Conjugate of a Complex Number

A number that can be represented in the form of (a + ib), where a is the real part, b is the imaginary part, and ‘i’ is an imaginary number called iota, which is equal to the root of negative 1. A complex number is basically a combination of a real part and an imaginary part of that number.

For example, 6 + i3 is a complex number in which 6 is the real part of the number and i3 is the imaginary part of the number.

What is a Conjugate?

A conjugate in Mathematics is formed by changing the sign of one of the terms in a binomial. For example, if the binomial number is a + b, the conjugate of this number will be formed by changing the sign of either of the terms. If we change the sign of b, the conjugate formed will be a – b. Therefore, in mathematics, a + b and a – b are both conjugates of each other.

What is the Conjugate of a Complex Number? 

The conjugate of a complex number represents the reflection of that complex number about the horizontal axis (real axis) on Argand’s plane. 

A complex conjugate or a conjugate of a complex number is also defined as another complex number with the same real part as the original number and the opposite sign of the imaginary part, keeping its magnitude the same, i.e ‘i’ of the original complex number changes to ‘-i’ in its conjugate.

The conjugate of a complex number is used to rationalize the complex number which makes solving problems quite easy.

Representation of Conjugate of a Complex Number 

We know, a complex number ‘z’ is of the form a + ib, where a, b are real numbers, and ‘a’ is the real part while ‘b’ is the imaginary part. 

Now, the complex conjugate of a + ib is given by a - ib whose real part is 'a' and imaginary part is '-b'. This complex conjugate number is represented by ‘\[\bar{z}\]’. 

Therefore, it can be said that (a - ib) is the reflection of (a + ib) about the real axis (X-axis) in the argand plane. Also, z and \[\bar{z}\] are called the complex conjugate pair.

For example, z = x + iy is a complex number that is inclined on the real axis making an angle of α, and \[\bar{z}\] = x – iy which is inclined to the real axis making an angle -α. 

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From the graph, the complex number z lies in the first quadrant, and its complex conjugate \[\bar{z}\], lies in the fourth quadrant because it the mirror image of z about the real axis i.e x-axis.

Properties of the Conjugate of a Complex Number

1. \[\overline{z_1\pm z_2}=\overline{z_1}\pm \overline{z_2}\]

Proof: Let z1 = p + iq and z2 = x + iy

So, \[\overline{z_1\pm z_2}\]

\[=\overline{p\pm x+iq\pm iy}\]

\[=\overline{p\pm x-i(q\pm y)}\]

\[=\overline{p-iq\pm x-iy)}\]

\[=\overline{z_1}\pm \overline{z_2}\]

2. \[\overline{z_1.z_2}=\overline{z_1}.\overline{z_2}\]

Proof: Let z1 = a + ib and z2 = c + id

Then, \[\overline{z_{}. z_{2}}\] =  \[\overline{(a + ib) . (c + id)}\]

= \[\overline{ac + iad + ibc + i^2bd}\]

= \[\overline{ac + iad + ibc - bd}\]

= \[\overline{ac - bd + i(ad + bc)}\]

= \[ac -bd-i(ad + bc)\]

= \[ac+i^2bd-iad-ibc\]

=\[ac-ibc+i^2bd-iad\]

= \[c(a-ib)-id(a-ib)\]

= \[(a-ib).(c-id)\]

= \[\overline{z_{1}. Z_{2}}\]

3. \[\frac{\overline{z_{1}}}{z_{2}}\] =  \[\frac{\overline{z}_{1}}{\overline{z}_{2}}\]

Proof: \[\frac{\overline{z_{1}}}{z_{2}}\] =    \[\overline{(z_{1}.\frac{1}{z_{2}})}\]

Using the multiplicative property of conjugate, we have

\[\overline{z_{1}}\] . \[\frac{\overline{1}}{z_{2}}\]

\[\frac{\overline{z}_{1}}{\overline{z}_{2}}\]

4. \[\overline{\overline{z}}\] = z

Proof: Let z= a + ib

\[\overline{\overline{z}}\] =  \[\overline{\overline{a + ib}}\] = \[\overline{a - ib}\] = a + ib = z

5. If z = a + ib

Then, \[z.\overline{z}=a^2+b^2=|z^2|\]

Proof: 

\[z.\overline{z}=(a+ib).\overline{(a+ib)}=(a+ib).(a-ib)=a^2-i^2b^2=a^2+b^2=|z^2|\]

6. \[z + \bar{z} = x + iy + ( x – iy)\]

Then,  \[z + \bar{z} = 2x\]

7. \[z - \bar{z} = x + iy - ( x – iy )\]

Then, \[z -\bar{z} = 2iy\]

8. For any z lying in the 1st quadrant, \[\bar{z}\] will lie in 4th quadrant and - \[\bar{z}\] will lie in the 2nd quadrant.

9. Real (z) = Real (\[\bar{z}\]) and Imaginary (z) = – Imaginary (\[\bar{z}\])

Therefore, the complex numbers also follow the subtraction, addition, multiplication and division mathematical operations that are applicable to real numbers. In complex numbers, just different methods and properties are used!