Origins of Multiplication 1

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We have seen that three of the basic operations on numbers could have evolved from daily activities of collecting, discarding and sharing.

The common understanding of multiplication as "repeated addition" does not appear to be a concept which would have arisen from the environment or daily life experience. It is more of a computational convenience. We will look at this issue later in this note. The present thinking among mathematicians is that the idea of multiplication arose from the idea of "scaling" which in turn arose from the study of how living beings "grow".

Scaling as Multiplying

Humans identified 2 properties while recognizing objects around them - size & shape. Two shapes could be different in size but have the same shape. Or they could be of the same size but have different shapes.

When most living beings grow, their size increases but their shape remains more or less the same. We can recognise our friends even after many years by their "shape".

Similarly trees and animals retain a recognizable shape through different stages of growth, from infancy to old age.

Greek geometers found that when size increases while maintaining "shape", the ratios of certain dimensions remained more or less same.

They defined this aspect as "scaling". They were able to relate the idea of scaling to the mathematical idea of multiplication.

Early Greek geometers refined these ideas into concepts of &ldquo;congruence&rdquo; and &ldquo;similarity&rdquo;. Congruence is when 2 figures have the same size &amp; shape. Similarity is when they have the same &ldquo;shape&rdquo; (and not necessarily the same size)

They discovered that the size of an object could be increased or decreased geometrically, without the object losing its shape. If the size of a rectangle is increased in a particular way (so that the diagonal of the new rectangle always coincides with the diagonal of the original rectangle) the resulting rectangle will have the same shape as the original rectangle.

In another mathematical perspective they could say that when any figure is &ldquo;scaled&rdquo; uniformly, it does not lose its &ldquo;shape&rdquo;. We can say that when the sides of a geometrical figure are increased or decreased by the same factor, then its shape does not change. Ideas of scale &amp; ratio were seen as related to the idea of multiples!

Nowadays even children are familiar with computers. While working with pictures on the screen, there is a way to drag one corner of the picture so that it becomes bigger or smaller while still looking same. This is an example of scaling the picture. Both the length & breadth of the picture increase in the same multiples. Here the multiples need not be whole numbers, but can even be fractions!

Hence, we can say that thinking in multiples also emerged through the science of measurement &amp; the study of scaling of objects.

Measurement &amp; Multiplicative Thinking

Ideas of measurement are of very ancient origin, dating from the hunting-gathering stage. Four of the earliest entities which were possibly measured were time, discrete quantities, distance and volume.

Time could be measured by keeping track of the movements of the Sun, Moon &amp; the seasons. The idea of numbers &amp; counting developed to keep track of possessions like the number of sheep, fruits or household effects.

Measurement of distance was very necessary to keep track of locations in moving (during hunting &amp; gathering) over vast territories. Measurement of volume was necessary to make exchange &amp; bartering of grains and produce easier. But these measurements were difficult. They could not be measured using numbers directly.

Gradually certain &ldquo;standard&rdquo; units were adopted which could convert ideas of distance &amp; volume to numbers. Distance could be thought in terms of the time taken to cover it by walking or running. So the distance to the next habitation could be thought as &ldquo;3 walking days&rdquo; or something similar. (This idea has been used in the last 2 centuries in astronomy with the invention of the new unit of distance, the &ldquo;light year&rdquo;)

Grains, berries &amp; produce could be exchanged by volume by using certain naturally available containers (shells of gourds or coconuts?).

Idea of Multiples

It was also realized that any distance could be seen as a &ldquo;multiple&rdquo; of the standard distance unit. Similarly, anything which was &ldquo;measured out&rdquo; was in multiples of the volume of the measuring bowl.

If grain was being poured from a granary into a bag using a bowl, the amount of grain in the bag increased one bowl at a time or additively. But the total amount of grain which was added to the bag was always a multiple of the volume of the bowl.

Through these experiences, &ldquo;multiplicative&rdquo; thinking emerged as an independent way of thinking.

Once the idea took root, many examples in nature and daily life were also seen as examples of multiples! Fruits kept in baskets, leaves in a stem, body parts of humans &amp; animals are some examples.

In early days, before the development of fractional numbers, multiples were seen always as &ldquo;whole number multiples&rdquo;. Hence the connection between &ldquo;multiples&rdquo; and &ldquo;repeated addition&rdquo; was also realized.

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