Squaring Cures All That Ails
The square of any nonzero number is always positive.
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5 solutions
This is a very misleading puzzle, since complex numbers are not endowed with an order relation. I think is an ill posed question.
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By defination what is a number? Google it, you will find that complex numbers are numbers.
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Yes they are, but only the subset of real numbers is defined with an order relation, hence if x is supposed to satisfy some order condition you have to assume x to be real. Otherwise you might get nonsense.
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@Juan García – we cannot compare non--rreal numbers(if they are greater or not). i agree. but we can see if the are equal, i.e i = i
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@Aareyan Manzoor – Yeah, but how you define a positive number in the complex plane???? You need an order relation, which only holds in the real line.
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@Juan García – We dont, however squaring some un-real numbers yield real numbers which can be positive or negative. We cannot have i < 0 but − 1 < 0
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@Aareyan Manzoor – Yes I agree. But that kind of relation only holds for a very small subset of complex numbers. If someone asks you something that relates a property that the elements of a given set could satisfy, you would expect that the property is well defined over the whole set. Even though what you are saying is right, the question is not properly asked. That's my point.
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@Juan García – Not really. Like:
Are squares of all real numbers positive?
ans: no, 0 2 = 0 which is not positive
so actually if you have 1 counter example then the statement is proven wrong.
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@Aareyan Manzoor – This was the question "The square of any nonzero number is always positive."
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@Juan García – This was an example to make you understand that 1 counterexample is enough.
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@Aareyan Manzoor – yeah but the counterexample belongs to the set where the property is well defined.
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@Juan García – Lets see the set of all the imaginary (pure) numbers, there squares will always be negative. So it is well defined to compare a square of pure imaginary number with real numbers.
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@Aareyan Manzoor – Let me put it more explicitly then. Let x be in R, for all element in R we can decide if it is positive or negative, hence makes sense to ask the question over this set. The question is well posed. Now I can only talk about positivity of a reduced set of numbers in the complex plane. So asking a question about the square of a number x in the complex plane, is not formulate right. Is (1+2i)^2 positive or negative? That's my point. The question was poorly formulated.
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@Juan García – purely imaginary means numbers of form b i for b real. We xan also always create a set s e t = { x ∣ x 2 < 0 } and then say that "for elements in set, the statement is false""
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@Aareyan Manzoor – Yes we can and if you give that piece of information that would be a much more coherent question than the one asked in this exercise.
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@Juan García – The question is fine, really, since all these are subset of "numbers"
Bro,these are imaginary numbers they doesn't really exist.
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That's why it says "number" and not "real number"
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So, then please enlighten me about what ''positive'' means for complex numbers? My point is that the use of that word implies using either reals or a subset thereof, making the solution to this question ''true''.
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@Thomas Wasserman – Well, i^2 is equal to -1, which clearly is not positive
@Thomas Wasserman – A complex number with a power of the multiply of 4 would be positive, eg. i^4=1
@Thomas Wasserman – 100% agree. Maybe say "numbers with magnitude greater than 0" because magnitude is something all numbers are endowed with.
@Abhishek Shukla Complex numbers exist, and they are numbers, hence they apply.
This doesn't make sense. You have to define the order: in an ordered field, no square can be negative. As it turns out, the field of complex numbers has no order that turns it into an ordered field.
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no, an ordering is not necessary for this question. you can't say IN GENERAL when a complex number is greater or not than something else, but here, squaring i makes it real and thus, we can talk about inequalities. What is important is that x^2 can be ordered, not necessarily just x.
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I see your point, but this is not a very well posed question. Since there is no clear definition of which "order" is being used. In the complex field, we cannot compare numbers, only for reals. It doesn't matter if the complex number becomes real after squaring it. See the book Principles of Mathematical Analysis, of Walter Rudin, I think it is more clear than my explanation. =D
This question is legit. It says "any nonzero number", which includes both real numbers and imaginary numbers. It doesn't limit it to only real numbers. Imaginary numbers exist and there's no escaping from it. It's part of math.
Edit: Whoever downvoted my comment should indicate and explain where I could be wrong at, not just downvoting and running away. You can't learn math this way.
a^2 is negative when a is imaginary. When a is complex the result is complex (neither positive nor negative) as (c+bi)^2 will always have an imaginary term when b not equal to 0.
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And the result of complex (c+bi)^2 is imaginary when c=b. Still non-positive.
The defination of a complex number is x = a + b i where a,b are real . that means both x = 1 , x = i are complex numbers. @Owen Berendes
At first glance you would say that any number squared is positive. However if you dig deeper you will remember complex numbers aka imaginary numberes. i^2= -1
Square of complex numbers is negative .
1+i is a complex number... its square is 2i. That isn't negative.
Complex numbers are exception
I wonder if you could also consider Z n 2 and choose n . So for example: 2 ∈ Z 4 gives 0 when squared in Z 4 ... Kind of begs the question as to what you consider a number I guess...
a 2 is negative for a complex value of a .
For eg; i 2 = − 1 ,where i = − 1 .