Non-Decimal Place Value Systems

Decimal Place Value System is Not Natural

We have been using the Decimal Place Value System for such a long time and have got so used to it that we almost think of it as "natural" or even "God-given". A little thought will clarify that this is not so.

Numbers are abstract ideas and representations are a means of communicating them them. The Place Value System in general and the Decimal Place Value System are just one, though very familiar, way to represent numbers. There are systems which use non-decimal place values also.

In the place value system which we use, a number like 34567 can be algebraically represented in the form 3X^4 + 4X^3 +5X^2 +6X ^1+ 7X^0 where x takes a value of 10.

X was taken as Ten was taken because we happen to use the Decimal Place Value system. We use the Decimal Place Value system "instinctively" since we have ten fingers. In the Decimal System we have ten numerals and the place values increase in powers of ten. Ten is called the Base of the Decimal System. We can remember it as the number of sticks that we use to make a bundle.

Non-Decimal Place Value systems

But when x takes other values, we get numbers represented in non-decimal number bases. This is equivalent to making bundles with numbers other than ten. I am deliberately writing "ten" and not "10" because 10 represents the decimal system too intimately and is likely to cloud our thinking.

Octal System

For example, if we think of a dozen we can represent it as a bundle of eight sticks and four sticks. We could write a dozen as 14, with the understanding that 1 represents eight!

Dozen is a concept and 12 or 14 are just different ways of representing this concept using numerals. We could call this system of representing numbers as a system with Base 8. It is called the Octal System. This system will have bundles of 8, 64 (82), 512 (83) etc.

If all humans were born with 8 fingers, we may be having an Octal System instead of a Decimal System! It is just one of those accidents in history.

Names of Non-Decimal Place Value Systems

Numbers can be and indeed are represented in different number bases, depending on the need. The non-decimal place value systems have been given the following names.

Place Value Systems Used in Computer Software

Those who are familiar with computer software will easily recognise Binary, Octal & Hexadecimal systems of representing numbers & data in computer software.

Binary system is very easy to represent in a machine which can represent 0 & 1 in terms of its physical state, either as Off (no current or voltage) and On (presence of current or voltage). By just having 2 physical states, the design of the machine becomes simpler and possibility of errors in interpretation is also minimized.

But this system would result in "daily use decimal" numbers being represented by long strings of numerals 0 & 1. Computers also have a requirement that the length of any string is fixed. Hence 5 would have to be represented as "00000011"

Humans would find it difficult to comprehend and work with such representations. But for a machine to store such long strings and operate upon them is just a matter of its structure and speed. Hence for writing the code and representing internal addresses, the Hexadecimal system is used.

Changing Numbers from one Base to another (visually)

Changing numbers from one system to a system with another base can be very confusing for students. With the visual imagery of bundles & sticks, this becomes very easy to visualise.

(picture of chaging a decimal number 34 into an octal number)

Let us take 34 in the decimal system. It can be visualised as 3 bundles (of ten sticks) and 4 sticks. How will it be written in Base Eight?

Base Eight means each bundle will have only 8 sticks. With 34 sticks we can make 4 bundles (of 8 sticks each) and 2 sticks would be left over. Hence in Base Eight, it would be written as 42.

Place Value System - a General Definition

Now we are in a position to give a general definition of a Place Value System.

It is a system where

One of the whole numbers is the Base
The count of the total number of digits is equal to the value of the Base
There are an infinite number of Places
Each place has a weightage
The weightage of the right-most place is always 1
The weightage of every place is more than the digit to its immediate right by a factor equal to the "Base"
Each place is occupied by a digit
The value of each digit is the product of the face value of the digit and the value of the place where it is situated
The value of the number is the total of the values of the digits