The Conjugate of a Complex Number is a number having the same real part as the original complex number, and the imaginary part has the same magnitude but the opposite sign.
Example: (5 + 3i) and (5 - 3i) are complex conjugates to each other.
Representation of Conjugate of Complex Number
Conjugate z is represented by
If z = x + iy is a complex number, then the conjugate of z is defined as
\bold{\bar{z}= x - iy}
Geometric Interpretation of Complex Conjugate
The geometrical meaning of the conjugate of a complex number
Multiplication of Complex Number with Its Conjugate
If we multiply a complex number with its conjugate, the result will always be a non-negative real number.
The product of the complex conjugate pair is given as (a + ib)(a - ib) = a2 - i2b2 = a2 + b2
Example:
For z = 3 + 4i, the conjugate is 3 − 4i.
The product is:(3 + 4i)(3 − 4i) = 32 + 42 = 9 + 16 = 25
So, the result is a real number: 25.
Properties of Conjugate
If z, z1, and z2 are complex numbers, then below will be the conjugate properties.
- Conjugate of a purely real complex number is the number itself (z =
{\overline{z}} ) i.e., a conjugate of (7 + 0i) = (7 - 0i) = 7
- Conjugate of a purely imaginary complex number is negative of that number (z +
{\overline{z}} = 0); i.e., the conjugate of (0 - 7i) = (0 + 7i) = 7i
{\overline{({\overline{z}})}} = z
z +{\overline{z}} = 2 Re(z)
z - {\overline{z}} = 2i. Im(z)
- z
{\overline{z}} = {Re(z)}2 + {Im(z)}2
{\overline{(z_1 + z_2)}} ={\overline{z_1} + {\overline{z_2}}}
{\overline{(z_1 - z_2)}} ={\overline{z_1} - {\overline{z_2}}}
{\overline{z_1.z_2}} ={\overline{z_1} \times {\overline{z_2}}}
- z = (z₁ / z₂), then
\overline{z} =\overline{z₁} /{\overline{z₂}}; z2 ≠ 0
Note: To find out the conjugate of a complex number that complex number must be in its standard form which is Z = (x + i y). If the complex number is not in its standard form then it has to be converted into its standard form before finding its complex conjugate.
Complex Conjugate Root Theorem
According to the Complex Conjugate Root Theorem, if p(x) is a polynomial in which coefficients are real numbers and its root is a + ib, then the conjugate of the root, a - ib, will also be the root of the polynomial.
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Solved Examples
Example 1: If z = (5 + 7 i) is a complex number, then its conjugate is given by.
Solution:
The conjugate of the complex number is obtained by changing the sign of the imaginary part.
Thus,
{\overline{z}} = (5 - 7 i).
Example 2: Find the conjugate of the complex number 3 + 4i.
Solution:
The conjugate of the complex number 3 + 4i is the complex number obtained by changing the sign of the imaginary part, i.e., (3 - 4i).
Therefore, the conjugate of 3 + 4i is 3 - 4i.
Example 3: If z = 1 / (4 + 3 i) is a complex number, then its conjugate is given by
Solution:
First z has to be converted into its standard form by multiplying the numerator and
denominator with the conjugate of (4 + 3 i)
z = (1 / (4 + 3 i)) × ((4 - 3 i) / (4 - 3 i))
z = (4 - 3 i) / (16 + 9)
z = (4 / 25) - (3 / 25) i
The conjugate of z is
{\overline{z}} = (4 / 25) + (3 / 25) i
Example 4: If (a + ib) is a complex number that is the complex conjugate of (8 - 3i), then find the values of a and b.
Solution:
Let z = a + i b
{\overline{(8 - 3 i)}} = (8 + 3 i)z =
{\overline{(8 - 3 i)}} z = (8 + 3 i)
Two complex numbers are equal only when their corresponding real & imaginary parts are equal
Equating the real and imaginary parts of z & (8 + 3 i)
Re(z) = a = 8
Im(z) = b = 3
Hence 8 & 3 are the respective values of a and b.
Example 5: Find the product of the complex numbers (2 - 3i) and its conjugate.
Solution:
The conjugate of the complex number (2 - 3i) is (2 + 3i).
Required number is (2 - 3i)(2 + 3i) = 4 -6i + 6i -9i2 = 4 + 9 = 13
Therefore, the product of the complex numbers (2 - 3i) and their conjugate is 13.