Origins of Multiplication 2

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We saw that thinking in multiples emerged through the science of measurement &amp; the study of scaling of objects. There was a crucial difference in both these approaches.

Multiples &amp; Whole Numbers

The earliest numbers to be invented were the Natural numbers. Multiples were seen as closely related to these numbers. It was easy to see that the distance to the nearest village was &ldquo;3 times&rdquo; the distance to the nearby hill. It was also easy to see that the big sack could hold a dozen bowls of grain!

It was also seen in nature as the pattern of leaves in a stem or petals of a flower. It was also seen as the grouped arrangements of things at home and in the house compound.

All these arrangements also revealed that multiples could be seen as cases of &ldquo;repeated addition&rdquo; of the same set of things. Hence the idea emerged that &ldquo;multiplication is repeated addition&rdquo;.

But the idea of scaling revealed another perspective.

Scaling as Multiplication

Scaling was more concerned with ratio of two numbers and not the numbers themselves. Scaling could be done geometrically without the use of any numbers. We can double the size of a figure without necessarily knowing its dimensions.

Greek geometers even arrived at a mathematical base for similarity of regular geometrical figures. They found that 2 geometrical figures are same shape when all their corresponding sides are in the same ratio! They did not bother about the exact magnitudes of the sides. In fact the number system they had was not sophisticated to express any magnitude of length. They only talked about their ratio. Even the original proof of the Pythagoras theorem was not about lengths of the sides of a right triangle but about the areas formed by squares on the sides.

Scale is always a &ldquo;multiplication factor&rdquo;. A photograph which is half the size of the original is thought of as having a scale factor of &frac12; and not as divided by 2!

A scaling factor can be any kind of number; whole, fraction or irrational. A scaling factor less than 1 indicates a decrease.

Modern Understanding of Multiplication

Extending the idea of scaling, multiplication can be seen as extending a line in a perpendicular direction, so as to form a surface.

This enables array concept of multiplication to be extended to the area concept of multiplication.

It extends the idea of multiplication from whole numbers to any type of number.

Repeated Addition &amp; Scaling

The idea of multiplication as repeated addition was an easier idea. The idea of multiplication as scaling was a more sophisticated one and emerged only much later.

Scaling is a more powerful idea which includes the idea of repeated addition within it as a special case. Students in middle school can certainly be introduced to this idea when they study ration and proportion.

Unfortunately, this is never done in the curriculum and the only idea that students and most teachers have of multiplication is that of repeated addition. They do not realize that many other examples of multiplication that they come across in geometry &amp; commercial arithmetic cannot really be understood through this metaphor!

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