The importance of mathematical equations and why they are our friends


In his science bestseller, A Brief History of Time, Stephen Hawking remarked that each equation he included would halve the book’s sales, so he only put one, the equation of Einstein relating mass and energy, E = mc2. This cynical view does science a disservice; we should realize that, far from being hostile, equations are our friends.

An equation says that whatever is to the left of the “equals” sign has the same value as whatever is to the right. An equation expresses a precise quantitative relationship.

The equals sign carries a profound message; it signifies mathematical equality, the essence of exactness. In ancient times, equalities were expressed in verbal terms. It was Robert Recorde, a Welsh-born mathematician, who introduced the symbol “=” for “equals”. In his book The Whetstone of Witte, written in 1557, Recorde writes that he chose this symbol consisting of two parallel lines “because 2 thynges cannot be moare equal”.

Equations have great power to clarify results. One side of an equation can be changed or transformed arbitrarily, and provided the same operations are performed on the other side, the equation remains valid. Great simplifications can be obtained by this means.

The term “algebra” is derived from “al-jabr”, meaning the addition of a number to both sides of an equation to simplify or cancel terms. This idea first appeared in the works of the great Persian polymath Muhammad ibn Musa al-Khwarizmi, The Compendious Book on Calculation by Completion and Balancing, written around 820 AD, a seminal work on algebra.

Pythagore’s theorem

We all remember struggling in school to prove the Pythagorean Theorem, described by science popularizer Jacob Bronowski as “the most important theorem in all of mathematics.” This geometric theorem concerns the squares erected on the sides of a right triangle and it is far from obvious why it is so important. If we note the lengths of the sides of the triangle by a, b and c, the result can be written in the form of an algebraic equation a2 + b2 = c2.

From this equation we can immediately calculate the length of any edge once given the lengths of the other two. It is the basis for all surveying, mapping and navigation.

This equation is fundamental in mechanics and, with the appropriate generalizations of Bernhard Riemann and others, it is the basis of spacetime physics. Such profound consequences have their origins in empirical observations in ancient Babylon.

It’s not uncommon these days to hear speakers at science seminars apologize for posting equations. This would not have happened in the past. It is akin to a composer apologizing for the use of a musical score or a biochemist asking forgiveness for presenting a chemical formula.

The “dots” allow a synoptic view of a composition and a chemical formula can clarify complex processes that might otherwise require many pages of text. They simplify life. So do equations.

Recorde is credited with introducing algebra to England with his book. In 1551, he was appointed surveyor of the mines and funds of Ireland. Alas, he ended his days in prison, for reasons that are unclear. Perhaps he found himself embroiled in religious controversy or caught up in political intrigue. Or maybe some of the “money of Ireland” has gone astray.

Peter Lynch is Emeritus Professor at the School of Mathematics & Statistics, University College Dublin – he blogs at thatsmaths.com

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