Properties of conjugate
Properties of Conjugate
In the context of complex numbers, the conjugate of a complex number is a number with the same real part and an imaginary part equal in magnitude but opposite in sign. If we have a complex number $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$), then the conjugate of $z$, denoted by $\overline{z}$, is $a - bi$.
Table of Differences and Important Points
| Property | Complex Number $z = a + bi$ | Conjugate $\overline{z} = a - bi$ |
|---|---|---|
| Real Part | $a$ | $a$ |
| Imaginary Part | $b$ | $-b$ |
| Magnitude | $\sqrt{a^2 + b^2}$ | $\sqrt{a^2 + b^2}$ |
| Multiplication with $z$ | $z \cdot z$ is not real | $z \cdot \overline{z}$ is real |
| Addition with $z$ | $z + z = 2a + 2bi$ | $z + \overline{z} = 2a$ (real) |
| Subtraction from $z$ | $z - z = 0$ | $z - \overline{z} = 2bi$ (purely imaginary) |
Formulas Involving Conjugates
Product of a complex number and its conjugate: $$ z \cdot \overline{z} = (a + bi)(a - bi) = a^2 + b^2 $$ This product is always a non-negative real number.
Sum of a complex number and its conjugate: $$ z + \overline{z} = (a + bi) + (a - bi) = 2a $$ This sum is always a real number.
Difference between a complex number and its conjugate: $$ z - \overline{z} = (a + bi) - (a - bi) = 2bi $$ This difference is always a purely imaginary number.
Division involving conjugates: To divide one complex number by another, we multiply the numerator and the denominator by the conjugate of the denominator: $$ \frac{z_1}{z_2} = \frac{z_1}{z_2} \cdot \frac{\overline{z_2}}{\overline{z_2}} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}} $$ This process is known as rationalizing the denominator.
Examples to Explain Important Points
Example 1: Product of a Complex Number and Its Conjugate
Let $z = 3 + 4i$. Find $z \cdot \overline{z}$.
$$ z \cdot \overline{z} = (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 + 16 = 25 $$
The product is a real number, and it is equal to the square of the magnitude of $z$.
Example 2: Sum and Difference of a Complex Number and Its Conjugate
Let $z = 5 - 2i$. Find $z + \overline{z}$ and $z - \overline{z}$.
$$ z + \overline{z} = (5 - 2i) + (5 + 2i) = 10 $$ $$ z - \overline{z} = (5 - 2i) - (5 + 2i) = -4i $$
The sum is a real number, while the difference is a purely imaginary number.
Example 3: Division Involving Conjugates
Divide $z_1 = 1 + i$ by $z_2 = 2 - i$.
$$ \frac{z_1}{z_2} = \frac{1 + i}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{(1 + i)(2 + i)}{(2 - i)(2 + i)} = \frac{2 + i + 2i - 1}{4 + 1} = \frac{1 + 3i}{5} = \frac{1}{5} + \frac{3}{5}i $$
By multiplying the numerator and the denominator by the conjugate of the denominator, we obtain a complex number in standard form as the result of the division.
Understanding the properties of conjugates is essential for performing arithmetic operations with complex numbers and for solving problems in complex analysis and other areas of mathematics where complex numbers are used.