Is Zero Considered a Pure Imaginary Number (as 0i)?Date: 12/02/2003 at 15:36:39 From: Andreas Subject: is zero purely imaginary as well as real? Representing complex numbers in the coordinate plane, the horizontal axis is the real axis, the vertical is the pure imaginary axis. They intersect at "0" and "0i". Does that intersection belong to both sets of numbers? Or is it only real, but not pure imaginary? I thought the pure imaginary numbers were numbers that were not real. I find one definition which says they consist of: "all numbers whose squares are negative" which excludes zero. Another says: "all numbers bi, where b is any real number." Date: 12/02/2003 at 16:22:36 From: Doctor Peterson Subject: Re: is zero purely imaginary as well as real? Hi, Andreas. Logically, one could make an argument that 0 is neither real nor imaginary, since is has neither an imaginary nor a real part. But excluding it from either the real or the imaginary axis would be extremely awkward; so we define "purely imaginary" in a negative way, not as a number that HAS only an imaginary part, but as one that DOES NOT have any (non-zero) real part: Purely Imaginary Number http://mathworld.wolfram.com/PurelyImaginaryNumber.html A complex number z is said to be purely imaginary if it has no real part, i.e., R[z] = 0. The term is often used in preference to the simpler "imaginary" in situations where z can in general assume complex values with nonzero real parts, but in a particular case of interest, the real part is identically zero. Therefore, 0 is considered to be a (pure) imaginary number. Your second definition is the better one. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 07/25/2015 at 12:26:42 From: Matthew Subject: What is the definition of a purely real number Above, Doctor Peterson said, "excluding [zero] from either the real or the imaginary axis would be extremely awkward." Why would it be awkward? Much like zero is neither positive nor negative because it sits in the center of the Cartesian plane, why can't it be neither imaginary nor real when it sits in the center of the imaginary plane? According to Wolfram's MathWorld, a purely imaginary number is a complex number z that has no real part, i.e., R[z] = 0. Shouldn't it follow that a purely real number is a complex number z that has no imaginary part, i.e., C[z] = 0? If I apply MathWorld's definition of a purely imaginary number together with my definition of a purely real number, the number zero would be both purely imaginary and purely real. Is it acceptable for a number to be both purely imaginary and purely real at the same time? Why or why not? What is the definition of a purely real number? I tried to search the Internet for a definition of what a purely real number is. I managed to find some with the same definition as I have given, but they did not address the "conflict" with the number zero. Date: 07/25/2015 at 19:52:55 From: Doctor Peterson Subject: Re: What is the definition of a purely real number Hi, Matthew. As Matthew wrote to Dr. Math On 07/25/2015 at 12:26:42 (Eastern Time), > According to Wolfram's MathWorld, a purely imaginary number is a complex > number z that has no real part, i.e., R[z] = 0. Shouldn't it follow that > a purely real number is a complex number z that has no imaginary part, > i.e., C[z] = 0? Yes, in fact this would seem to be obvious, more so than the definition of a pure imaginary number: we all know that 0 is a real number, and "pure real" should not mean anything different. It's interesting that MathWorld does not bother to define "pure real." I presume, again, that this is because, when used, it means nothing different from "real," which of course is very commonly used. (In my experience, "pure" is usually used, rather than "purely," so I'll stick with that.) > I tried to search the Internet for a definition of what a purely real > number is. I managed to find some with the same definition as I have > given, but they did not address the "conflict" with the number zero. I don't see any real "conflict" in saying that zero is both pure imaginary and pure real. Zero is clearly on the real axis, so it is a real number; and it is clearly on the imaginary axis, so it should be a pure imaginary number. It is the intersection of the two. This is not much different from saying that a square is both a rhombus and a rectangle. The awkwardness I referred to is in part due to the fact that in math, we tend to name things inclusively; a word like "irrational," emphasizing what something is NOT, is unusual. The word "imaginary" was initially used for "non-real" numbers (and it is still used that way at times); Gauss introduced the word "complex" for the ENTIRE complex plane, reserving "imaginary" to refer to the ENTIRE imaginary axis. These usages are more consistent with usual mathematical practice (so that we can talk about properties that apply to those entire sets). To require an "imaginary" number to be anything on the axis EXCEPT 0 would mess that up. Because the word "imaginary" historically has two quite different usages (any number that is NOT real, and any number that has no real PART), when we need to refer to the former it is better to say "non-real," and for the latter, "PURE imaginary." Both of these terms are clearer than merely "imaginary." But you will certainly find disagreements about the meaning of "pure imaginary," and people who share your uncertainty about whether zero should be considered both imaginary and real; I found debates about it on various forums, as is true of just about anything that is not commonly taught in an explicit way. I suspect part of the reason is that mathematicians don't need to talk about pure imaginary numbers very often, because they are not closed under any operation except addition (and if we exclude 0, they are not even closed under addition). For that reason also, I don't think it matters much whether we define "pure imaginary" to include or exclude 0. Looking around for definitions from reputable sources, I find this one that agrees with me and MathWorld on the definition, but shows a Venn diagram that does not include 0 (perhaps inadvertently): http://www.mathwords.com/i/imaginary_numbers.htm Imaginary Numbers, Pure Imaginary Numbers Complex numbers with no real part, such as 5i. Wikipedia also currently agrees, though there has been some debate among its contributors on this point: https://en.wikipedia.org/wiki/Imaginary_number An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i^2 = -1. The square of an imaginary number bi is -b^2. For example, 5i is an imaginary number, and its square is -25. Except for 0 (which is both real and imaginary), imaginary numbers produce negative real numbers when squared. ... The term "imaginary number" now means simply a complex number with a real part equal to 0, that is, a number of the form bi. This site gives a definition that implicitly excludes zero, and does not give zero as an example, but that could be an oversight: http://www.regentsprep.org/regents/math/algtrig/ato6/imagineles.htm An imaginary number is a number whose square is negative. But, as I said, this issue is mostly restricted to pedagogy, where pure imaginary numbers are introduced in preparation for the "real" concept of complex numbers. It is not really important how we define imaginary numbers once we have the whole set of complex numbers to discuss. Can you suggest some contexts in which this matters? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 07/26/2015 at 01:11:27 From: Matthew Subject: Thank you (What is the definition of a purely real number) Thank you so much, Doctor Peterson. This was a very enlightening answer. Accepting that the definitions of both a pure imaginary and pure real number were true met with much resistance inside my head. I couldn't wrap my mind around the concept of a number being both pure imaginary and pure real. And when I wasn't able to find a definition in MathWorld for pure real numbers, I assumed that not defining it was done intentionally to avoid the "conflict" with zero. Then with that assumption, I concluded that there can't be a pure real number, because no definition for it was given. I was under the impression that no number could be both real and imaginary. I now understand the mistake of the assumption and conclusion I made. It seems I just have to accept that zero is both pure real and pure imaginary, and there is nothing conflicting about that. I became curious about pure real numbers while I was going over some concepts of electronics for a blog post. In electronics, as I'm sure you know, we use imaginary numbers in impedance -- specifically, in the reactive part of impedance. I remember defining certain loads to be purely reactive and purely resistive. Of course, I immediately equated them with pure imaginary and pure real. I then became curious about the technical definition of both terms because sometimes the technical definition will give more information than just what we commonly know about the terms. Finding a definition for pure imaginary was easy. It was with finding a definition for pure real numbers that caused all of this confusion. Thanks again for the help, Doctor Peterson. Zero is really a very complicated number! Date: 07/26/2015 at 14:22:46 From: Doctor Peterson Subject: Re: Thank you (What is the definition of a purely real number) Hi, Matthew. Zero is indeed complicated! Special cases like this tend to mess up our thinking wherever they show up. Another example that comes to mind is "degenerate conic sections," where an equation almost meets the requirements to be, say, a circle, but turns out to be a single point (x^2 + y^2 = 0). In your application, you are finding that a zero impedance is not really resistive or reactive. In some sense, it is an extreme case of each: you can dial down a resistance until it reaches zero, or you can dial down an inductance until it reaches zero. So it belongs in each "world," yet is not really "of" either. Mathematically, we just like to include such a special case within each for consistency. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/pre |
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