Difficulties with the Place Value Concept

Place Value System of Mayans, Sumerians & Egyptians

Mayans of Mexico had place value system with a base of twenty. Hence their place values were one, twenty, four hundred and eight thousand and so on. They used shells to represents zero, sticks to represent five, and beans or seeds to represent one.

Sumerians invented a place value system, which we can call as a "partial" one as compared to the system which we use today.

Sumerians used a system with a "number base" of sixty. There is no consensus on why the base sixty was chosen by Sumerians. One reason could be that sixty had five single digit factors (2,3,4,5,6) and hence made fractional representations easier.

But the system survives today in the way we measure daily time intervals in hours, minutes & seconds. The system had many numerals.

But they did not have a numeral to indicate “nothingness” or “zero as we call it today.” They could not think of “nothingness” as a quantity to be represented with a number.

This caused a problem for all users.

The problem of “places” with no numerals

All place value systems faced a common problem of “places” which had no numerals.

To understand the problem let us take a modern situation of representing an amount of four hundred and five. This amount can be paid with 4-hundred-rupee notes & 5 one-rupee notes. We can write this number as 4 in the “hundreds” place and 5 in the “units” place.

The question for the ancient civilizations was “how to convey the information that there are no ten-rupee notes used in this transaction.” Or “how to convey the information that there is no number in the ten’s place?”

Using a "blank space" as a Place Holder to indicate Nothingness

Many of the civilizations solved the problem of representing "four hundred and five" by leaving a “blank” at the ten’s place.

They wrote 4 hundred & five as 4 5. Please note the blank space left between 4 & 5.

Today this system, using modern vocabulary, is referred to as using “zero as a place holder.”

Sumerians did not think of zero. They only thought that there is no number in a particular place.

What we mean today by “zero as a place holder” is that “the absence of a number in a particular place, is indicated by placing a "blank space" in that place.”

Problems With a Blank Place Holder

The use of a blank space as a place holder caused several problems.

Blank spaces were used in several other places apart from their use as a place holder.

While writing we sometimes leave a blank where we do not intend to or do not leave a blank where necessary.

In case of reading a string of numerals as names, the blank space was critical.

In many situations, the existence or otherwise of a blank space had to be “guessed” from the context in which the number is used.

This is the modern equivalent of deciding whether a number was 45 or 405. If the context was age of a person, it was likely to be 45.

In spite of this advantage the idea of “zero as a place holder” was used for many centuries until a better idea was invented.

Thinking of Zero as a Number

This new idea was the Hindu idea of considering zero as a number and representing it with a numeral shape.

This changed the entire place value system to an easily understandable one.

Before understanding the place of zero in the place value system, let us understand the evolution of the different perspectives with which the idea of zero was viewed.