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The concept of conjugate seems to exist in many fields of mathermatic such as complex conjugate, group conjugate, etc.

I search through many websites about what exactly is the conjugate. Most of them always claim that "just change the sign".

Suppose there is $a+b$, the conjugate is $a-b$.

While in the group, the conjugate of $s$ is $x*s*x^{-1}$, where $s$ is element of group $G$ and $x$ is fixed element of group $G$.

  • How do we will know that the conjugate will be in what form?

  • What is the conjugate told us or use for?

I already read this What is the conjugate?. It doesn't help at all.

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    $\begingroup$ I may be wrong about this, but I think the word "conjugate" has been used to refer to many distinct notions that are sometimes related and sometimes not. Usually, context dictates which particular notion is meant (e.g., complex conjugate in complex analysis, group conjugation in group theory), and often the definition will be restated somewhere in the paper/article/book so that there is no ambiguity. $\endgroup$
    – angryavian
    Oct 31, 2019 at 6:32

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As angryavian says in their comment, the word conjugate gets used slightly differently in different areas of mathematics, but there is an underlying theme. The word conjugate itself comes from Latin and means (literally) "to yoke together", and the idea behind the word is that the things that are conjugate are somehow bound to each other. In (most European) languages verbs are conjugated by tying endings to the infinitival stem, and in mathematics things are conjugate when they are somehow bound together so that having one of them gets you the other.

Examples then: the conjugate of a complex number is obtained by changing the sign of the imaginary part: $a+ib \longrightarrow a-ib$ but the thing that really makes it conjugate is what happens when you multiply those two things together: $(a+ib)(a-ib) = a^2 +b^2$ -- you get a real number. So any real number can be expressed as the product of a complex number and a highly-related complex number, so we call one of them the conjugate of the other.

You can see this perhaps more directly by staying the in non-zero reals: take any number $a\in {\mathbb R}\setminus \{0\}$ and consider the conjugation operation of inversion. So we would say that $\bar{a} = 1/a$ (note this is well-defined by excluding $0$) and that $a\bar{a}=1$. The conjugacy here comes from the fact that there is a unique inverse for any non-zero real.

For a more advanced example, the dual space of a Banach space is sometimes (in older literature) referred to as the conjugate space: if you have a Banach space $X$ you obtain the conjugate (dual) space by taking the collection of all linear functionals $x^*:X \rightarrow {\mathbb K}$. Getting the original space back is a little trickier, as typically $X \subseteq X^{**}$, but $X^* = X^{***}$ so there's only one special case.

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