Talk:Imaginary unit

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Origin of term "imaginary"[edit]

The last sentence of the first paragraph reads 'The term "imaginary" is used because there is no real number having a negative square.' I think that is wrong; Gauss coined the term "real" after "imaginary" had been coined and in contradistinction to it. Maybe. Others will know better than I. 81.131.23.34 (talk) 16:08, 27 July 2012 (UTC)

Constant not variable[edit]

In this encyclopedia all variables are italicized. But numerals and the constant e are not rendered in the italic font. Why is the imaginary unit written with italics ? Rgdboer (talk) 19:49, 25 September 2012 (UTC)

This has been discussed several times in the past. See e.g. Talk:Complex number/Archive 1#The non-italic writing style of the imaginary unit. And Wikipedia:Manual of Style (mathematics)#Choice of type style currently says
“Special care is needed with subscripted labels to distinguish the purpose of the subscript (as this is a common error): variables and constants in subscripts should be italic, while textual labels should be in normal text font (Roman, upright). … On the other hand, for the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font … Some authors prefer to use an upright (Roman) font for dx, and Roman boldface for i. Both forms are correct; what is most important is consistency within an article. It is considered inappropriate for an editor to go through articles doing mass changes from one style to another. This is much the same principle as the guidelines in the Manual of Style for the colour/color spelling choice, etc.”
So it basically comes down to convention. — Tobias Bergemann (talk) 07:23, 26 September 2012 (UTC)

Thank you for the clarification on convention, consistency, and practice.Rgdboer (talk) 20:58, 26 September 2012 (UTC)

Note that ISO 80000-2 mathematical standards code now specifies that i is a constant, not to be rendered in italics.Rgdboer (talk) 00:36, 24 September 2013 (UTC)

Since this discussion concerns many mathematics articles, it might be a good idea to raise this point at Wikipedia talk:Manual of Style/Mathematics rather than here. Isheden (talk) 18:34, 25 September 2013 (UTC)
Agreed. However, Tobias's comment still stands: italic constants are the predominant convention in WP. And IMO the distinction between a constant and a variable is not needed, whereas distinguishing between text/name and a variable/constant in WP is much more useful. — Quondum 21:28, 25 September 2013 (UTC)

Alternative notations - PROBLEM[edit]

Regarding this statement:

  • Some sources define[citation needed] j = −i, in particular with regard to traveling waves (e.g., a right traveling plane wave in the x direction) e^{ i (kx - \omega t)} = e^{ j (\omega t-kx)} \,.

This is not accurate: no one defines j = −i. What does arise is the need to have a complex sinusoid reference for use with the phasor representation of time-harmonic quantities (e.g., voltages, currents, signals, fields, etc.). Engineering-oriented sources tend to use the complex sinusoid e^{ j \omega t}, while physics-oriented sources tend to use e^{ -i \omega t}. However, in no way does this imply that j = −i.

i and −i[edit]

The section "i and −i" discusses the interchangeability of +i and −i, noting that both are roots of −1, and can be used interchangeably without loss of generality. One root is arbitrarily deemed the "positive root", +i, and the other the "negative root", −i. However, while it's useful to note that the two roots are algebraically equivalent, I think it's a little confusing to mention "positive" and "negative" in this context. Simply declaring i as a positive root of −1 and −i as its additive inverse is probably sufficient. I'm a bit rusty on the details of complex arithmetic; can it be said that −i < 0 < +i , or are pure imaginary numbers not ordered in this way? (This article seems to say they are not.) If they cannot, then it's not possible for one root to be "more positive" or "more negative" than the other (they are simply additive inverses of each other), and it's just confusing for the reader. — Loadmaster (talk) 21:34, 2 December 2013 (UTC)

I don't see any problems with the passage as is. It's longer than it needs to be for some perhaps but it covers the ground pretty well. On 'positive' and 'negative' I think that it's clear that these are in a sense labels. You label one of the roots positive +i and the other negative −i, but you could have done it the other way around. In practice as soon as you write down the roots you are making one positive and one negative, because every method we have to represent imaginary numbers makes that distinction. But the distinction is arbitrary and the 'positive' and 'negative' are just convenient labels.--JohnBlackburnewordsdeeds 21:57, 2 December 2013 (UTC)
I understand the inherent interchangability of +i and −i, but my point is that neither one is actually a negative or positive value, even though they are additive inverses of each other. Yes, −i is the "negative" value of i, but it is not actually a negative value itself (in the sense of being comparable to, or ordered with respect to, zero). This is what you mean by "sense label", right? Perhaps we could explicitly mention this fact in that section, and perhaps also tighten up the verbiage a bit. — Loadmaster (talk) 23:13, 2 December 2013 (UTC)
The equation x^2 = -1 has no solution in the field of the real numbers but two distinct solutions in the field of complex numbers, namely (0, 1) and (0, -1). Since complex numbers have no natural ordering, it is not meaningful to speak of positive and negative complex numbers or saying that one is greater than the other. If a pure imaginary number is defined as a complex number with real part 0, then the same applies to imaginary numbers. In contrast, the imaginary part of the complex number may be positive or negative (or zero). With -i as the imaginary unit a complex number (a, b) would be denoted (a, -b) instead, i.e. the sign of the imaginary part changes. Isheden (talk) 10:14, 3 December 2013 (UTC)
I think that Loadmaster has a very valid point. JohnBlackburne says that "it's clear that these are in a sense labels", and yes, it is clear to me, and probably to any competent mathematician, but it is not necessarily clear to all readers of Wikipedia, the vast majority of whom do not have an advanced knowledge of mathematics. I also think it is a fundamental mistake to say that by writing one root as i (or even +i) and the other as −i "you are making one positive and one negative". If x is -2, then -x is not negative, and it is not true that by writing it with a minus sign we are "making it negative". We are labelling it as the negative of another number that we call x, but that is a different matter altogether, using a different meaning of "negative. In exactly the same way, writing one root as -i labels it as the negative of i, but does not label it as negative. Having spent decades teaching mathematics, I know damned well that this sort of thing does confuse many people, and the fact that "negative" is being used in two different meanings is not by any means clear to everyone. The whole point of the relevant section of the article, as I read it, is to explain that i and -i are not respectively positive and negative, and using those two words in this context (as, for example, referring to one root as "the positive i" is likely to mislead or confuse people with a limited knowledge of the subject. (And let us remember that it is people with a limited knowledge of the subject who are likely to be reading this article to try to find out about it. I doubt that any graduate mathematician has ever referred to Wikipedia to find out whether -i is negative or not.) The very fact that an editor who is clearly intelligent and articulate, and evidently has some understanding of the topic, can ask the questions that Loadmaster has asked, and can express the opinion that "it's just confusing for the reader", is proof that it is confusing for at least one reader, and I have not the slightest doubt that the same will be true of many other readers. Like very many Wikipedia articles on mathematical topics, this one suffers from being written too much from a mathematician's point of view, and those of us who are mathematicians should always take seriously any statement from any non-mathematician that an article is confusing. I have changed the wording of the article, in ways that I hope have helped to address the problems. Finally, Loadmaster, in case it is not already clear from the comments above, the answer to your question "can it be said that −i < 0 < +i , or are pure imaginary numbers not ordered in this way? " is "No, pure imaginary numbers are not ordered in this way." In fact, the concepts "less than" and greater than" have no meaning in complex numbers that are not purely real, just as the concepts of a complex negative being positive or negative have no meaning. JamesBWatson (talk) 11:18, 3 December 2013 (UTC)
Lest any confusion arises about my motives, let me make clear that I understand complex algebra fairly well, having a degree in Mathematics. However, my vocation is computer programming, so there are quite a few topics in math that I have not dealt with in detail for some time. I realized (remembered) after I posted my question that pure imaginary numbers are not ordered. I don't want to convey that I was confused as a reader, but I think that other less competent readers may get the wrong idea about "negative" imaginary numbers. We need to make it clear that i and −i are arithmetic inverses of each other, but that neither is more "positive" or "negative" than the other because they are not reals. Like you say, we're using the term "negative" for two different meanings here. We might also mention that i could be considered to be "larger" than zero, at least in the restricted sense that it has a non-zero magnitude (although this might open a whole different can of worms). — Loadmaster (talk) 20:39, 3 December 2013 (UTC)
Right. I had taken what you said as meaning that you yourself were confused by the article, but the essential point is that I am sure you are right in believing that many less knowledgeable readers might be misled or confused by it. The sense in which you say that i could be considered to be "larger" than zero is of course perfectly valid, but I don't think it would help to put that in the article. "Larger than" different from "greater than" would be just one more issue to confuse non-mathematical readers, and besides there are already more than enough different ways of describing this concept: having a greater modulus, a grater magnitude, a greater absolute value ... JamesBWatson (talk) 09:54, 4 December 2013 (UTC)
It seems the polar form of the imaginary unit is nowhere mentioned in the article. How about mentioning that the absolute value and argument are 1 and π/2, respectively, in the Properties section? Then it would be clear that its magnitude is larger than zero (in fact a unit, as implied by its name). Isheden (talk) 10:05, 4 December 2013 (UTC)
I added a paragraph mentioning the complex and polar forms in the "Definition" section. Feel free to improve on my text. — Loadmaster (talk) 22:04, 4 December 2013 (UTC)

Parse Errors need to be fixed !![edit]

Several of the equations in the article are rendering as 'failed to parse unknown function ...' Otherwise, it's an excellent discussion.50.72.137.1 (talk) 08:03, 9 February 2014 (UTC)

It looks fine now. There was a problem with the WP software, see Wikipedia talk:WikiProject Mathematics#Problem with multiline equations but that has now been fixed.--JohnBlackburnewordsdeeds 08:14, 9 February 2014 (UTC)

Uses[edit]

There is nothing in this article demonstrating how i is useful. It mentions it can be treated as an unknown value and removed but that's it. How is it used in engineering etc.. There needs to be actual practical applications of the use of i in the article.