If $a$ and $b$ are positive rational numbers, why might we want to call the $√a$+$√b$ and $√a$-$√b$ are conjugates? If $a$ and $b$ are positive rational numbers, why might we want to call the $√a$+$√b$ and $√a$-$√b$ are conjugate?
I know the definition of conjugates. Using that I can see this. What is the purpose of calling these two numbers as conjugates, that I don't know. How can I give the accurate answer for this question. Please help me. I am just the beginner in pure Mathematics.
 A: Assuming the numbers $a,b,ab$ are not perfect squares of rational numbers, the four numbers
$$\pm \sqrt{a} \pm\sqrt{b}$$
are indistinguishable algebraically over the rationals. More precisely, if one of them satisfies $f(x) = 0$ where $f$ is a polynomial in $x$ with rational coefficients, then so does the other.

More generally, if $K$ is a subfield of a field $L$, two elements $u,v \in L$ are called conjugates over $K$ if there exists nonzero $f \in K[x]$ (i.e., $f$ is a polynomial in $x$ with coefficients in $K$) such that


*

* $f(u) = f(v) = 0$.

*$f$ is irreducible over $K$ (i.e., $f$ cannot be expressed as $f=gh$ where $g,h \in K[x]$, and $g,h$ are non-constant).


Thus for example, the complex numbers $2+3i$ and $2-3i$ are conjugates over $\mathbb{R}$ since they both satisfy the polynomial $x^2-4x+13$, which is irreducible over $\mathbb{R}$.

Another example . . . 

Let $L$ be the field over $\mathbb{R}$ generated by the expressions $\cos(\theta),\sin(\theta)$ (i.e., $L = \mathbb{R}(\cos(\theta),\sin(\theta))$, where $\theta$ is an indeterminate).

Let $K$ be the smallest subfield of $L$ such that $R \subset K$ and $\cos(\theta) \in K$ (i.e., $K = \mathbb{R}(\cos(\theta))$.

Then $1+\sin(\theta)$ and $1-\sin(\theta)$ are conjugates over $K$ since they both satisfy the polynomial $x^2 - 2x + \cos^2(\theta)$, which is irreducible over $K$.
A: This is because $(\sqrt a + \sqrt b)(\sqrt a - \sqrt b) = a-b$. So multiplying those two irrational numbers gives a rational result. This is often a very useful step in algebraic manipulations such as "rationalising" the denominator (i.e. to make the denominator of a surd expression rational without changing the value of the whole expression). Hence the two irrational numbers are called conjugates.
This is analogous to complex numbers, where multiplying a complex number $a+bi$ by its conjugate $a-bi$ gives a real number, also a very useful step in complex number manipulation. Even though it's known by the same name, note that it's a different concept here (albeit related).
