Real Square Roots
Why
We want to solve equations with squares.
Definition
Given a real number $a$, we call any number
$r \in \R $ satisfying
\[
r^2 = a
\]
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.
Notation
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. ↩︎