Today we take for granted the numerous symbols that appear in the mathematical literature, but each has a history of its own, often reflecting the state of Mathematics at the time.
-Eli Maor
The symbols used in modern Mathematics tend to be understood exclusively by those who deem to use them. In school, the beginner Mathematician uses $+, -, \times$ and $\div$. These operations are the basis of arithmetic; and subsequently, further complex problems arise from here. For example, a combination of multiplication and addition can be written in indices, such as: $3^3=3 \times 3 \times 3=27$ where the smaller number tells you how many times to multiply the big number by itself.
These operations are our instructions that we lay out to show an explanation, i.e. how a problem develops stepwise into a solution. It is crucial for a symbol to give a universal understanding of what it is showing, and especially to not be mistaken for a different use. When reading a long line of calculations, we read ‘=’ simply as ‘equals’ like it’s second nature. However, hundreds of years ago, it would’ve taken much longer for a writer to translate someone else’s notation into their own.
The Equals sign – Early History
Equality in Mathematics is defined by being a relationship between two expressions holding the same value. Written as two parallel horizontal lines with equal length, it is difficult to imagine the equals sign being displayed as anything else. The long history of the equals sign dates back thousands of years…
Ancient Greek & Arabic: 3-15 A.D
Greek – Diophantus – $\iota^{\sigma}$
*The symbol is an abbreviation of the Greek word for equals (ίσος)!
Arabic – Al-Qalasâdî – ﻝ
*Diophantus wrote from left to right, and Arabic is written right to left, so the curved part in each symbol is in the direction of what we know as the RHS of the equation.
European: late 1400s
Translation:
$$ \begin{align} 84 \ – \ 2x &= 42 \\ 42 &= 2x \\ 21 &= 1x \end{align}$$
Translation:
$$ \begin{align} 2x \ + \ 6 = \ &216 \\ 2x = \ &210 \\ \text{Value of x} \quad &105 \end{align}$$
The dash was used by Pacioli for many other purposes besides equality, as well as some Mathematicians that succeeded him. Though there are many recorded uses of the dash to show equality, this was gradually phased out to avoid ambiguity – no doubt there would have been some confusion up until this point!
A manuscript from the University of Bologna, written sometime between 1550-68, was discovered by Professor E. Bortolotti, which shows the modern equals sign being used for equality in an equation. This extract could potentially be from a lecture of Pompeo Bolognetti. (seen in image to the right)
It is believed to be a coincidence that around the same time we see Recorde’s notation!
Robert Recorde and the modern Equals sign
Although many variations have been used to show equality, it wasn’t until 1557 when Robert Recorde is credited to have invented the familiar ‘=’ used today. In his book “The Whetstone of Witte” (one of the first books to be printed in the English Language), Recorde includes some examples of addition between two numbers, followed by the words ‘is equalle to’. A quote from his book gives his reasoning for using the new symbol:
(The book may seem to have quite an obscure name at first glance, as it is a play on words. A whetstone was a stone used to sharpen tools like knives and axes – to sharpen your wits!)
Even Thomas Harriot, who is known to be the one of the greatest Mathematicians before Isaac Newton, used 2 different symbols for equality. In the early 1600s, while working on a problem about the ratio between the values of silver and gold, Harriot writes his equals sign as 2 horizontal lines connected by 2 vertical lines, which can be seen throughout this extract:
The 2 vertical lines symbol for equality became surprisingly popular all over Europe, even after Recorde’s published version! The German Mathematician Wilhelm Xylander used a double vertical line (||) as the sign for equality, and many followed in his footsteps for a while. However, it is thought that this symbol was soon switched over due to its similarity to the capital pi! ($\Pi$)
There have been countless different ways to express the same thing. Symbols have come and gone or have been assigned new meanings, though different signs for equality were still used by Mathematicians for hundreds of years after Robert Recorde, it withstood the test of time. Finally, its popularity stuck around after Leibniz used the ‘=’ sign at the end of the 1600s.
It is incredible to think that Recorde’s creation was less than 500 years ago, and only a century after this did we see Newton’s differentiation and integration! A whole different story…
Related references and links
- Regiomontanus extract taken from ‘Documents on the History of Mathematics in the Middle Ages and the Renissance’ – page 291.
- Pacioli extract taken from ‘Summa de Arithmetica’ – part 1, page 91a.
- https://archive.org/details/Galaxy_v23n04_1965-04/page/n61/mode/2up?view=theater – Discussing Thomas Harriot’s use of symbols.
- https://archive.org/details/Galaxy_v23n04_1965-04/page/n61/mode/2up?view=theater – Discussing the history of Thomas Harriot.
- https://www.caltech.edu/about/news/question-week-who-invented-equal-sign-and-why-171 – An article about Robert Recorde and the equals sign
Leanne is a 2nd year Mathematics Student at UEA. Her learning consists of Inviscid Fluid Flow, Linear Algebra and other complex subjects, though it was the Differential Equations module that won her over! Integration and Differentiation are her cup of tea, and the methods are ‘just fascinating’ (when it works!)
On the side of her degree, Leanne plays for the Norwich Devils Women’s American Football team. It’s intense but has an incredible fast-paced environment.
She believes the best way to grow is to speak to new people, ask questions, and to become what you respect. She loves writing about people, and telling their stories.