and the place values increase in powers of ten.

The decimal place value system is also called a number system with a Base of 10.

We mirror this by making bundles of ten sticks. Higher powers of ten are mirrored either by larger bundles or by manipulatives called FLPs (Flats, Longs & Pieces)

Deci is the Latin name for the adjective form of Ten. "Deci" has entered our common vocabulary through the word "decimate" which means destroy. Mathematically it would mean reducing a quantity to one tenth's of its magnitude.

Non-Decimal Number Systems

In theory it is possible to have a number system with basses other than ten also. For example, The 2 dozen pencils can also be represented with bundles made with numbers other than ten. Chapter 6.5 shows the various possibilities.

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&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 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). We need to remember that 456 is &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). A person who has not learnt this code cannot understand the magnitude of a string of numerals.

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 India invented and proposed that zero can 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 &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.

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.

Let us now 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) >