imaginary number or imaginative number

[shahar]

Why imaginary number is called imaginary number and not imaginative number? or both of forms are correct?

 

[binbots]

I think Euler wanted them to be called alternative numbers which is much more fitting.

 

[Dr.Peterson]

Why imaginary number is called imaginary number and not imaginative number? or both of forms are correct?

An imaginative number (in English) would be a number with a good imagination, not one that can only be perceived if you have a good imagination!

An imaginary number, like an imaginary friend, is one that exists only in our imagination -- it isn't real.

Imaginary number - Wikipedia

en.wikipedia.org

At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie in which he coined the term imaginary and meant it to be derogatory. The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).​

Of course, real numbers, too, exist only in the mind (What color is three? How tall is four?). And as late as the nineteenth century, even negative numbers were often regarded as false.

 

[blamocur]

And as late as the nineteenth century, even negative numbers were often regarded as false.

At this time, many programming languages interpret zero as False and everything else as True

 

[fresh_42]

Dieudonné wrote (emphases mine):

History offers a whole range of examples that demonstrate how much the ratio of progress achieved by collectives to that achieved by individuals can vary from case to case. Thus, it is almost impossible to attribute to a specific mathematician the first conception of imaginary numbers, which appeared from various sources in the sixteenth century.

The term "number" initially encompassed positive real numbers, of which the Greeks already had a clear understanding (they called them "ratios of magnitudes"), then zero and negative numbers, which were used purely empirically before the nineteenth century, but finally, after long being avoided (cf. the fundamental objections as in Viète), were used continuously since Descartes; furthermore, imaginary numbers, the bold creation of the Italian algebraists of the sixteenth century, whose nature remained mysterious until around 1800, but which were increasingly used in algebra and analysis from the seventeenth century onwards because calculating with them brought simplifications.

 

[fresh_42]

The thing is that people operated with some concepts long before an "official" term has been coined. Many concepts were originally of a geometric nature, or in this case imaginary roots of polynomials. But who was first? The usual candidates are Euler and Gauß, but have you ever tried to find something specific in the Disquisitiones? I have only found a facsimile that doesn't allow an automatic search. And reading 480 pages in Latin isn't really an option.

The only example I know of that can be exactly located is the prefix "eigen-". Eigenvalues have been considered long before Hilbert introduced the term "eigenfunction" in 1904 for the first time. But "eigen-" is a germanism and can easily be identified. People invented and used their own names before 1904. "Imaginary" which is the same in all languages that have been used is way harder to determine who was first.

 

[binbots]

The thing is that people operated with some concepts long before an "official" term has been coined. Many concepts were originally of a geometric nature, or in this case imaginary roots of polynomials. But who was first? The usual candidates are Euler and Gauß, but have you ever tried to find something specific in the Disquisitiones? I have only found a facsimile that doesn't allow an automatic search. And reading 480 pages in Latin isn't really an option.

The only example I know of that can be exactly located is the prefix "eigen-". Eigenvalues have been considered long before Hilbert introduced the term "eigenfunction" in 1904 for the first time. But "eigen-" is a germanism and can easily be identified. People invented and used their own names before 1904. "Imaginary" which is the same in all languages that have been used is way harder to determine who was first.

What I’m curious about is does i still have anything to do with the sqrt of -1? It feels like once we starting seeing the use of the complex plane it took on some other sort of meaning. I mean what does 2i represent? Would that be the sqrt of -2? Wouldn’t that just be the same as sqrt2?

 

[fresh_42]

I think people first took imaginary numbers as solutions to polynomial equations: $x^2+1=0$ simply because they were very practicable and the newly introduced Cartesian coordinates certainly helped, too. We attribute the complex number plane to Gauß but was he really the first and did he call it the imaginary axis?

I have found in Dieudonné's book (History of Mathematics between 1700 and 1900)

Also worth mentioning is the treatise "De insigni usu calculi imaginariorum in calculi integrali": Euler performed complex variable transformations, demonstrating that, with the help of complex numbers, one can combine or derive integral calculus formulas, particularly those related to logarithms and the arctangent; these formulas, in fact, appear completely different in the real world. Euler's contemporaries took careful note of these phenomena, which contributed to the widespread use of complex numbers in analysis.

So Euler definitely used the term and calculated with it. But did he also invent the name?

 

[Dr.Peterson]

What I’m curious about is does i still have anything to do with the sqrt of -1? It feels like once we starting seeing the use of the complex plane it took on some other sort of meaning.

We define i (one way or another) as a number whose square is -1; therefore it is a square root of -1 (not really the square root of -1).

I mean what does 2i represent? Would that be the sqrt of -2?

So 2i is not the square root of -2; rather, since $(2i)^2=2^2i^2=(4)(-1)=-4$, it is the square root of -4.

Wouldn’t that just be the same as sqrt2?

The square root of 2 is a number whose square is 2, not -2. It has nothing to do with i.

 

[binbots]

We define i (one way or another) as a number whose square is -1; therefore it is a square root of -1 (not really the square root of -1).

So 2i is not the square root of -2; rather, since $(2i)^2=2^2i^2=(4)(-1)=-4$, it is the square root of -4.

The square root of 2 is a number whose square is 2, not -2. It has nothing to do with i.

Is this a good way to think of the of the ones on the unit circle on the complex plane?

 

[Dr.Peterson]

Not quite. Check the details!

 

[binbots]

Not quite. Check the details!

Better?

 

[fresh_42]

View attachment 39275
Better?

Right. However, you should reconsider using roots to express complex numbers. This can lead to mistakes because the rules you know about roots from the real numbers do not apply to complex numbers. The most famous "equation" that shows it is
$$1=\sqrt{1}=\sqrt{(-1)\cdot (-1)}\stackrel{?}{=}\sqrt{(-1)}\cdot \sqrt{(-1)}=i\cdot i = -1$$which is obviously wrong. The equation in the middle is wrong.

There are two possible ways to avoid mistakes like that. You can either work with a variable $x$ instead of $i$ and replace $x^2=-1$ at the end of the calculation, or you can use Euler's identity
$$e^{i\pi}=-1.$$

Note that $\sqrt{x}$ isn't defined in the first variant and can not be used. The equation would become
$$1=\sqrt{1}=\sqrt{(-x^2)}\neq \sqrt{x}\sqrt{x}=\sqrt{x}^2=x$$and the error is obvious. In the second case, we would get
$$1=\sqrt{1}=\sqrt{e^{i\cdot 0}}=\left(e^{i\cdot 0}\right)^{\frac{1}{2}}=e^{i \cdot 0\cdot \frac{1}{2}}\cdot e^{i \cdot 0\cdot \frac{1}{2}}=e^0\cdot e^0 = 1\cdot 1=1$$without the occasion to make the mistake.

Of course, there is also a third solution. You have to learn which rules about roots you know are still applicable and which are not. But that would be quite confusing. Euler's identity is easier to use. The first version has an algebraic background and is based on the fact that complex numbers can be written as polynomials - which is why roots of $x$ are undefined - plus the condition $x^2+1=0.$ It's probably a bit inconvenient to do calculus this way.

 

[blamocur]

This can lead to mistakes because the rules you know about roots from the real numbers do not apply to complex numbers.

Actually, the rules for real numbers ignore the fact that $\sqrt{}$ is not really a well-defined function and use its positive values only. But complex numbers don't have a notion of positive or negative: is $i$ positive or negative? how about $1-i$?

The first version has an algebraic background and is based on the fact that complex numbers can be written as polynomials - which is why roots of x x x are undefined - plus the condition x2+1=0. x^2+1=0. x2+1=0. It's probably a bit inconvenient to do calculus this way.

This is exactly how algebraic field extensions are constructed in algebra.

 

[fresh_42]

Actually, the rules for real numbers ignore the fact that $\sqrt{}$ is not really a well-defined function and use its positive values only. But complex numbers don't have a notion of positive or negative: is $i$ positive or negative? how about $1-i$?

... or one could read the following articles ...

Things Which Can Go Wrong with Complex Numbers

At the first sight, there are many paradoxes in complex number theory. Here are some nice examples of things that don't seem to work...

www.physicsforums.com

Exploring the Multifaceted Nature of Complex Numbers

Discover the diverse perspectives and applications of complex numbers, from historical views and kinematics to analytical, topological, and algebraic insights, breaking the traditional association with the Gaussian plane.

www.physicsforums.com

This is exactly how algebraic field extensions are constructed in algebra.

Seeing the basic statements about $\pm \sqrt{-1}$ I decided to describe the procedure rather than refer to maximal ideals in polynomial rings.