Representation of the complex conjugate

Question

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?

Answer 1

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.

Answer 2

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

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

Answer 3

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.