Complex Conjugate

Every complex number has a complex conjugate, where the real part remains the same, but the imaginary part has the opposite sign.

The formula we use for a complex number is \(a + bi,\) where:
 

• 'a' and 'b' are real numbers
   
• 'i' is an imaginary number equal to the square root of – 1
   
• 'a' is the real part
   
• 'b' is the imaginary part
   

The complex conjugate of \(a + bi\) is \(a – bi,\) where 'a' is the real part and the imaginary part is –b.

The conjugate of a complex number can be mathematically denoted as z. Since z and z are each other’s conjugates, they are together called a complex-conjugate pair.

Consider \(z = x + iy \) as a complex number, so the conjugate of z can be represented as:

\(z = x – iy\)

The relationship between z and z shows us that the conjugate is found by simply changing the sign of the imaginary part.

In geometry, the conjugate of a complex number z is its reflected or mirrored version across the x-axis in the complex plane, also called the Argand plane, as shown below:

Multiplication of complex number with its conjugate
 

Multiplying a complex number by its complex conjugate always results in a real number. 

For example,

We multiply the complex number \(a + bi\) by its conjugate \(a – bi\)

\((a + bi)(a – bi) = a² – (b²)(i²) \\[1em] (a + bi)(a – bi) = a² + b²\)

The complex conjugate of a matrix A with complex entries is a matrix in which each element is replaced by its complex conjugate. 

For example, 

Let us consider the row matrix \(A=[2+3i, 5−4i, 1+7i].\) 

Its complex conjugate is \(B=[2−3i, 5+4i, 1−7i]\)

Here, each entry in B is the conjugate of the corresponding entry in A. 

The complex conjugate of A is denoted by \(\overline{A}\),

Therefore,  \(B = \overline{A} \).

This demonstrates how we can find the complex conjugate for any matrix with complex entries.

If z, \(z_1,\) and \(z_2\) are complex numbers, their properties are as follows:

• When the conjugate of a complex number is equal to the number itself, it indicates the number is real. We represent it as: \(z = \bar{z}.\) 
  
  For example: \((5 + 0i) = (5 – 0i) = 5.\)
   
• If the sum of a complex number and its conjugate is zero, the number is considered to be imaginary. This can be mathematically expressed as: \((z + \bar{z} = 0).\)
  
  For example: the conjugate of \((0 – 5i) = (0 + 5i) = 5i\)
   
• \(\overline{(\bar{z})} = z\)
   
• \(z +  \bar{z} = 2 \ \text{Re} \ (z)\)
   
• \(z – \bar{z} = 2i \cdot{ } \text{Im}(z)\)
   
• \( z \cdot \overline{z} = (\text{Re}(z))^2 + (\text{Im}(z))^2 \)
   
• \(\overline{(z_1 + z_2)} = \bar{z_1} + \bar{z_2}\)
   
• \(\overline{(z_1 - z_2)} = \bar{z_1}- \bar{z_2}\)
   
• \(\overline{(z_1 \times z_2)} = \bar{z_1}\cdot{}\bar{z_2} \)
   
• \( z = \frac{z_1}{z_2},\quad \rightarrow \quad \bar{z}= \frac{\bar{z_1}}{\bar{z_2}};\ z_2 \neq 0 \)
   

Keep in mind that the conjugate of a complex number can only be found if the complex number is in the form:

\(z = (x + iy)\)

If not, convert it to this form before finding the conjugate.

The complex conjugate root theorem states that if p(x) is a polynomial with real coefficients and \(a + bi\) as its root, then \(a – bi\) is also a root of the polynomial.

Let the polynomial be p(x), where,

\(p(x)=a_n x^n + a_{n-1}x^{n-1}+\cdots+a_1 x + a_0\)

All the coefficients are real, \(p(r) = 0.\)
 

Let’s take the complex conjugates of the equation \(p(r) = 0,\)
 

\(\text {Using} \ \overline{z + w} = \overline{z} + \overline{w}, \ \overline{zw} = \overline{z}\,\overline{w}, \ \text{and} \ \overline{a_k} = a_k, \text{we get},\)

\(0=\overline{p(r)}\\[1em] 0  =\overline{a_n r^n + a_{n-1} r^{\,n-1} + \cdots + a_1 r + a_0}\\[1em] 0  = a_n \overline{r}^n    + a_{n-1}\overline{r}^{\,n-1}    + \cdots    + a_1\overline{r}    + a_0\\[1em]  0 = p(\overline{r}).\)

Therefore, \(p(r) = 0.\)

A complex conjugate is an important concept in complex numbers. Here are some tips and tricks to help learners master complex conjugates.
 

• Teachers can start the class by teaching learners that a complex number is z = a + bi, and then stating that its complex conjugate is z = a - bi. Ask them to keep the real part, and flip the sign of the imaginary part.
   
• Teachers can use the complex plane to show learners the mirror image. Ask students to plot points like 2+3i and 2-3i on the Argand plane. This shows them that the conjugate is the reflection across the x-axis. And that the real part stays the same, but it is the imaginary part that becomes the opposite.
   
• Learners can use color coding to highlight parts. We can use two colors, such as red and blue, to highlight the real and imaginary parts, respectively. While keeping the blue number, change the sign of the red number.
   
• Parents can help learners by connecting the concept with real-life analogies. Use some simple analogy like, “think of a conjugate as undoing the imaginary twist.”
   
• Learners can gain more knowledge by trying to explain their thinking to the teachers. Explain how you get the conjugate of any number. Or explain why multiplying by the conjugate helps.

Complex conjugates play a vital role in different fields. Students must understand how complex conjugates simplify expressions. Here are some applications:

• Complex conjugates are widely utilized in studying alternating current circuits. For example, engineers apply complex conjugate in calculating impedance, current, and voltage.
   
• They are used in aerodynamics for structuring airplane wings and analyzing the airflow around objects.
   
• In the medical field, they enhance image quality and assist in detecting medical conditions in MRI scans.
   
• We use complex conjugates to analyze sound waves and other mechanical vibrations. 
   
• They are utilized in signal processing in mobile networks to enhance the transmission of messages.