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Non-Standard Representations
The standard manner in which "expanded representation" is taught in schools is that 3425 = 3000+ 400+ 20 + 5. For children this is a mechanical exercise which does not give any insight into the Place Value System. This is known as the Standard Representation, where at each "place" only numbers <=9 can be exist.
We can think of the standard representation as a restriction that a person can pay Rs 3425 only in terms of 3 Thousand Rupee notes, 4 Hundred Rupee notes, 2 Ten Rupee notes and 5 One Rupee notes. But in reality, we can pay Rs 3425 is many other combinations as shown below.
For example, Row 4 represents a situation where 34 Hundred-rupee notes, 2 Ten rupee notes and 5 One rupee notes are used to pay an amount of Three Thousand Four Hundred and Twenty Five rupees.
These are called Non-Standard representations and the table above gives a number of such representations. With the structure of the Place Value Concept, writing down of non-standard representations becomes very easy. The same string of numerals, 3425 appear in the same sequence but in various "places".
These representations are very useful make additions & subtractions easier, as will be seen later in Section (11) on addition & subtraction operations. These are the basis of the "carry over & borrow" techniques.
What we do when we "borrow or carry over" is actually re-writing that number in a non-standard representation.
When subtracting 38 from 45, when we borrow 1 (actually ten) from 4 what we are actually doing is rewriting 45 in a non-standard form 3 (tens) 15 (ones). Subtracting 8 from 15 is easy.
Other Number Bases
A clear understanding of the decimal place value system makes it easy for students to understand other number bases & conversions between bases. We will see this in detail in the next chapter.
Major Disadvantage
In the place value system, the value of a number can be drastically altered by inserting an additional numeral within a number. For example, 200 can be changed to 2000 or 345 can be changed to 3045.
Keeping this in mind, most financial documents, like a cheque, require that the number, though stated in numeral form, also has the word form in brackets.
We have become so familiar with the number system that we use that we have almost forgotten that it is a code.
234 actually means 2 x 10^2 + 3 X 10^1 + 4 X 10^0. These powers of 10 are not shown, They are just assumed to be these but hidden from our view.
This clarification would also help students become familiar with decimal representation.
23.4 is just 2 x 10^1 + 3 x 10^1 + 4 x 10^-1 ( or 4/10).
We also think that this representation is "natural". We do not realise that it has become natural because we have got completely used to it.
We can think of two hundred and thirty four as a quantity of toffees made of twenty dozen less six!
In the decimal place value system is written as 234 which means 2 hundreds 3 tens & 4 ones.
This is equivalent to paying this amount with 2 hundred rupees, 3 ten rupees & 4 one rupee notes.
But in real life there are many other way of paying this amount.
In 20 ten rupees & 34 one rupee notes.
Or 234 one rupee notes.
These can also be represented with the place value concept.
We can call these non-standard representations.
The same number can be expressed in different ways using non-standard representation of the place value system, as per the example given below.
Six different ways of representing 34981 - Thirty Four Thousand Nine Hundred Eighty One
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