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# Introduction to surds

The roots of most rational numbers cannot be written as fractions. They are irrational numbers. A surd is the root of a rational number that is irrational. A surd is also called a radical.

$\Tred{\sqrt{2}}$, $\Tred{\sqrt{3}}$, $\Tred{\sqrt{1/2}}$, $\Tred{\sqrt{8}}$ are surds.

$\Tblue{\sqrt{4}} = 2$, $\Tblue{\sqrt{1/9}} = 1/3$ and $\Tblue{\sqrt{9/16}} = 3/4$ are not surds because the roots are rational numbers.

In practice, we don't simplify surds and keep the roots $\sqrt{a}$ in the expressions.

We can simplify $\Tblue{\sqrt{4}}$ to $2$, but not $\Tred{\sqrt{2}}$.

$$1-\Tblue{\sqrt{4}} +\Tred{\sqrt{2}} = 1-\Tblue{2}+\Tred{\sqrt{2}} = -1 + \Tred{\sqrt{2}}.$$

To check if a number is a surd, write it in decimal form with a calculator. If you find recurring decimals, it is not a surd. Otherwise it is.

$\Tred{\sqrt{5}}$ is a surd because it has no recurring decimals.

$$\Tred{\sqrt{5}}\simeq 2.2360679774998$$

The sides of a square with area two have length equal to the square root of two. This is a surd.