Radicals

Here's something that trips people up immediately: every positive number actually has two square roots. The number 25 has square roots 5 and −5, because both 5 × 5 and (−5) × (−5) equal 25. When you write $\sqrt{25}$, though, you're specifically asking for the principal square root , the positive one. The negative version gets written as $-\sqrt{25}$.

The symbol ​ is called the radical sign, and whatever sits inside it is the radicand. So in $\sqrt{5}$, the radicand is 5. When a radical can't be simplified to a clean integer or fraction, the result is called a surd , something like $\sqrt{2}$ or $3\sqrt{7}$.

Surd: An irrational number expressed using a radical sign, such as $\sqrt{3}$ or $5\sqrt{2}$, that cannot be written as a terminating or repeating decimal.

Think of rational numbers like songs with a predictable structure , they either end cleanly or loop in a repeating pattern. Formally, a rational number can be written as $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$. Their decimal expansions either terminate (like 0.25) or repeat (like 0.333...).

Irrational numbers are the opposite , their decimals run forever with no repeating block. $\pi$ is the famous example, but $\sqrt{2}$ is just as irrational.

Here's the part that catches people out: a radical sign doesn't automatically mean irrational. $\sqrt{9}=3$, which is perfectly rational. The test is whether the radicand is a perfect square (1, 4, 9, 16, 25, 36...). If it is, the result is a whole number and therefore rational. If it isn't, you've got a surd.