Numbers and Magnitudes

Concept of Numbers Took Time to Develop

Greeks developed geometry to a more sophisticated level than arithmetic.

Their arithmetic stopped with the development of Natural numbers. They did not accept zero as a number. They used Natural numbers for representing discrete quantities.

The Concept of Magnitudes

For representing continuous quantities like like fractions, lengths, angles, time intervals, area & volumes, they used the concept of "magnitudes" rather than numbers.

Any property which could be compared or added or subtracted, was considered a magnitude. Magnitude of lines (lengths) could be compared by keeping them together. Volumes could be compared by physical means. Areas could also be compared through geometrical means.

Only as number systems developed fractions, decimals and real numbers, the above magnitudes could be represented with an universal medium i.e. numbers.

The Concept of Ratios

Pythagoreans believed that the entire universe could be explained with rational numbers. Hence discovery that the diagonal of an unit square could not be expressed with whole numbers was a shock to them.

When Greek mathematicians used the term ratio, they did not mean those using numbers as we do today. They were comparing magnitudes i.e magnitudes of lines (what we call as length today), areas, volumes etc. They were abstractions outside the realm of numbers.

They could say that in 2 similar triangles, the ratio of the 3 pairs of corresponding sides is the same. They could say that without knowing the magnitudes of the 6 sides in question.

For example, for Euclid, Pythagoras theorem was about areas of squares. His method of proof was that the 2 smaller squares can be cut and rearranged to fit the larger square. Our way present of representing it algebraically as c^2 = a^2 + b^2 would have been totally alien to him.

As per Euclid, the 2 terms of a ratio should be of the same magnitude, an idea which we continue to accept and call as scale.

Only after the development of the number system, we could also define a ratio of two magnitudes also as a rate, where the 2 magnitudes could be of different units of measurement!

The ratio of 2 magnitudes of a particular kind can be equated to a ratio of 2 magnitudes which are of a different kind. The ratio of different quantities of a product can be same as the ratio of their total costs. This is the scale approach to doing a problem which can also be done through a rate approach or unitary method.

Simon Stevin extended the decimal notation to fractions also. With this, the idea of “real numbers” which represented "continuous" magnitudes emerged. Real numbers could now represent continuous “magnitudes”.