We want to solve equations with squares.

Given a real number $a$, we call any number $r \in \R $ satisfying

\[ r^2 = a \]

a square root of $a$.
We can show that any positive real number has
*two* square roots.

Suppose $a \in \R $ and $a > 0$.
Then the set

\[ \Set{r \in \R }{r^2 = a} \]

has size 2.
Furthermore, these two roots are each the
additive inverse of each other.^{1}

Suppose $a \in \R $ and $a > 0$.
If $r_1, r_2 \in \R $, $r_1 \neq r_2$ are
the two distinct square roots of $a$, then
$r_1 = -r_2$.

When speaking of *the* square root of a
real number, we mean to reference the
*positive* square root.
We also speak of the square
root function which associates a real
number to its positive square root.

As with natural numbers, we denote *the*
(positive) square root of the real number $x
\in \R $ by $\sqrt{x}$.
Some authors refer to both the roots of $x$
by writing $\pm\sqrt{x}$.

- Future editions will include the account. ↩︎