# Origin of infix notation

Wondering where the infix notation of things like 1 + 2 came from, when roughly it came about, and if it was before/after prefix or postfix notation. I know the summation and function notation came about potentially from Euler, but I haven't seen where infix notation comes from. Maybe if it's top of mind, knowing about if there were any controversies or competitions regarding which became the standard practice for math, that would be interesting.

The process of writing "equations" in symbols was very long.

We can start from an Ancient Greece specimen, due to Diophantus :

$$\Delta \beta M \alpha \pitchfork ς \gamma \iota^{\sigma} M \delta$$

that reads approximately as : $$2 \Delta +1 - 3 x = 4$$, where $$\alpha, \beta, \gamma, \delta, \ldots$$ are the naturals : $$1,2,3,4,\ldots$$

$$\Delta$$ is square, $$M$$ stand for units and $$ς$$ is the unknown. $$\pitchfork$$ is subtracts and $$\iota^{\sigma}$$ is equals.

Justapoxition is sum and negative terms are collected, so that a single $$\pitchfork$$ suffices for all terms following it.

In moder symbols : $$2x^2-3x+1=4$$.

We can see some Medieval and Renaissance examples into Victor Katz, A History of Mathematics: An Introduction (Addison-Wesley, 2009).

Page 386 :

Paolo Gerardi, in his Libro di ragioni of 1328, gave the rule for adding the fractions $$\dfrac {100} {x}$$ and $$\dfrac {100} {(x + 5)}$$:

You place $$100$$ opposite one cosa [$$x$$], and then you place $$100$$ opposite one cosa and $$5$$. Multiply crosswise as you see indicated, and you say...

and page 405 for Rafael Bombelli (1526 1572) :

he used R.q. to denote the square root, R.c. to denote the cube root, and similar expressions to denote higher roots. He used $$\downharpoonright$$ $$\downharpoonleft$$ as parentheses to enclose long expressions, as in $$R.c. \downharpoonright 2 p R.q.21 \downharpoonleft$$, but kept the standard Italian abbreviations of $$p$$ for plus and $$m$$ for minus. His major notational innovation was the use of a semicircle around a number $$n$$ to denote the $$n$$ th power of the unknown. Thus, $$x^3 + 6x^2 − 3x$$ would be written as

$$1^{\frac 3 ˘} \text { p } 6^{\frac 2 ˘} \text { m } 3^{\frac 1 ˘}$$.

In conclusion, IMO, in Western mathematics the infix notation was an obvious choice, because it rised slowly from writing "descriptions" in abbreviated form.

For further examples, see also : Jens Hoyrup, Jacopo da Firenze and the beginning of Italian vernacular algebra, HistMat (2006), regarding the Tractatus algorismi, written in 1307 in Montpellier by a certain Jacopo da Firenze, that contains the earliest extant algebra in a European vernacular and probably, the first algebra in vernacular Italian.