The importance of mathematical equations and why they are our friends

That’s Math: Stephen Hawking’s cynical view of equations is a disservice to science

Equations have great power in clarifying results

Equations have great power in clarifying results

 

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

An equation indicates that whatever is to the left side 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 deep message; it signifies mathematical equality, the essence of exactitude. 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 wrote that he chose this symbol consisting of two parallel lines “bicause no 2 thynges can be moare equalle”.

Equations have great power in clarifying results. One side of an equation can be modified or transformed in an arbitrary manner and, provided the same operations are performed on the other side, the equation remains valid. Great simplifications may be achieved by this means.

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

Theorem of Pythagoras

We all recall struggling at school to prove the theorem of Pythagoras, described by science populariser Jacob Bronowski as “the most important single theorem in the whole of mathematics”. This geometric theorem relates the squares erected on the sides of a right-angled triangle and it is far from obvious why it is so important. If we denote the lengths of the sides of the triangle by a, b and c, the result may be written as 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. This is the basis for all surveying, map-making and navigation.

This equation is fundamental in mechanics and, with appropriate generalisations by Bernhard Riemann and others, it is the basis for the physics of space-time. Such profound consequences have their origin in empirical observations in ancient Babylon.

It is not uncommon these days to hear speakers at scientific seminars apologise for displaying equations. This would not have happened in the past. It is akin to a composer excusing the use of a musical score or a biochemist asking pardon 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 into England with his book. In 1551, he was appointed surveyor of the mines and monies of Ireland. Alas, he ended his days in prison, for reasons that are unclear. Perhaps he became embroiled in a religious controversy or ensnared in some political intrigue. Or perhaps some of the “Monies of Ireland” went astray.

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

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