Question

What is a complex conjugate?

Answer 1

Hint: This is a subjective question based on the concepts of complex numbers. The basic topics like the real and imaginary part of the complex number should be sound and clear. One should be able to understand these concepts for answering this particular question.

Complete step-by-step answer:
 Let us start this question,
let a and b be real numbers, and the standard value \[i = \sqrt { - 1} \].
Now, the complex number formed by it is given as
\[ \Rightarrow a + ib\]
After observing the complex number we can find the complex conjugate by simply reversing the sign of the imaginary part as shown below,
\[ \Rightarrow a - ib\]
Now, sum properties of the complex conjugate are given below,
The sum of the complex number and the complex conjugate gives
\[ \Rightarrow a + ib + a - ib\]
\[ \Rightarrow 2a\]
Or, the sum of the complex number and its conjugate gives twice the real part.
Now, similarly subtracting the two complexes to obtain
\[ \Rightarrow (a + ib) - (a - ib)\]
\[ \Rightarrow a + ib - a + ib\]
\[ \Rightarrow 2ib\]
Or, the difference of the complex and its conjugate gives twice the imaginary part.
Now, let us multiply the two complex numbers
\[ \Rightarrow (a + ib) \times (a - ib)\]
\[ \Rightarrow {a^2} - {b^2}\]
Clearly, the multiplication of the two complexes gives us a real number.
Now, let us try to divide the two complexes to obtain the equation given below,
\[ \Rightarrow \dfrac{{(a + ib)}}{{(a - ib)}}\]
Rationalising the denominator we get,
\[ \Rightarrow \dfrac{{(a + ib)}}{{(a - ib)}} \times \dfrac{{(a + ib)}}{{(a + ib)}}\]
\[ \Rightarrow \dfrac{{{a^2} - {b^2} + 2iab}}{{{a^2} + {b^2}}}\]
Now, finally a complex number is obtained.

Note: This concept of complex and its conjugate can be applied to quadratic equations whose discriminant is less than zero. Then the two roots of the equation can be easily calculated or some other analysis can also be drawn.