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

Question
Complex numbers
asked 2020-11-08
Determine whether H is a subgroup of the complex numbers C with addition
\(\displaystyle{H}={\left\lbrace{a}+{b}{i}{\mid}{a},{b}\in{R},{a}{b}\ge{0}\right\rbrace}\)

Answers (1)

2020-11-09
Clearly, the complex number 0=0+0i in H as a=b=0 and \(\displaystyle{a}{b}={0}\ge{0}.\)
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 \(\displaystyle{a}{b}={0}\ge{0}\)
Hence, \(\displaystyle{1}+{0}{i}\in{H}\)
Consider the complex number 0−i.
For the complex number 0−i, a=0, b=−1 and \(\displaystyle{a}{b}={0}\ge{0}.\)
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
Hence, \(\displaystyle{1}−{i}\notin{H}.\)
Thus, \(\displaystyle{1}+{0}{i},{0}−{i}\in{H}\), but their \(\displaystyle\sum{1}−{i}!{n}{H}.\)
Therefore, H is not closed under addition.
So, H is not a subgroup of C under addition.
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