This question made me keep thinking deeply for an hour , but I couldn't get it ? Can someone give me a satisfactory answer to the question .
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where can we find the question? – Dr. Sonnhard Graubner Apr 14 '17 at 07:45
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@Dr.SonnhardGraubner In the title, I believe. – Arthur Apr 14 '17 at 07:48
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what is the smallest possible number? – Dr. Sonnhard Graubner Apr 14 '17 at 07:49
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That's the one. Now we need to figure out what that means. Perhaps whoever wrote the question can help us? Xucel, what do you mean by "the smallest number"? $0$ is quite small, for instance. Is that small enough for you? – Arthur Apr 14 '17 at 07:49
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@xucel As you can see from the other responses, "complex number" has a very specific meaning in mathematics. Did you mean that or just the day to day sense of "complex"? – badjohn Apr 14 '17 at 08:39
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Since you tagged this 'complex numbers', it's important to understand that there is no natural order on the complex numbers in the sense you may know from real numbers. So the usual way of comparing (real) numbers "$x_1 \ge x_2$" has no direct equivalent in the complex numbers.
You can of course compare the sizes of complex numbers: for any complex number $z=a+bi$ you have the modulus $|z| =|a+bi| = \sqrt{a^2+b^2}$ which is of course a real number again. The complex number with the smallest possible modulus is $0 = 0+0i$. It is the only complex number with modulus $0$, for all non-zero $z \in \mathbb{C}$, you have $|z| > 0$.
See also:
Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?
StackTD
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Same first sentence as the other answer. What is the probability of that happening ? – Apr 14 '17 at 07:52
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@A---B Indeed! Although considering the question (and limited context), it's not that unlikely... :-). – StackTD Apr 14 '17 at 07:53
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You seem to define a number as a complex number, which is tricky since there is no natural ordering. What do you mean by small?
We could make our own definitions of course...
Definition (Smaller) A complex number $x$ is said to be "smaller" than another complex number $y$ if and only if $|x|<|y|$.
Definition (Smallest number) A complex number $x$ is said to be "smallest" if we have $|x|\leq|y|$ for all $y\in\mathbb{C}$.
We could perhaps go on to show that the only number which meet these definitions is $x=0+i0$. Hence $0+i0$ is the "smallest" number.
This really just goes back to the ordering of the reals since $|\cdot|$ always produces a real number.
You could define your own meaning of "small" and "smallest" to suite your needs !
See this link also.
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Since you tagged it "complex numbers", let me first note that there's no natural order on the complex numbers, so from that point of view there cannot be a smallest number.
Having said that, one usually considers a complex number as small if its absolute value is small. With that definition, the smallest possible complex number is $0$.
celtschk
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