# Determine whether H is a subgroup of the complex numbers C with addition H = {a+bi|a,b in R, ab>=0}

Determine whether H is a subgroup of the complex numbers C with addition
$H=\left\{a+bi\mid a,b\in R,ab\ge 0\right\}$
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d2saint0
Clearly, the complex number 0=0+0i in H as a=b=0 and $ab=0\ge 0.$
So, H is a non empty subset of C.
In order for H to be a subgroup of C, H must be closed under addition.
Consider the complex number 1+0i.
For the complex number 1+0i, a=1, b=0 and $ab=0\ge 0$
Hence, $1+0i\in H$
Consider the complex number 0−i.
For the complex number 0−i, a=0, b=−1 and $ab=0\ge 0.$
Hence, 0−i in H.
Now, (1+0i)+(0−i)=(1+0)+(0−1)i=1−i.
For the complex number 1−i, a=1, b=−1 and ab=−1<0.
Hence, $1-i\notin H.$
Thus, $1+0i,0-i\in H$, but their $\sum 1-i!nH.$
Therefore, H is not closed under addition.
So, H is not a subgroup of C under addition.
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