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Last time you wrote down an equation you probably used some mathematical symbols to represent operations and objects. Sure, you would say, it does not seem like much. After all, you learnt how to do this at school. But this notation has been developed over the course of thousands of years, and is what allows most of our calculations to be done so easily. Not impressed? Let us go way back, to 2000-1800BC in ancient Egypt when the Rhind Mathematical Papyrus was written.

Problem 50 of this exercise sheet reads something in the spirit of “A circular field has diameter 9khet[approx. 50m]. What is its area?” and the written answer is “Take away thou 1/9 of it, namely 1; the remainder is 8. Make thou the multiplication 8 times 8; becomes it 64; the amount of it, this is, in area 64 setat”.

This was an application of a formula, A=(d-\frac{d}{9})^2 which is an approximation to the area of a circle =\frac{64}{81}d^2\approx \pi d^2/4. Much easier to understand when we use our familiar symbols!

We skip across the pond and through many years to the ancient Greeks, c.350BC, and examine what Eudoxus had to say; “Magnitudes are said to be of the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than the latter equimultiples taken in corresponding order”.Which we can simply translate as:

“The ratios a/b and c/d are equal when for any integers m and n:

if ma<nb then mc<nd also,

if ma=nb then mc=nd also,

if if ma>b then mc>nd also.”

This is a rather significant definition, Huxley writes of it “It is difficult to exaggerate the significance of the theory, for it amounts to a rigorous definition of real number.”

Eudoxus was followed by perhaps the gretest mathematician of ancient times, Archimedes, who went as far as calculating the area under part of a parabola, a feat usually taught at schools when integration is introduced.

So when did our familiar mathematical notation begin to appear? We know the origin of the misnamed Arabic numerals, they were first seen by europeans in the book “Concerning the hindu art of reckoning” by al-Khwarizmi. Islamic mathematics built on the ideas from central Asia, and spread them to the west. But it was in the 14thcentury that things began to look familiar, when symbols such as “+” began to be used for addition by Nicole Oresme. In the 16thcentury, equalis, minus, square root and parenthesise appeared, and things took off.

Leibniz is famous for his developments of differential and integral calculus, introducing the \int, elongated S, for his infinite summations of small quantities. Although Newton famously also developed calculus, he appears to have been less consistent with notation. Leibniz’s notation is still in use today, as for example in the equations \int ydy=y^2. It became clear to Leibnitz, that a clear and precise notation was necessary, and, this has been a major focus of mathematics. Not least because mathematics is about communication as well as solving problems, a good notation helps one see the problem and solve it more easily!

The 16th century saw the arrival of great mathematicians such as Eurler, who’s notation provides the foundation for most of mathematics for the next 400 years. Much of Eurlers work is perfectly readable today, if you are willing to discard a few conventions. But many things will be perfectly familiar, the use of \pi, denoting imaginary units as i and summations with \Sigma.

Matrix and vector notation was also an important development, just think, Maxwell wrote his equations component by component! We will not detail all the developments from here onwards, but simply give an example of some very compact notation, that of tensors. Suppose you have a grid of numbers called \eta, which is n by n. The entry in cell (m,n) is \eta_{mn}. Suppose also two lists of n numbers each, X and Y, where like before X^m is the mth entry of X. One writes X^\mu Y_\mu to mean X_\mu Y_\nu \eta^{\mu\nu} where repeated indices have sums on them, meaning,

    \[\sum_\mu \sum_\nu X_\mu Y_\nu \eta^{\mu\nu} =  X_1 Y_1 \eta^{1 1}+...+X_n Y_1 \eta^{n 1} +  X_1 Y_2 \eta^{1 2}+...+X_n Y_2 \eta^{n 2} +… +X_n Y_n \eta^{n n}\]

Imagine writing out every element of this every time!

Closing remarks:

From words to compact notation for complicated and complex ideas in mathematics, we see the usefulness of the most precise language humanity has developed. For further reading, see a list of mathematical symbols here [http://jeff560.tripod.com/mathsym.html]. Another online reference for history of mathematics is [http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html].

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