My question is very simple that is: Is Complex number closed under division? Can we consider this 0+0i as complex numbers? or it is not a complex number.
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$0+0i$ is indeed a complex number and is the only one without a multiplicative inverse. – Darth Geek Aug 11 '14 at 15:42
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So division by this complex number give us not a complex number. Right? – user165385 Aug 11 '14 at 15:43
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Right. Dividing by zero (as a complex number) is undefined, just like for real numbers. – Cameron Williams Aug 11 '14 at 15:43
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Complex numbers are not closed under division. Can I say that? – user165385 Aug 11 '14 at 15:44
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For any nonzero $z=a+bi$, we have a multiplicative inverse $z^{-1}=\frac{a-bi}{a^2+b^2}$. This being the case, you can divide by $z$ by multiplying with $z^{-1}=\frac{a-bi}{a^2+b^2}$.
But the additive identity zero (given by $0+i0$ in this case) cannot have a multiplicative inverse, so you cannot divide by it in this ring, nor in any field.
In this sense, fields (actually even division rings) are as closed as possible under division as you can get.
Added after your question persisted in the comments:
It is not sensible to ask if a set is closed under an operation that isn't even defined on all the set. If there are elements that can't participate in the division, then it is hardly fair to say that the set "isn't closed" under an operation.
The best one can say is that division (where defined) always results in another element of $\Bbb C$. Said another way, the result of a division in $\Bbb C$ will not fall outside of $\Bbb C$. But this is not exactly saying that "$\Bbb C$ is closed under division."
rschwieb
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oh.. nice.. so Complex numbers are closed under division? can you explain it more? – user165385 Aug 11 '14 at 15:50
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2@user165385 You are asking an off-kilter question. It does not make sense to ask if a set is closed under an operation when the operation is not defined for all pairs of elements of the set. The best you can say is that where division is defined, the result is in $\Bbb C$. – rschwieb Aug 11 '14 at 16:27
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$0+0i$ is the same as $0$. $0$ is a complex number. It is the zero element of the field $\mathbb{C}$. In a field the zero element cannot have a multiplicative inverse because the equation $$ 0 \cdot x = 1 $$ cannot not have a solution. This means one cannot divide by $0$.
miracle173
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