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