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Natural Square Roots
Real Squares
Needed by:
Matrix Squares
Normal Densities
Outcome Variable Covariance
Quadratic Equation Solutions
Real Equation Solutions
Real Inner Product Norms
Standard Deviation
Sheet PDF
Graph PDF

Real Square Roots


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 $\pm\sqrt{x}$.

  1. Future editions will include the account. ↩︎
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