# Why Euler spoke of them as "complex" numbers?

• mcastillo356
In summary, complex numbers are numbers that have two axes, like real numbers but with different names. They were first identified by Euler in 1777 and are used in many different ways than real numbers.

#### mcastillo356

Gold Member
Hi PF, this is just for fun...Or not; I don't know. In 1777 Euler set up the notation ##i## to identify any roots of ##x^2-1##, which are indistinguishable, and verified ##i^2=-1##. This way, the set of real numbers grew larger, to a bigger set called complex numbers.
This is a translation made by me from a book for absolute beginners like me. Isn't this description...complex? I mean they are not more complex than real, natural, irrationals...Or are them?.
Greetings!

How come that you say Euler used ## i ## first? It was Descartes 1637 who first called imaginary numbers imaginary, hence ## i ## is a natural choice. It is hard to believe that it took 140 years before someone introduced ## i ## as name for the root. Btw. roots that have been known for another hundred years before Descartes.

See the references in
https://www.physicsforums.com/threa...ary-if-they-really-exist.996269/#post-6419479

Klystron
Hey...! Descartes could have spoke of them as imaginary, but the word, begins with i either in french and Euler's language? I'm going to investigate it. In spanish it is imaginario; in euskera (the other language I speak, rather bad) I'm not sure.
Thanks. This night owl who is me has get something to look at.

• imaginaire - french - Descartes 1637
• imaginär - (swiss) german - Euler 1777
• imaginarium - latin - usual scientific language back then
• вообража́емый - russian - Euler was in Saint Petersburg, and the russian word starts with a "w", but they spoke french at the Tsar's court
• imaginary - english - irrelevant in that context

I can only guess why they are called complex numbers, and would like to see a historic document where it first was used. I'm a bit uncertain whether it was really Euler, and not Gauß later on.

The numbers have been known long before Euler, but not called complex. It takes ##\mathbf{i} ## to see why they are complex (latin for closely related or connected), namely ##\mathbb{R} \oplus \mathbf{i} \mathbb{R}##: You tie two real numbers into a pair which is a new complex number.

Klystron
I guess it is as complex as I want. Ah...in euskera is irudikari

mcastillo356 said:
I guess it is as complex as I want. Ah...in euskera is irudikari
Google says imajinarioa, which I think make sense, because the word is probably taken from another language into basque, and all others around are latin languages.

fresh_42
mcastillo356 said:
I can also offer alegiazko or irudimenezko from other translators, but I only asked for imaginär, not imaginary number.

Balizko maybe...but I think imaginarioa should be the most popular. From my point of view, your argue about the appearance of the word is the most reasonable I see. The others are synonims, but a few pretentiouse.

mcastillo356 said:
Isn't this description...complex? I mean they are not more complex than real, natural, irrationals...Or are them?.
I think it comes form the fact that complex means consisting of more than one part, real and imaginary.

epenguin and mcastillo356
Hi martinbn, I think the same. In other words, real numbers need only one axis; imaginary numbers need two. Don't you think?

mcastillo356 said:
Hi martinbn, I think the same. In other words, real numbers need only one axis; imaginary numbers need two. Don't you think?
Complex numbers, imaginary numbers also need one axis.

mcastillo356
The basic difficulty is, that we cannot draw a complex number line as we do with the real, although ##\dim_\mathbb{C}\mathbb{C}=1##. It is ##\dim_\mathbb{R}\mathbb{C}=2## which we can draw in a plane. But a vector space is not a field, so some information is inevitably lost.

I still think that the invisibility of imaginary zeros in a polynomial equation ##p(x)=0## or specifically in ##x^2+1=0## was the reason Descartes coined the term imaginary. Gauß observed the possibility of the representation in a plane, which is complex in the sense of combined, tied to a pair.

Wikipedia has some interesting remarks on the history of the terms:
https://en.wikipedia.org/wiki/Complex_number#History
I haven't checked whether all claims are referenced, so some caution is due.

mcastillo356 said:
Hi martinbn, I think the same. In other words, real numbers need only one axis; imaginary numbers need two. Don't you think?
I've written it too quickly; I meant complex numbers like ##a+bi##, for example, need a plane to show them graphically, clearly. I'm I wright?

mcastillo356 said:
I mean they are not more complex than real, natural, irrationals...Or are them?.

What is in a name? That which we call a complex number by any other name would behave the same.

BillTre and mcastillo356
But it would be a far different world if we had to use the Montague axis and the Capulet axis.

hutchphd said:
But it would be a far different world if we had to use the Montague axis and the Capulet axis.
But we could call an Argand diagram Verona.

hutchphd
mcastillo356 said:
In 1777 Euler set up the notation ##i## to identify any roots of ##x^2-1##, which are indistinguishable, and verified ##i^2=-1##.
Small typo, OP: the roots of ##x^2+1##.

mcastillo356
gleem said:
What is in a name? That which we call a complex number by any other name would behave the same.
hutchphd said:
But it would be a far different world if we had to use the Montague axis and the Capulet axis.
Ibix said:
But we could call an Argand diagram Verona.