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 = [Tex]{\overline{z}} [/Tex]) i.e., conjugate of (7 + 0 i) = (7 – 0 i) = 7

• Conjugate of a purely imaginary complex number is negative of that number (z + [Tex]{\overline{z}}  [/Tex] = 0) i.e. conjugate of (0 -7 i) = (0 + 7 i) = 7 i

• [Tex]{\overline{({\overline{z}})}}  [/Tex] = z

• z + [Tex]{\overline{z}}  [/Tex] = 2 Re(z)

• z – [Tex]{\overline{z}}  [/Tex] = 2 i . Im(z)

• z [Tex]{\overline{z}}  [/Tex] = {Re(z)}² + {Im(z)}²

• [Tex]{\overline{(z_1 + z_2)}}  [/Tex] = [Tex]{\overline{z_1} + {\overline{z_2}}}[/Tex]

• [Tex]{\overline{(z_1 – z_2)}}  [/Tex] = [Tex]{\overline{z_1} – {\overline{z_2}}}[/Tex]

• [Tex]{\overline{z_1.z_2}}  [/Tex] =  [Tex]{\overline{z_1} \times {\overline{z_2}}}[/Tex]

• z = (z₁ / z₂) then [Tex]{\overline{z}}  [/Tex] = [Tex]{\overline{z₁}}  [/Tex] / [Tex]{\overline{z₂}}  [/Tex]; z₂ ≠ 0

Note: In order to find out the conjugate of a complex number that complex number must be in its standard form that 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.

Multiplication of Complex Conjugate

The multiplication of complex conjugate is given below:

Let a+ib be a complex number then its conjugate will be a – ib.

Hence, the product of complex conjugate pair is given as (a + ib)(a – ib) = a² – i²b² = a² + b²

Complex Conjugate Root Theorem

According to 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.

Read More

• Addition of Complex Numbers
• Additive Identity and Inverse of Complex Numbers
• Multiplicative Identity and Inverse of Complex Numbers

Conjugate of Complex Number is a complex number obtained by changing the sign of the imaginary part. In simple words, conjugate of a complex number is a number that has the same real part as the original complex number, and the imaginary part has the same magnitude but opposite sign.

In the world of mathematics, complex numbers are one of the most important discoveries by mathematicians as it helps us solve many real-life problems in various fields such as the study of electromagnetic waves, engineering, and physics.

Complex Numbers is defined as a pair of real numbers and represented as a+ib, where a and b are the real numbers, and i = √(-1) is called iota (an imaginary unit). Some Examples of Complex Numbers include (5 + 3 i), (0 + 7 i), (-4 + 9 i), (1 + 0 i), (0 – i), etc. Now, let’s discuss the conjugate of complex numbers or complex conjugates in detail.