1.3 Complex Conjugate

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We learn properties of the complex conjugate.

We have seen that the complex conjugate is defined by $\overline {a+bi} = a-bi$

The conjugate of the conjugate is the original complex number:

$\overline {\overline {a+bi}} = \overline {a-bi} = a+bi$

The conjugate of a real number is itself: $\overline {a}=\overline {a+0i} = a-0i = a$

The conjugate of an imaginary number is its negative: $\overline {bi}=\overline {0+bi} = 0-bi = -bi$

1 Real and Imaginary Part

If we add a complex number and it’s conjugate, we get $(a+bi) + (\overline {a+bi}) = 2a = 2\Rep (a+bi)$ Thus, we have a formula for the real part of a complex number in terms of its conjugate: $\Rep (a+bi) = \frac 12 \left [(a+bi) + (\overline {a+bi})\right ]$ Similarly, subtracting the conjugate gives $a+bi - \overline {a+bi} = 2bi,$ and so $\Imp (a+bi) = -\frac {i}{2}\left [(a+bi) + (\overline {a+bi})\right ]$

2 Modulus

If we multiply a complex number by its conjugate, we get the square of the modulus: $(a+bi)(\overline {a+bi}) = (a+bi)(a-bi) = a^2 + b^2 = |a+bi|^2$ Thus, we have a formula for the modulus of a complex number in terms of its conjugate: $|a+bi|= \sqrt {(a+bi)(\overline {a+bi})}$

3 Multiplicative Inverse

For a non-zero complex number, $a+bi$ , its multiplicative inverse is its conjugate divided by the square of its modulus: $\frac {1}{a+bi} = \frac {1}{a+bi}\cdot \frac {\overline {a+bi}}{\overline {a+bi}} = \frac {\overline {a+bi}}{\ |a+bi|^2}$

4 Addition and Multiplication

The conjugate of a sum is the sum of the conjugates: $\overline {(a+bi) + (c+di)} = (a+c) - (b+d)i = \overline {a+bi }+\overline {c+di}$

The conjugate of a product is the product of the conjugates: $\overline {(a+bi) \cdot (c+di)} = \overline {a+bi }\cdot \overline {c+di}$ To see this, we can calculate the left and right sides separately and see that they are the same: $\overline {(a+bi) \cdot (c+di)}= \overline {(ac-bd) + (ad+bc)i} = (ac-bd) - (ad+bc)i$ and $\overline {a+bi }\cdot \overline {c+di} = (a-bi)(c-di) = (ac-bd) + (-ad-bc)i = (ac-bd) - (ad +bc)i$

(problem 1) Prove that the conjugate of a difference is the difference of the conjugates.

(problem 2) Prove that the conjugate of a quotient is the quotient of the conjugates.

Here is a video solution for problem 2:

2024-09-27 14:07:21