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## Monday 4 March 2024

### 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.

• Lesson slides (.ppt format)
• Lesson slides (.pdf format)

## 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 GrammateusAyn 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 GrammateusAyn 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.'