The First Equation

The 'First Equation' is a story about the evolution of the notation we use for addition (+), subtraction (–), and equality (=), which led to the first time an equation was written using the modern notation we use today, 14x + 15 = 71. Interestingly, the equals sign was introduced some two centuries after the introduction of the addition and subtraction signs, but some 50 years before the multiplication sign and a century before the division sign. 

The 'First Equation' is a story that will help students appreciate how the invention of mathematical notation fuelled the development of mathematics. Moreover, it is a story that may help students appreciate the diversity inherent in the development of mathematics; let them frame their own learning not just a function of that diversity, but also as an intrinsic part to the continuing story of mathematics. It can be used by teachers to support their teaching of setting up and solving algebraic equations to solve problems.

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The introduction of the + and – signs

In her 2000 paper A Brief History of Algebraic Notation, Lucy Stallings identifies 'one of the earliest known symbols for addition and subtraction' to come from ancient Egypt, as seen in the Rhind papyrus (c. 1650 BC). The symbols used were 'a pair of legs walking forward [for] addition and a pair of legs walking away for subtraction' (p233). Howard Eves, in his 1983 book An introduction to the history of mathematics, describes a symbol for subtraction being used by the Greek mathematician Diophantus in the 3rd century, and the use by Hindu mathematicians of yu (यु ) as an abbreviation of yuta, meaning 'added', seen in the Bakhshali manuscript (which also featured the first recorded use of 0 for zero). 

But none of these symbols were universally adopted. The + and − signs that we use universally today were first introduced by French polymath Nicole d'Oresme in his Algorismus Propotionum (1356-61). However, as Edward Grant notes in his 1965 translation of d'Orseme's Algorismus, d'Oresme somewhat confusingly for us today called multiplication addition, and division subtraction. (For more about d'Oresme, the 'French Einstein of the 14th century', see this short biography.) 

Both the + and − signs were first used in print to indicate 'surplus' and 'deficit' ('what you add' and 'what you minus') by German mathematician Johannes Widman in his 1489 book 'Behende und hüpsche Rechenung auff allen Kauffmanschafft' (following the link will take you to the page where the symbols were first used; see the excerpted image above.) The subsequent universal adoption of + and − for addition and subtraction is evident from Heinrich Grammateus' Ayn new Kunstlich Buech (1521).

It is widely agreed that d'Oresme and subsequently Widman used + as an abbreviation for et, the Latin for 'and'. Eves, in describing Widman's use of +, and the subsequent universal adoption of such evident from Heinrich Grammateus' Ayn new Kunstlich Buech (1521), reinforces this belief 'as [et] was cursively written in manuscripts just before the time of the invention of printing' (p139). We remain unsure, however, of the reasons behind the adoption of the − symbol for subtraction.

The introduction of the equals sign

Welsh Doctor and mathematician Robert Recorde, born in 1510 in Tenby, Pembrokeshire, South Wales, was a popular author of a number of mathematical books — which he wrote, unusually for the time, in the English vernacular, thus making his writing more accessible than most scholarly books of the age, which were usually written in Latin. In his 1557 book, The Whetstone of Witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Nombers [1] (which you can peruse electronically here), Recorde was the first to use the plus and minus signs in English: 'There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse'. In the same book, Recorde also introduced the '=' symbol, 'to avoide the tediouse repetition of these woordes : is equalle to :', which he had already used some 200 times in the book [2].

During Recorde's time, much of the mathematical notation we take for granted today was not yet in use. In designing the symbol '=' — as 'a paire of paralleles, or Gemowe [twin] lines of one length, thus: =====, bicause noe 2 thynges can be moare equalle' — Recorde's initial motivation to abbreviate was quickly overtaken by something more profound, more enduring.  As Joseph Mazur (2014) eloquently puts it, 'the concise character of the symbol came with an unintended benefit: it enabled an unadorned picture in the brain that could facilitate comprehension'.

(As an aside, neither was there an easy way in the 16th century of denoting the powers of numbers, so Recorde coined the now unsurprisingly obsolete term ‘zenzizenzizenzic' [3] to ‘doeth represent the square of squares squaredly’, or in other words to denote the square of the square of a number's square: \({\left( {{{\left( {{n^2}} \right)}^2}} \right)^2} = {n^8}\).  Recorde also used another word (which didn't quite catch on), the 'sursolid', meaning to be raised to a prime number greater than three.  So a power of five would be the first sursolid, a power of seven the second sursolid, a power of eleven the third, and so on.)

The First Equation

In The Whetstone of Witte, using his newly invented '=' sign, Recorde published what is thus the first equation to appear in the symbolic form we use today. The equation was 14x + 15 = 71. Before Recorde, mathematicians would state an equality in words. But the introduction of the equals sign was more than just a function of efficiency (or laziness!). Recorde used the equals sign to express the fundamental role of equality in algebra, namely that if we perform the same manipulations on two equal expressions, i.e., to what we now call an equation, the results must still be equal. In Recorde’s words, 'if you abate even portions from thynges that bee equalle, the partes that remain shall be equall also.'

Notes, References & Links:

The Absurd Equation

The 'Absurd Equation' is a story about the work of Greek mathematician Diophantus, the problems he posed and the developments that they provoked. It s a story that will help students appreciate the diversity inherent in the development of mathematics; let them frame their own learning not just a function of that diversity, but also as an intrinsic part to the continuing story of mathematics. It can be used by teachers to support their teaching of setting up and solving algebraic equations to solve problems.

Diophantus and his Arithmetica

Diophantus was a Greek mathematician from Alexandria who lived in the 3rd century AD. He is known to us through his work Arithmetica, a series of texts containing over 100 mathematical problems that survived the destruction of the Library of Alexandria in 641AD. 

Arithmetica has had a resounding influence on the development of mathematics, particularly on number theory and the solution of algebraic equations. It was translated into Arabic in the 10th century by the Persian mathematician Abu al-Wafa' Buzjani (بوژگانی) and spread through Europe after the Italian mathematician Rafael Bombelli's 1572 book L'Algebra, and the publication of Arithmetica in full in 1621 by French mathematician Claude Gaspar Bachet de Méziriac. 

Arithmetica contains the earliest known use of algebraic symbolism, which had a huge influence on Islamic mathematics and the subsequent development of algebra as we know and love it. Indeed, the word Algebra itself comes from the Arabic ‏الجبر‎ (al-jabr), which itself came from the title of an early 9th century book Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing) by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c780–850). The development of algebra as a discipline independent of arithmetic and geometry by al-Khwārizmī was taken further by the French mathematician François Viète (1540-1603), and — with contributions to notation from Italian, German, Dutch and Welsh mathematicians — led to the algebra we take for granted today. 

The Four Square Theorem

Diophantus was undoubtedly ahead of his time. He knew, for example, that every number can be written as the sum of at most four squares:

\[\begin{array}{c}\begin{align}1 &= {1^2}\\2 &= {1^2} + {1^2}\\ &\vdots \\43 &= {5^2} + {4^2} + {1^2} + {1^2}\\ &\vdots \\999,999,999 &= {30985^2} + {6319^2} + {3^2} + {2^2}\\ &\vdots \end{align}\end{array}\]

It was remarkable that Diophantus knew this, given that it took another 1500 years before it was proven — by Joseph-Louis Lagrange in 1770. Even Leonhard Euler, widely considered to be the greatest mathematician in history, and certainly the most prolific, was unable to prove that it was true. (You can use this applet to try some numbers out for yourself (maybe try your birthdate), and if you'd like to see and explore Lagrange's proof of the Four Square Theorem, this Wikipedia article isn't a bad place to start.)

As is always the case in mathematics, Diophantus' work provoked more questions that paved the way for more profound developments. For example, with respect to the four square theorem: what numbers can be written as the sum of four squares in only one way, excluding \({0^2}\)? (In fact, there are only 138 such numbers.) How many ways can each number be written as the sum of at most four squares (see here)? Do these numbers have certain characteristics? What numbers can be written as the sum of five squares, excluding \({0^2}\)? (In fact, only 12 numbers can't!) What about cubes...? And so on. 

Arithmetica and Fermat's Last Theorem

Artihmetica is also renowned in the history of mathematics because it led to Fermat's Last Theorem, and the awe-inspiring story of its eventual proof by the Abel Prize Winner Sir Andrew Wiles (see Simon Singh's wonderful book, short video, and documentary). Fermat had a copy of Arithmetica and when working through it he wrote a tantalising note in the margin next to one problem: 'I have discovered a truly remarkable proof [but] this margin is too small to contain it.' Fermat was asserting that he could prove that there are no positive integers for which \({x^n} + {y^n} = {z^n}\) when n is greater than 2, but he died before he gave his proof. Fermat's son published his father's note in a 1670 edition of Arithmetica, after his father's death, and the mathematical world was subsequently transfixed for centuries.  

The Absurd Equation

Diophantus was also the first Greek mathematician to recognise fractions as numbers. It may seem incredible to us to say this today, but before Diophantus, fractions simply weren't 'allowed' to be solutions to problems. Only positive integers were 'allowed'. This was because it was felt that numbers had to have a geometric sense; they were representations of lengths, areas, and volumes.

But the incredible Diophantus, the genius who opened the mathematical door for so many who followed, would not 'allow' negative numbers to be solutions to problems. He described the concept of \(\lambda \varepsilon \iota \psi \iota \varsigma \), meaning 'deficiency,' and the rules that 'deficiency multiplied by deficiency yields availability' (the product of two negative numbers is positive), and 'deficiency multiplied by availability yields deficiency' (the product of a negative and a positive number is negative). But even though he was happy to manipulate negative numbers in order to get to a solution, as a solution themselves Diophantus considered negative numbers 'useless'. By way of illustration he described the equation \(4 = 4x + 20\) as 'absurd,' because it leads to the 'useless' negative solution:

\[\begin{array}{c}\begin{align}4 &= 4x + 20\\ - 16 &= 4x\\ - 4 &= x\end{align}\end{array}\]

As a result, all of the problems in Arithmetica had positive integer solutions, and we now call problems of these type, i.e., that lead to equations whose only solutions of interest are integer solutions, Diophantine Equations.

The Problem of Diophantus' Age

Although we lack information about Diophantus’ life, we can work out his age upon his death from an algebraic problem inscribed on his tombstone. There are several versions of the epitaph, including this from Sir Thomas Little Heath's 1910 study of Diophantus: 

'His boyhood lasted \(\frac{1}{6} \) of his life; his beard grew after \(\frac{1}{12}\) more; after \(\frac{1}{7}\) more he married, and his son was born five years later; the son lived to half his father’s age, and the father died four years after his son.'

\[\begin{array}{c}\begin{align}x &= \frac{x}{6} + \frac{x}{{12}} + \frac{x}{7} + 5 + \frac{x}{2} + 4\\x &= \frac{{14x}}{{84}} + \frac{{7x}}{{84}} + \frac{{12x}}{{84}} + 5 + \frac{{42x}}{{84}} + 4\\x &= \frac{{75x}}{{84}} + 9\\\frac{{9x}}{{84}} &= 9\\9x &= 756\\x &= 84\end{align}\end{array}\] 

Notes, References & Links: