Place Value Systems

The second type of system which has stood the test of time are the “place value” systems. They can also be called "positional value" systems.

All number systems need a set of numerals.

In the ‘aggregating systems’ the value of a numeral was the same irrespective of the place it occurred.

In contrast, in the “place value” system, the value of the numeral depends on its position or place in relation to the other numerals in the number. That is the origin of the name of the system.

Place Value Concept is a Code

The place value system is actually a secret code to interpret a series of numerals which make up a number.

The value of each place and its relation to other places is determined by a number which is called the "base" of the place value system.

In addition, the value of the places increase or decrease in terms of "powers" of the base. If the base is 10, then the value of the places increases as 1 (10^0), 10^1, 10^ 2 etc. When we deal with the details of the system, these issues will become clearer.

Many civilizations have invented a variety of “place value” systems each with a different secret code necessary to interpret it. We will briefly look at the Sumerian/ Babylonian system & the Mayan system.

Partial Place Value Systems

For want of a better word, we call them "partial" place value system. This is because they were unable to achieve the full potential of the system for want of a concept of "xero as a number" without any inherent value.

Sumerians/ Babylonians used sixty as the base and had place values which in our notation were 1, 60, 3600 etc. This possibly made it easier for them to represent big numbers which were needed to represent their astronomical observations. In place of a zero, they left a "blank space" to indicate a place which had not number/ value in it.

Plimpton 322

The earliest recorded arithmetical artifact is the fragmentary clay tablet known as Plimpton 322, dating back to around 1800 BC in Mesopotamia. This tablet contains a list of “Pythagorean triples” (a, b, c) satisfying, the relation stated by the theorem.

The tablet indicates advanced mathematical knowledge. The table’s layout suggests the use of an implicit identity related to Old Babylonian exercises. The purpose of the table remains unknown, but it might have served as a source of numerical examples for educational purposes.

The tablet also shows that the Babylonians had a sophisticated understanding of right-angled triangles and could solve problems related to their sides with exceptional accuracy. It reveals that Babylonians had a sophisticated understanding of Trigonometry, at least 1500 years before the Greeks.

Their Sexagesimal Place Value system made representation of fractions much simpler. Their trigonometric ratios were more accurate and avoided irrational numbers. Sixty is a highly composite number. 1/3 in decimal base yields a recurring decimal. But in Sumerian system it can be represented as 0.20 which in their system would be “twenty/ sixty” which is 1/3.

It also shows that the Babylonians had a sophisticated understanding of right-angled triangles and could solve problems related to their sides with exceptional accuracy.

Mayans (of Mexico) on the other hand used a base of twenty, possibly because they used both fingers and toes to count. So their place values were 1, 20, 400, 8000 etc.

But they had a symbol (conch shaped) as a "place holder" in places that had no number. But it was just a symbol without any idea of number attached to it.

Difficulty with the Place Value Concept

The idea of a secret code is difficult for children to understand.

In addition, children, and even adults, find it difficult to relate to the place value system in mathematics for 2 other reasons.

One is that the mathematical idea is something they have never experienced in life before.

Second is that this abstract idea is taught to them using several similar- sounding & confusing terms. They are taught, for example, that in 23, 2 is in the ten's place and 3 is in the one's place!

They are not able to differentiate between the different terms "one", "one's" ,"ten" and "ten's" and the idea of a "place" in the context of numbers.

It would be better if children are first familiarised with the idea of place value, as it occurs in daily life but which has never been perceived as a place value system.

We will deal with this aspect in more detail, in the next chapter.

Place Value in Daily Life - Postal System

Let us take examples from Postal System and Date System. Unfortunately, with the invention of email & SMS, even the idea of a postal system may become alien!

Imagine you are a postman and that you have to deliver a letter whose address is written in the manner shown under "Address A".

Most likely you will interpret that the letter has to be delivered to one Anuradha living in a house called Lakshmi. However, if the address is written as given in "Address B", then it would be delivered to Lakshmi living in a house called Anuradha.

This is because the postal system has a (unwritten) place value system or convention where the first line of the address has the value "Name of the Person" and the second line has the value "Name of the House".

The way "Anuradha" is interpreted depends on the "place where it occurs in the address". We can say that the value or the interpretation of "Anuradha" depends on its place, relative to the rest of the address.

Place Value System in Daily Life - Writing Dates

Let us now look at how dates are written in different countries. 11thSeptember is written a 9/11 in the US and 11/9 in India.

In India we write a date in the dd/mm/yyyy format, whereas in the US it is written in the mm/dd/yyyy format. The value of a number in a date sequence depends on its position or place in the date which again is dependent on the convention adopted by a country.

To be able to operate effectively in a new situation (or country in this context) we need to be familiar with its conventions including the place value systems. Unlike the postal place value, the date place value is formal and documented.

Place Value Concept in Ancient Literature

Patanjali, in his Yogasutra (2nd century BC) explains the idea of the place value system using the example of the same person being referred to by different names. (Crest of the Peacock by George Ghevarghese Joseph)

He said "Just as the same sign is called a hundred in the “hundreds” place, ten in the “tens” place, and one in the “units” place, so is one and the same woman referred to (differently) as mother, daughter, or sister."

Sankara in his bhashya also uses similar examples. Quoting such examples to make a point in philosophy, is an indication that by their time the idea of place value was fairly well established.

Google Sites

Report abuse