Arithmetic Square Root
📐 What is a Square Root?
Before we define the arithmetic square root, let's recall what squaring a number means:
When we compute $3^2 = 9$, we say "3 squared equals 9." The square root is the reverse operation — it asks: "What number, when squared, gives us 9?"
⚠️ Key Observation: Both $3^2 = 9$ and $(-3)^2 = 9$. So the number 9 has two square roots: $3$ and $-3$. This is where the arithmetic square root becomes important!
✅ Formal Definition
Definition: For a non-negative real number $a \geq 0$, the arithmetic square root of $a$ is the unique non-negative number $b \geq 0$ such that:
$$b^2 = a$$
We write this as: $\quad b = \sqrt{a}$
Three Critical Properties
- Non-negative radicand: $a \geq 0$ (the number under the radical must be non-negative)
- Non-negative result: $\sqrt{a} \geq 0$ (the arithmetic square root is always non-negative)
- Defining equation: $(\sqrt{a})^2 = a$
Common Mistakes to Avoid
- $\sqrt{9} = 3$, NOT $\pm 3$
- $\sqrt{(-5)^2} = 5$, NOT $-5$
- $\sqrt{a^2} = |a|$, NOT simply $a$
- $\sqrt{-4}$ is undefined in real numbers