# Introduction of $\imath$ and $\jmath$ notations for the imaginary unit

The imaginary unit is generally denoted $i$ or $\imath$. I have learned that the term imaginary ("imaginaires") was coined by R. Descartes in 1637, and the "i" notation was introduced by L. Euler (cf. Short History of Complex Numbers). In engineering and physics, the notation $j$ or $\jmath$ is often used. Some say it was used to limit confusions with the current, often denoted $i$ or "I", as noted at Electric current:

The conventional symbol for current is I, which originates from the French phrase intensité de courant, meaning current intensity .

My questions are:

1. Was the $\imath$ (or $i$) notation used before Euler?
2. Who introduced the $\jmath$ (or $j$) notation?

EDIT: I just found, answering on a different topic about frequency, a mention of the word "cisoid" (abbreviated as $\mathrm{cis}$), I just found Cisoidal Oscillations, 1911, by George A. Campbell, where he writes:

The use of $i$ (or Greek $\imath$) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol $j$ is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using $j$ for $+i$ and others using $j$ for $- i$

So the story is not fully new, and we could date the $j$ before 1911.

• In my experience, the imaginary unit is generally denoted $i$ (or in Europe $\mathrm{i}$) and almost never denoted $\imath$. – Gerald Edgar Aug 10 '16 at 13:43
• @Gerald Edgar The dotless version seems more compatible with hats, overbars or exponents. I have edited accordingly, I am interested in the letter, dotted or not. – Laurent Duval Aug 10 '16 at 14:08
• @PeterDiehr -- The use of Phasor notation using the lower case letter j was first used in Power Systems Analysis texts by Charles P. Steinmetz in the early 1900s. Steinmetz died in 1923 so his work certainly predated your reference specifying 1940-1945. All Power System EEs learn this bit of history in their course work. – K7PEH Aug 11 '16 at 2:37
• @K7PEH I have added a reference of a similar discussion from 1911. – Laurent Duval Aug 11 '16 at 20:58
• it wasn't intensity, Laurent. it was current. as in $$v \ =\ iR$$ or $$p \ = \ v \ i \ = \ i^2R$$ . that's what i have been told, from the beginning, why EE's use "$j$" for the imaginary unit instead of "$i$". $$i(t)=I\ e^{i \omega t}$$ might be more confusing than $$i(t)=I\ e^{j \omega t}$$ – robert bristow-johnson Aug 29 '16 at 18:24

In answer to your second part of the question regarding $j$ for $\sqrt{-1}$, this was introduced into text books describing Power System Analysis of AC power circuits in the early 1900s by Charles P. Steinmetz. I am not sure of the earliest date but my guess is between late 1890s and 1920s but certainly no later than 1923 as Steinmetz died in 1923.
Steinmetz is to Power Systems Engineers (EEs) as Einstein and $E=m c²$ is to physicists (and everyone else for that matter).
I doubt it that the $i$ notation would have been used for $\sqrt{-1}$ before Euler because even Euler himself did not start using it until a rather late date, and moreover used $i$ in a different sense namely for an infinite integer, in his Introductio and Institutiones.