The World's Greatest Invention

The design of virtually everything we use in the modern world from the bricks and mortar that keep us warm and safe to the sophisticated electronic gadgets we now use every day of our lives depends on some form of calculation.

Builders, engineers (both mechanical and electronic), ship builders, suppliers and a host of other people involved in the design of the world around us use calculations constantly to make our environment suitable for the way we live today.
So the design of the number system we use is crucial. To see how flexible our modern system of numbers is, try multiplying the Roman numbers LXV11 and XXIV without converting them to our numbers first.
It's almost impossible (certainly in any meaningful way) and this accounts for the fact that the Romans made no developments in the field of mathematics, despite their other great achievements. So what is it that makes our number system so powerful and so easy to use? The most important thing is that our system is a 'place value system'; in other words the value of a digit is determined by its position in the number.

For example, the digit '5' in the number 35,789 represents 5,000 while the digit '5' in the number 56 represents 50.

The system is easily extended to decimals (which do not seem to have been represented in Roman times).

The digit '5' in 0.
056 represents 5/100.

Because of this, the system has another great advantage and that is that when you multiply by 10 the whole number moves to the left one position, so 34.
673 x 10 = 346.
73.
An extension of this is that multiplying by 100 moves the whole number to the left two places, by 1000 three places and so on.

Similarly, when you divide by 10 the number moves to the right one place, by 100 it moves two places etc.

These are sometimes referred to as 'left shift' and 'right shift'.

This simplifies the multiplication by a multiply of ten. Multiplying by 20 is equivalent to multiplying by 2 followed by a left shift of one place. Dividing by 300 is equivalent to dividing by 3 followed by a right shift of two places.
This idea of left and right shifting is extremely important in the multiplication and division of numbers and should be emphasised when teaching children these operations.
Multiplying a whole number by should be seen as a left shift which in turn reveals a blank in the units column. We fill this with a zero, of course. Never teach children to 'add a nought' because this is only correct in certain circumstances.
It is not true, for example, when multiplying 3.4 by 10. 3.4 x 10 is definitely not 3.40! Whenever 'add a nought' is taught, another teacher further down the line is going to have to unteach it.
It's one thing to teach a concept that will be given a more detailed analysis when children are older.
It's quite another to teach a concept that has to be untaught later.
One last point that will aid teaching of these ideas (when the children have an understanding of the concept of decimals) is that every number floats in a sea of zeroes. For example, the number 67.89 could be written as...
0000000000067.89000000000..
. with an infinite number of zeroes to the left and to the right.
This makes it easier to understand where the zero comes from when we multiply a whole number by 10.
As an example, multiplying 56 by 10 moves every digit (including that infinite number of zeroes on either side) to the left.
The zero immediately to the right of the decimal point jumps across into the units column. Easy, don't you think? Multiplying a whole number by 100 moves the two zeroes immediately to the right of the decimal point across into the tens and units columns. Once this idea has been demonstrated, children soon get the idea of multiplying and dividing by multiples of 10.