The Conjugate of a Complex Number
In the realm of complex numbers, a number's conjugate is derived by simply switching the sign of its imaginary component. If you start with a complex number represented as z = a + bi, its conjugate will be z' = a - bi. $$ z' = a-bi $$
To elaborate, a complex number is traditionally written in algebraic form as a + bi, where "a" and "b" are real numbers, and "i" represents the imaginary unit, defined by the condition i2 = -1.
Let's consider z = a + bi.
$$ z = a+bi $$
The conjugate of this complex number will be z' = a - bi.
$$ z' = a-bi $$
This is the complex conjugate of complex number z.
Of course, this relationship is reciprocal. That is to say, if z' is the conjugate of z, then it follows that z is the conjugate of z'. For example, if you have a complex number z = a - bi $$ z = a-bi $$ its conjugate would then be z' = a + bi $$ z' = a+bi $$ This results in a complex number with the identical real part but an inverted imaginary part.
Example
Consider the complex number z=3 + 4i
$$ z = 3+4i $$
To obtain its conjugate, all you need to do is flip the sign of the imaginary part (4i), yielding the conjugate of z = 3 + 4i, which is the complex number z' = 3 - 4i.
$$ z' = 3+4i $$
Thus, the conjugate of z=3+4i is the complex number z'=3-4i
So, why exactly is this called a complex conjugate? The answer lies in its geometric representation. The conjugate of a complex number bears a symmetry relationship to the original complex number, reflected along the real axis in the complex plane. If you plot the complex number z = 3 + 4i and its conjugate z' = 3 - 4i on an Argand-Gauss diagram—a Cartesian plane where the horizontal axis stands for the real part and the vertical axis for the imaginary part—you'll find the two numbers are mirror images about the real axis. Hence, the term "conjugate" encapsulates this symmetry.
Characteristics of Conjugate Complex Numbers
Conjugate complex numbers come with some properties:
- The sum of a complex number and its conjugate yields a real number
$$ z+z' = 2a $$
Therefore, for a complex number z = a + bi and its conjugate z' = a - bi, the sum z + z' equals 2a, which is a real number. $$ z+z' = a+bi + a - bi = 2a $$ - The difference between a complex number and its conjugate is an imaginary number
$$ z-z' = 2bi $$
The difference between a complex number z = a+bi and its conjugate z' = a - bi is an imaginary number: z - z' = 2bi. $$ z-z' = a+bi - (a - bi) = 2bi $$ - The product of a complex number and its conjugate is a real number
$$ z \cdot z' = a^2+b^2 $$
Multiplying a complex number z = a + bi and its conjugate z' = a - bi produces a real number z·z'=a2+b2 $$ z \cdot z' = (a+bi) \cdot (a - bi) = a^2-abi+abi-b^2i^2 = a^2 - b^2 i^2 $$ You already know that the square of the imaginary unit is i2=-1 $$ z \cdot z' = a^2 - b^2 i^2 = a^2 - b^2 \cdot (-1) = a^2+b^2 $$
What's the Purpose of a Complex Conjugate?
The conjugate of a complex number is incredibly useful in mathematics and engineering because of its various practical applications.
For example, it simplifies the division between two complex numbers.
When you need to divide two complex numbers in algebraic form z1/z2, you multiply the numerator and the denominator by the conjugate of the denominator (z'2).
$$ \frac{z_1}{z_2} = \frac{z_1}{z_2} \cdot \frac{z'_2}{z'_2} $$
This action turns the denominator into a real number, simplifying the calculation.
Example. Consider the two complex numbers $$ z_1 = 4+5i $$ $$ z_2 = 2+3i $$ Now calculate the division between the two numbers $$ \frac{z_1}{z_2} = \frac{4+5i}{2+3i} $$ To simplify the calculation, multiply the numerator and the denominator by the conjugate of the denominator z'=2-3i $$ \frac{z_1}{z_2} = \frac{4+5i}{2+3i} \cdot \frac{2-3i}{2-3i} $$ Then carry out the multiplications at the numerator and the denominator using the traditional rules of algebra $$ \frac{z_1}{z_2} = \frac{8-12i+10i-15i^2}{4-6i+6i-9i^2} $$ $$ \frac{z_1}{z_2} = \frac{8-2i-15i^2}{4-9i^2} $$ You already know that the square of the imaginary unit i2=-1 equals negative one. $$ \frac{z_1}{z_2} = \frac{8-2i-15 \cdot (-1)}{4-9 \cdot (-1)} $$ $$ \frac{z_1}{z_2} = \frac{8-2i+15}{4+9} $$ $$ \frac{z_1}{z_2} = \frac{23-2i}{13} $$ $$ \frac{z_1}{z_2} = \frac{23}{13} - \frac{2}{13} i $$ The final result is the quotient of the division between the two complex numbers.
Also, the conjugate of a complex number allows you to calculate the modulus (or absolute value) of the complex number.
According to the Pythagorean theorem, the modulus |z| of a complex number z = a + bi is given by the following formula:
$$ |z| = \sqrt{a^2 + b^2} $$
In turn, the sum a²+b² is equal to the product z·z' between the complex number z and its conjugate z'
This is one of the properties of complex conjugates.
$$ |z| = \sqrt{a^2 + b^2} = \sqrt{ z \cdot z' } $$
Therefore, you can also obtain the modulus of a complex number z by calculating the square root of the product of the complex number z and its conjugate z'.
Example. Consider the complex number $$ z_1 = 4+5i $$ To get the modulus of the complex number, calculate the square root of the product between the complex number z1=4+5i and its conjugate z1'=4-5i. $$ |z_1| = \sqrt{z_1 \cdot z_2} $$ $$ |z_1| = \sqrt{(4+5i) \cdot (4-5i)} $$ $$ |z_1| = \sqrt{16 - 20i + 20i +25i^2} $$ $$ |z_1| = \sqrt{16 - 25i^2} $$ The square of the imaginary unit is i2=-1 $$ |z_1| = \sqrt{16 - 25 \cdot -1 } = \sqrt{16+25} = \sqrt{41} $$ The final result is the modulus of the complex number z1=4+5i. It is the same result you would obtain using the Pythagorean theorem, where a=4 and b=5 are the coefficients of the real and imaginary parts of the complex number z1=4+5i. $$ |z_1| = \sqrt{a^2 + b^2} = \sqrt{4^2 + 5^2} = \sqrt{16+25} = \sqrt{41} $$
Lastly, here are a few more practical uses of conjugate complex numbers:
- Multiplying a complex number by its conjugate yields a real number. This characteristic proves to be useful whenever you wish to eliminate the imaginary component of an expression.
- In the field of electrical engineering and control systems, complex conjugates are utilized for analyzing transfer functions and system frequency responses.
- In quantum physics, the complex conjugate plays a pivotal role in computing probabilities based on the wave function of a system.
