Question

Identify a necessary and sufficient condition for two complex numbers a, b to have a real sum and a real product

1. Either both numbers are conjugated or both numbers are real numbers.

### What are complex numbers?

• Complex numbers are made up of two parts: a real number and an imaginary number.
• Complex numbers serve as the foundation for more complex math, such as algebra.
Consider two complex numbers q and p such that,q=c+id and p=a+ib.
• A.T.Q. PQ is real and p + q real.
Since PQ is real, the imaginary part must be zero, hence bc+ad=0 or:
Then,
• p+q=(a+c)+(b+d)i
Here also b+d must be zero or:
• d=−b
So putting d=−b in bc=−ad, we will get,
• b(c−a)=0
So either,
• b=d=0
• or, a=c,b=−d
So either p and q are conjugated
• p=a+bi,q=a−bi
or both are real
• p=a,q=c
Therefore, either both numbers are conjugated or both numbers are real numbers.
Know more about complex numbers here:
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The correct question is given below:
If the product and the sum of two complex numbers are real, what can we say about the numbers? Prove it.