# Representation of the complex conjugate

The conjugate of the complex, is simply the opposite additive of the imaginary part of the complex number.

so, z = a + bi , its conjugate: z = a - bi

In an exercise they asked me for the complex conjugate (2, 4), which is in Cartesian form.

Which I have represented as:

z = 2 - 4i

(2, -4)


Which is in Cartesian form, but my answer is in its binomial form. Are they totally equivalent? or is one of the two more correct than the other?

• You cannot be "more correct". Yes, both of these ways of expressing the conjugate are perfectly fine. – user532449 Feb 20 '18 at 23:16

The two representations of complex numbers, $(2,-4)$ and $2-4i$ are equivalent.

The complex number $a+bi$ and its ordered pair form $(a,b),$ and also its polar form, $re^{i\theta}$ are all the same.

Each representation has its own advantages and is used accordingly.

they asked me for the complex conjugate (2, 4), which is in Cartesian form.

Such questions usually expect the answer to be in the same form as the given one i.e. (2,-4).

Your answer $\,2-4i\,$ is mathematically correct (and entirely equivalent), but then so would be $\,2 \sqrt{5} e^{-i \arctan(2)}\,$ for example.

• how you get that last form ? – Jav. dev Feb 20 '18 at 23:30
• @Jav.dev Lookup the polar form of a complex number. In this case $\,2+4i=2 \sqrt{5} e^{i \arctan(2)}\,$, and you get the conjugate by changing the sign of the argument. – dxiv Feb 20 '18 at 23:33
• Wish the downvoter had left a comment why. – dxiv Feb 22 '18 at 17:22

Since there is a bijective correspondence between point in $\mathbb{R}^2$ and $\mathbb{C}$ both expression are correct and equivalent each other, thus

$$\overline {(2,4)}=(2,-4)\equiv \overline {2+4i}=2-4i$$