Surds are numbers that we leave as square roots because they give us irrational numbers. An irrational number can't be written as a fraction, and in decimal form is infinitely long with no recurring pattern.

For Example: $\sqrt{5} \approx 2.23606...$

(2.23606…) is an irrational number. The decimal form is infinitely long with no recurring pattern.

This means that:

The square root of 5 is a surd.

Result:

This number,$( \sqrt{5} )$, is a surd because it can't be exactly written as a simple fraction or a neat decimal. The square root of 5 is an irrational number.

For example, $( \sqrt{3} )$ and the cube root of 2 $( \sqrt[3]{2} )$ are both surds.

For example: $\text{2, 100, -3, } \frac{3}{4}, \text{ and } \frac{1}{9}$ All of these are rational numbers because we can write them as fractions or whole numbers.

Irrational numbers in decimal form are infinite, with no recurring or repeating pattern.

Why do we use surds? We use surds to write numbers precisely.

Let's look at an example: We know that: $\sqrt{2} \approx 1.4142135623...$

If we round this to 1.41 and use it in our calculations, the answer might not be accurate.

Instead, we should keep it as$( \sqrt{2} )$ to use the full value of the number. This is called leaving the number in its exact form.

For example, Find the value of$(x)$. Give your answer in exact form.

You're given a right triangle where one side is 4, another side is 6, and you need to find the hypotenuse $( x )$.

We need to use Pythagoras' theorem.

$a^2 + b^2 = c^2$

So,

$4^2 + 6^2 = x^2$

$16 + 36 = x^2$

$52 = x^2$

To find$( x )$, take the square root of both sides:

$x = \sqrt{52}$

And simplify the surd if possible:

$x = 2\sqrt{13}$

Now, $( x = 2\sqrt{13} )$ is the exact form of your answer.

Relevance?

It is much more precise than using an approximate decimal.