state the condition under which the product of two complex number is purely imaginary.
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The sum of two complex numbers is a pure imaginary number when either of the two conditions happen:
The real parts of two complex numbers are both zero.
The real parts of two complex numbers have the same absolute value but have different signs.
In other words, if
z1=a1+ib1
z2=a2+ib2
then given that z1+z2=0+ic
⟹(a1+a2)+i(b1+b2)=0+ic
⟹(a1+a2)=0
⟹a1=−a2 or a1=a2=0
The real parts of two complex numbers are both zero.
The real parts of two complex numbers have the same absolute value but have different signs.
In other words, if
z1=a1+ib1
z2=a2+ib2
then given that z1+z2=0+ic
⟹(a1+a2)+i(b1+b2)=0+ic
⟹(a1+a2)=0
⟹a1=−a2 or a1=a2=0
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