Place Value System - 1

After understanding the basics of the place value system and the evolution of zero as a number, we are ready to explore the details of the decimal place value system.

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 related to various powers of ten, depending on where it occurs in relation to the other numerals used to represent a number.

The value of the number is the product of the numeral and a particular power of ten.

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.

We must have adopted the idea of bundling in tens 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 Decimal Place Value System

We used 24 to represent the quantity 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.

Why the name Decimal?

Ten is called Deci as derived from the Latin word Decem for ten. Since the place values are in powers of ten and the number of numerals needed in this system are ten, the system is called Decimal Place Value System.

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- hundreds, Longs- tens & Pieces- ones)

"Deci" has also 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 bases 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.