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B Is 0 real?

  1. Feb 21, 2016 #1
    I was just thinking about this question, and I see 3 possible answers:
    1) 0 is a purely real complex number. This seems to be the most intuitive, however the one problem is that it shows up on the imaginary numberline.
    2) 0 is not real nor imaginary. I understand this one, but I have found one problem with this: the absolute value of a complex number is modeled by the equation |x+yi|=√(x2+y2), where x and y must be real, and either can be equal to 0, and therefore 0 must be real. However, 0=0i, just as 0=-0, and 0 is thought of as neither positive or negative.
    3) 0 is both real and imaginary. I'm leaning towards this one, because it appears on the real and imaginary numberlines (and other degrees of imaginary numberlines), and it can satisfy the absolute value equations as 0 can be thought of as real. I am not sure about this, however, which is why I ask.
    I ruled out just imaginary because it just doesn't make sense at all, but if I'm wrong tell me.
    Thanks!
     
  2. jcsd
  3. Feb 21, 2016 #2

    K41

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    I'm guessing it must be a real number because it obeys all the rules (not sure about the Dedeking-complete rule) which define whether a number is real or not:
    https://en.wikipedia.org/wiki/Real_number#Definition

    It is also a complex number because the only definition of that is that a and b (from a + bi) must be real numbers and 0 is a real number as just discussed.

    So it forms part of the real numbers and it forms part of the complex numbers.

    https://en.wikipedia.org/wiki/0_(number)#Mathematics
     
  4. Feb 21, 2016 #3

    Svein

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    The real numbers are a subset of the complex numbers, therefore it does not matter what you call it.
     
  5. Feb 21, 2016 #4
    Hi

    Here's a quick extract from Wiki - never guaranteed to be accurate, but generally a good start! Be careful when using the word "real" as you'll see from the first line, Real numbers include "types" of number such as natural and rational; and indeed the Complex plane includes the Real numbers!

    Elementary algebra
    The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is a whole number and hence a rational number and a real number (as well as an algebraic number and a complex number).

    The number 0 is neither positive nor negative and appears in the middle of a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors).[39] Zero is, however,even.

    Other branches of mathematics
     
  6. Feb 21, 2016 #5

    PeroK

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    This sort of question is answered by looking at the precise definitions. How would you define the real and imaginary subsets of the complex numbers? The most obvious way would be:

    A real number is a complex number whose imaginary part is 0.
    An imaginary number is a complex number whose real part is 0.

    What does that make 0?
     
  7. Feb 21, 2016 #6
    I am with you on this (3) that 0 can be considered both real and imaginary. And I think the complex plane supports your thought of this; zero is where both real and imaginary axes meet.
     
  8. Feb 21, 2016 #7
    Actually, ##0+0i## shows up on the imaginary number line. This normally isn't such an important distinction, since the two objects are, for some purposes, "the same", but it pays to be pedantic here, since it seems to be causing confusion.

    Zero is real. It is an element of the real numbers. The claim that ##0 = 0i## is technically false, because one is real and one is complex. There is an isomorphism between the real numbers and a set of complex numbers which maps zero to ##0+0i##, but they are not technically "the same".

    The answer, again, is that ##0## is real and ##0+0i## is complex.
     
  9. Feb 21, 2016 #8
    I have thought about that too, but I have heard this definition that supports number 2:
    A real number is a complex number whose imaginary part is 0 and whose real part is not zero.
    An imaginary number is a complex number whose real part is 0 and whose imaginary part is not zero.

    These two conflicting definitions (when it comes to zero) are one of the sources of my confusion, and then there is
     
  10. Feb 21, 2016 #9

    Samy_A

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    Do you have any serious reference for this "definition"?
     
  11. Feb 21, 2016 #10
    I'm pretty sure that technically, both are complex and 0=0i=0+0i.
     
  12. Feb 21, 2016 #11
    The difficulty here is that completely answering your question needs tools that you probably haven't studied yet.
    People say "real numbers are complex numbers with zero imaginary part", but this technically isn't true -- it's just a casual shorthand way of saying "the real numbers are isomorphic to the subfield of complex numbers with zero imaginary part". What this means is that the complex numbers with zero imaginary part behave "just like" the real numbers, and so people often talk about them as if they're "the same". They are not technically the same, though, in the sense that, if you're working with the real numbers, there's not necessarily any notion of complex or imaginary at all; the complex numbers and the real numbers are entirely different sets.

    So, to answer your question, zero is a real number, but the complex numbers contain an element ##0+0i## that behaves "just like" zero.

    As I explain above, this technically isn't true, they they're members of completely different sets.
     
  13. Feb 21, 2016 #12

    PeroK

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    Once you've defined and proved your isomorphism, you can drop the formal distinction. It's not a casual shorthand.
     
  14. Feb 21, 2016 #13
    The distinction is useful here, where the OP is confused about the relationship between real and complex numbers. The p-adic numbers contain a subfield isomorphic to the rationals, but I would still correct a student if they came to class confused because they didn't know if zero was real or rational or complex or p-adic or neither or all at once because of something they read on wikipedia.
     
  15. Feb 21, 2016 #14

    PeroK

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    Without wanting to get into an argument, you now have to explain isomorphisms to the OP, which isn't really necessary to get a feel for complex numbers.
     
  16. Feb 21, 2016 #15

    fresh_42

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    Maybe it can be answered this way:
    There is a neutral element ##0_ℝ## of real addition and a neutral element ##0_ℂ## of complex addition.
    The moment you regard ##ℝ ⊂ ℂ ## as a subfield or field extension you have to identify the two: ##0_ℝ ≡ 0_ℂ##.
     
  17. Feb 21, 2016 #16
    Ok, I'm confused. Is this statement true: 0=0i=0+0i?
     
  18. Feb 21, 2016 #17

    PeroK

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    Yes.
     
  19. Feb 21, 2016 #18
    @Isaac0427 No argument with @PeroK on this but be careful on the question you ask...

    Does 0 = 0i ?
    Yes - if you evaluate both sides, then you get the same answer; and you could substitute 0i for 0 in an equation and get the same answer; etc. It's s similar question to
    Does 4 = (4 + 0i) ? Yes.

    However...
    Is 0 the same as 0i ? Well you're back to your original question!!! No... 0 is a whole number in the set of real numbers; 0i is a complex number.
     
  20. Feb 21, 2016 #19

    PeroK

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    If you really believe that, then are not the whole numbers only isomorphic to a subset of the real numbers? How are you going to denote ##0## the integer, ##0## the rational number and ##0## the real number? It seems to me that you have three more isomorphisms there that you are forgetting about.

    The point is: once you have defined and proved the properties of your isomorphism, you can omit it. You don't have to persevere with the isomorphic relationship. Otherwise, you'd never get clear of the theoretical clutter.
     
  21. Feb 21, 2016 #20

    fresh_42

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    To distinguish between ##ℝ⊂ℂ## and an embedding ##φ : ℝ → ℂ## or ##ℂ ≅ ℝ[x]/(x^2+1)## is more than artificial when dealing with no other properties than pure numbers.
     
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