Why Euler spoke of them as "complex" numbers?

[mcastillo356]

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!

 

[fresh_42]

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

 

[mcastillo356]

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.

 

[fresh_42]

• 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

 

[fresh_42]

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.

 

[mcastillo356]

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

 

[fresh_42]

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.

 

[mcastillo356]

Google...

 

[fresh_42]

Google...

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

 

[mcastillo356]

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.

 

[martinbn]

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.

 

[mcastillo356]

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

 

[martinbn]

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.

 

[fresh_42]

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]

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?

 

[gleem]

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.

 

[hutchphd]

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

 

[Ibix]

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.

 

[TeethWhitener]

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 $x^2+1$.

 

[fresh_42]

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

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.

Objection! From my link above:

If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, √−1 positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. - Gauß