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I have read that addition, subtraction, multiplication, exponentiation and division are considered the fundamental operations in math.

Has this notion been made precise? Ie. is an operation considered fundamental if it cannot be defined in terms of other fundamental operations?

Can we know these are all the fundamental operations?

Is it possible to define complex conjugation in terms of the other five operations?

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These are usually considered the four fundamental operations of arithmetic. I don't think I would describe them as the fundamental operations of mathematics. – Jim Belk Jun 18 '11 at 18:16
There are an enormous number of different operations used in mathematics, acting on many different sets of objects. You mention complex conjugation, but there are also things like square root, logarithms, absolute value, composition of functions, matrix transpose, vector norm, matrix determinant, dot product, cross product, and so forth. Some of these can be described in terms of the four basic arithmetic operations, and some of them can't. – Jim Belk Jun 18 '11 at 18:48
$x - y = x + ((-1)\times y)$ or $x/y = x \times y^{-1}$ show that subtraction and division are not fundamental in the sense of not being defined in terms of the others. – GEdgar Jun 18 '11 at 20:10
and $x+y=x-(0-y)$... – Mariano Suárez-Alvarez Jun 19 '11 at 5:57
@JimBelk Are there fundamental operations of mathematics? – Voyska Sep 13 '12 at 0:40

In general, what is fundamental depends on what one chooses as foundations. Your question, in that sense, has no answer.

As for conjugation: there is no expression of a complex number $z$ involving only scalars and the five operations you listed which equals $\overline z$. A speedy argument is that any such expression denotes a function analytic on an open set of the complex plane, and $\overline z$ does not have that property; this has the disadvantage of requiring a bit of mathematical technology which maybe you don't have available :( Hopefully someone can come up with a simpler argument.

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These are important operations on structures in which they are defined, however to say they are fundamental operations in math isn't a very precise statement. In fact, using your definition, one can derive subtraction and division from addition and multiplication respectively. Abstract algebra provides a framework for studying various algebraic operations, but no operations are given precedence over the others.

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