Brief account of the history of 'imaginary numbers'?

[Kyriakos]

As dr. Merkel noted, i am by collective non-virtue lazy, and indeed i thought i should not bother to check around the web (or even worse, books....!) for this info when i can just ask here from the comfort of my chair

I do know that the 'i' came to be due to late medieval identities in algebra inherently leading to a solution involving the root of a negative number (eg the root of -1). But how did this come to be deemed as important? And does it exceed the inherent duality issues here (the + and - duality, while a square root is defined as leading to positive outcome in real parts).

Be rational and help

 

[Borachio]

The answer is naturally complex.

But practically speaking, you couldn't solve the basic equations of electromagnetism without imaginary numbers. And if you didn't, where would we be now?

Really, though, if you can get over negative numbers, and you aren't bothered that multiplying two fractions together gives you a smaller number, then why would you baulk at imaginary numbers?

Maths is just a matter of getting used to stuff, imo.

 

[warpus]

Imaginary numbers are required for many things and lead to many beautiful things as well:

e^{i x} = cos(x) + i sin(x)

1. They fit in quite snuggly into our understanding of mathematics
2. They are required to solve a lot of types of equations completely - Without them we just wouldn't see the full picture of many types of problems and their implications.
3. They are very useful, such as for example making Euler's formula possible (the one I posted). In this case imaginary numbers helped us see a connection between two other concepts.

So basically they're an integral part of math and they're here to stay.

 

[Borachio]

Also, elegantly: e^(i*pi) = -1.

Rude word, xkcd:

Spoiler :

 

[uppi]

Imaginary numbers have their origin in the 16th century, when people noticed that although sqrt(-1) does not appear to make any sense, you can still get useful results if you just continue your calculation.

But the real breakthrough was in the 18th century, when it was discovered that one can greatly simplify calculations by introducing complex analysis.

But practically speaking, you couldn't solve the basic equations of electromagnetism without imaginary numbers. And if you didn't, where would we be now?

I do not see, why you could not solve Maxwell's equations without imaginary numbers. After all, electromagnetic fields are always real, so you can write everything down using sines and cosines. It would be much harder and would involve a much better knowledge of trigonometric functions, but it should be possible.

Quantum mechanics would be much harder to reformulate. There, i already appears in the Schrödinger equation itself.

 

[Perfection]

I own this book which I've never read:

http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004

 

[bhavv]

According to father, India is the greatest country in the world because they invented the 0!!!

I'm like nope, because Mayans.

 

[Borachio]

I own this book which I've never read:

http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004

In idle moments, I sometimes like to indulge myself in Amazon's 1 star customer reviews.

Llibre interessant malgrat tot, si bé l'autor utilitza un nivell massa alt de matemàtiques, la qual cosa fa que la comprensió de l'obra estigui únicament a l'abast de persones amb una formació específica en aquesta matèria. No recomanable per a lectors sense formació superior en matemàtiques.

I thought this volume was going to be an entertaining, but stimulating, history and development of i--well, I gave up in anger and frustration after the first three chapters.

Mr. Nahin suffers from two of the maladies of many writers of math textbooks, the inability to express himself clearly to the intended reader and the inability to separate the chaff from the wheat. Chapters 2 and 3 are illustrative of the author's shortcomings, especially the latter on the geometric interpretation of i as developed by Wessel. Here, vector addition is clearly explained both in words and by a diagram, but vector multiplication and how Wessel derived it (actually the heart of the matter) is not. We are forced to try to figure it out based on the author's verbal explanation with no examples (and I mean simple and meaningful ones) given. The reader feels the full force of the inadequate presentation when the author tries to build on concepts which he has not clearly enunciated in the first place.

Negative comments, and in a foreign language, are the only ones to trust, imo. 5 star reviews are usually written either by the author or his friends.

 

[Perfection]

Well, there's a reason I didn't read it.

 

[Agent327]

So far no answer then. Googling might have been faster. Y'know, from the comfort of your chair.

 

[Rashiminos]

two of the maladies of many writers of math textbooks

This is where I begin to wonder whether the reader/student is misplacing the blame.

Robert Allen says (regarding comments pertaining to another review):
At this point, I think it only fair to inform you that I am not a student, but rather someone trying to learn math on his own--and the little I have learned is despite the textbooks, not because of them, the same with the respect in which I hold the discipline.

Yep.

 

[Kyriakos]

^Math autodidacts suck, just look at useless Fermat

 

[Leoreth]

According to father, India is the greatest country in the world because they invented the 0!!!

I'm like nope, because Mayans.

I don't understand. As far as I'm aware, the Mayans did not use positional number systems.

 

[Rashiminos]

^Math autodidacts suck, just look at useless Fermat

Fermat was more than willing to engage the texts available in his time, plus his marginal comments are famously not always clear . Even so, many modern math textbooks assume a classroom setting.

 

[Kyriakos]

Fermat was more than willing to engage the texts available in his time, plus his marginal comments are famously not always clear.

If the criterion is who is willing to engage concurrent texts, then Gauss fails.

/Abeltroll

 

[Borachio]

Even so, many modern math textbooks assume a classroom setting.

This is certainly true. It's not immediately obvious to me why it should be so, though. There must surely be a market out there for maths books which are clearly written, and represent more than just lecture notes for a lecturer.

 

[Kyriakos]

^ Yes. +i

Infact the most boring thing (for me) in reading math stuff is that i have to lose time to examine what the obscure symbols mean. It would be far better if there were categories of math concepts tied to each other in any significant ways up to now, instead of an endless sea of symbols and catch-words.

Btw, from reading a couple of university papers by graduating students, it also seems that in that setting the math graduate-to-be still has to name the history of the notions he/she is using, at least up to a level. So it is only next to us downtrodden autodidacts that monocles are worn by the math-uni crowd

 

[warpus]

Infact the most boring thing (for me) in reading math stuff is that i have to lose time to examine what the obscure symbols mean.

Isn't that like someone who has little understanding of world history reading world history textbooks, every once in a while having to look up historical figures because he/she doesn't know who they are?

You can't just dive into math and expect for everything to make sense, mate. (for some reason "mate" works here, I'm going with it)

 

[bhavv]

I don't understand. As far as I'm aware, the Mayans did not use positional number systems.

http://www.scientificamerican.com/article/what-is-the-origin-of-zer/

History of the zero:

Babylon > Mayans > India > Cambodia > China > Arabia > Europe.

 

[Leoreth]

Yeah, looks like I was wrong. Interesting.