Origins of Multiplication

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

But origins of multiplication are not so apparent.

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 in Real Life

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 also retain a recognizable shape through different stages of growth, from infancy to old age.

Scaling, Shape & Ratio

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".

When a body scaled ideally, the "internal" ratio of certain dimensions or "shape" remains the same.

The "scale factor" by which the dimensions of an object need to be scaled is worked out by the concept of ratio. Hence scaling and ratio are reverse ideas.

We will study about ratios when studying division.

Scaling is concerned with ratio of two numbers and not the magnitude of 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. Scaling a picture on the computer screen is an example of this.

Congruence & Similarity

Early Greek geometers refined these ideas into concepts of "congruence" and "similarity".

Congruence is when 2 figures have the same size & shape. Similarity is when they have the same "shape" and not necessarily the same size.

They found that 2 geometrical figures have the 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 ratios.

In another mathematical perspective they could say that when any figure is "scaled" uniformly, it does not lose its "shape". 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.

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.

Scaling on the Computer Screen

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.

insert picture from FB post

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!

Scaling increases or decreases size of an object in a "continuous manner" which could involve scale factors which could be any continuous "real" number.

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

Scaling as Multiplication

Through these ideas mathematicians were able to relate the idea of scaling to the mathematical idea of multiplication.

Scale factor is always considered a “multiplication factor”. A photograph which is half the size of the original is thought of as having a scale factor of ½ and not as divided by 2!

Multiplication as Repeated Addition

Scaling was a difficult concept of multiplication for common people & school children to understand.

However, when the dimensions of an object are in whole numbers and the scale factor is also a whole number, the entire process could be thought of as “finding multiples” or "addition of equal quantities".

In this situation, multiplication could be thought of as repeated addition (of equal quantities). Scaling is a more powerful idea which includes the idea of repeated addition within it as a special case

The special case of "multiplication as repeated addition" is a much easier concept to understand than the general idea of "multiplication as scaling".

Hence possibly the tradition of introducing multiplication to primary school students as "repeated addition" started.

Multiplication in the School Curriculum

Scaling is an universal idea of multiplication as compared to "repeated addition". It is understandable that primary school children are introduced to this idea first.

But at middle & high schools, the idea of "multiplication as scaling" should replace the idea of "repeated addition".

Unfortunately, this is never done explicitly in the curriculum and the only idea that students and most teachers, all over the world, have of multiplication is that of repeated addition.

They do not even realize that many other examples of multiplication that they come across in geometry & commercial arithmetic cannot really be understood through this metaphor! One easy example is the multiplication of fractions!

Extending the Concept of Scaling 

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