Contents

# Surd|Definition & Meaning

## Definition

A **surd** is any number whose **nth root** (square root, cube root, etc.) cannot be **simplified** so as to remove the **radical** sign, e.g., $\sqrt{2}$ and $\sqrt[3]{4}$. Unless the **radicand** can be **factorized** into parts that are **all **perfectly expressible as a **power** of the radical’s index (perfect squares for square root, cubes for cube roots, etc.), the radical’s **simplification** will always contain a surd.

**Surds** are a special class of numbers that cannot be expressed as a simple **fractional number.** As the above **definition** suggests, one of the easiest ways of **defining** such numbers is through the n^{th} root route. The following **figure** shows some **examples** of surds that are most **elementary** in nature.

**Figure 1: Some Examples of Elementary Surds**

## Examples of Surds

As per the **example** given above, we can easily explain surds with the help of some **elementary roots.** As a litmus test, we will see if the **given root** can be written without the **square root.** Let us consider all the above numbers one by one:

$\sqrt{1}$ = 1 $\to$ Not Surd

$\sqrt{2}$ $\to$ **Surd**

$\sqrt{3}$ $\to$ **Surd**

$\sqrt{4}$ = 2 $\to$ Not Surd

$\sqrt{5}$ $\to$ **Surd**

$\sqrt{6}$ = $\sqrt{2}$ x $\sqrt{3}$ $\to$ **Surd**

$\sqrt{7}$ $\to$ **Surd**

$\sqrt{8}$ = $2\sqrt{2}$ $\to$ **Surd**

## Types of Surds

The different **types** of surds are as follows:

**Simple Surds:**Surds comprising of a**single term**are called simple surds. Examples in the figure 1 are all simple surds.**Pure Surds:**Surds that are**completely irrational**and can’t be written as a fraction are termed as pure surds.**Similar Surds:**If two or more surds have a**common factor**that they are called similar surds.**Mixed Surds:**If a surd can be represented as a**product**of a**rational and****irrational number**then it can be called a mixed surd.**Compound Surds:**If a surd is made up of**addition**or**subtraction**of any two or more surds, then its called a compound surd.**Binomial Surds:**If a surd is constituted by**other simpler surds,**its called a binomial surd.

The following figure **summarizes** all these different types:

**Figure 2: Types of Surds**

## History of Surds

The concept of surds was initially **proposed by** the Ancient **Greeks,** who studied the properties of **right triangles** and discovered that some **length ratios** could not be expressed as **simple fractions.** These ratios were termed as **surds** and were represented by the **square root sign,** as explained by the above example.

## Explanation of Surds

A specific type of **irrational number** that cannot be expressed as a simple fraction is called a **surd.** A square root is **represented** by the symbol $\sqrt{}$ **(square root),** which is then followed by a **number.** Because it is a **non-terminating, non-repeating decimal.** For example, the surd $\sqrt{2}$ signifies the square root of 2, which cannot be expressed as a **simple fraction.**

The concept can also be considered by way of **rational numbers.** In mathematics, a number is considered to be rational if it can be expressed as the **ratio of two integers.** However, a number is considered **irrational** if it cannot be expressed as the **ratio of two integers.** Surds are examples of irrational numbers.

In addition to **square roots,** there are a number of other types of roots, including **cube roots** (represented by the sign ∛), **fourth roots** (represented by the symbol ⁴√), and so on. However, square roots are the type of surd that is most **usually explored.**

One of the most important features of surds is that they cannot be expressed as **straightforward fractions.** They differ in this sense from **rational numbers,** which may be expressed as **simple fractions.** As a result, surds cannot be expressed as decimals with a **finite number** of **digits.** Instead, they are shown as **non-terminating, non-repeating** decimals.

Another key feature of surds is their capacity to be **combined** with **other surds** and rational numbers to form more **complex assertions.** For instance, the expression $\sqrt{2}$ + $\sqrt{3}$ **combines two surds** to represent the sum of the square roots of 2 + 3.

To summarize, surds are **irrational numbers** that cannot be expressed as **simple fractions.** They are frequently studied in **mathematics,** and their symbol is (square root$\sqrt{ }$). Surds have a variety of **important properties,** such as the fact that they cannot be expressed as **simple fractions** and that they may be **combined** with other surds and rational numbers to form more **complex assertions.**

In several areas of **mathematics,** such as **geometry, algebra,** and **calculus,** surds are also **extensively employed.**

## Applications of Surds

In addition to their special properties, **surds** have many **applications** in **geometry, algebra,** and **calculus.** For example:

**Calculus:**Surds are used in**calculus**to obtain the**derivatives**of**functions**having**square roots.****Geometry:**Surds are a tool used in**geometry**to calculate the**lengths**of the**sides**of**right triangles.**It can further be used in many**real-world contexts,**including**construction and surveying,**as right triangle side**lengths**may contain surds. For instance, a surd may be found in the hypotenuse of a given triangle. Another example is when calculating the**area**of a**circle**involves using the**square root**of**pi.****Trigonometry:**Trigonometric operations like**sine, cosine,**and**tangent**may make use of surds when computing**certain angles.****Physical measurements:**Some physical measurements, like the**distance**between them, may require surds if the**route**linking the two locations is**not straight.****Logarithmic operations:**Operations like**log base****2**or**log base 10**may call for surds when working with**certain values.**

Among many others.

## Some Famous Surds

Surds are very **commonly seen** in mathematics. However, some of these have **gained** **significance** over time. Following are some **examples:**

### The Golden Ratio

The** golden ratio** (also known as the golden number, the golden fraction, or the **divine proportion),** which is often symbolized by the greek letter **phi** ( $\phi$ ), is the ratio between two numbers **defined** as follows:

Golden Ratio = $\phi$ = $ \dfrac{ 1 + \sqrt{5} }{ 2 } $

This ratio is significantly **associated** with the **Fibonacci Series.** It is widely used in **trading strategies, finance,** and **modeling.** This so-called golden ratio manifests itself in many **natural phenomena,** including **animals, plants, celestial systems** etc.

**Figure 3: The Golden Ratio**

### The Quadratic Equation

The quadratic equation, or the **2nd order polynomial,** is used to model many real-world **objects** and **phenomena.** It also serves as the basis for **parabolic, hyperbolic, elliptical,** and **circular** shapes. The solution to such equations very often results in roots that are actually **surds.**

**Figure 4: The Quadratic Equation**

## Key Features of Surds

Some key properties of surds are as follows:

**Irrationality:**Since surds are irrational numbers, they**cannot**be expressed as**simple fractions.****Non-terminating:**Surds cannot be expressed as a**finite decimal**since they are treated as**non-terminating, non-repeating**decimals.**Simplification:**Finding the**optimal square roots**of a surd’s values will make it easier to understand.**Addition and subtraction:**Surds can be**added**to and**subtracted**from other surds and rational numbers to form more**complex statements.****Division and Multiplication:**Division and Multiplication of Surds may be made**simpler**by applying the**exponent**and**square root**methods.

There are quite a few others, but these are common ones you are likely to use.

## Mathematical Rules for Solving Surds

Working with surds requires the **ability** to **convert** them into more **basic forms.** For instance, since $\sqrt{2 \times 2}$ equals $\sqrt{4}$, the **surd** $\sqrt{4}$ can be **reduced** to 2. Just as the surd $\sqrt{9}$ may be **reduced** to 3 since $\sqrt{3 \times 3}$ = $\sqrt{9}$.

Since surds are very common in many mathematical **problems** and **numerical** questions, some important rules may ease their **solution.** Following are some **commonly used mathematical rules** or so-called surd identities:

\[ \sqrt{x \times y} = \sqrt{x} \times \sqrt{y} \]

\[ \sqrt{ \dfrac{x}{y} } = \dfrac{ \sqrt{x} }{ \sqrt{y} } \]

\[ \dfrac{x}{\sqrt{y}} = \dfrac{ \sqrt{x} \times \sqrt{y}}{ y } \]

\[ x \sqrt{z} \pm y\sqrt{z} = ( x \pm y ) \sqrt{z} \]

## Numerical Examples of Surds

**(a) Multiply** the surds $\sqrt{2}$ and $\sqrt{5}$.

**(b) Divide** the surd $\sqrt{8}$ by $\sqrt{2}$

**(c) Simplify** the surd $\sqrt{2} + 5 \sqrt{2}$

### Solution

#### Part (a)

\[ \sqrt{2} \times \sqrt{5} = \sqrt{2 \times 5} = \sqrt{10} \]

#### Part (b)

\[ \dfrac{ \sqrt{8} }{ \sqrt{2} } = \sqrt{ \dfrac{ 8 }{ 2 } } = \sqrt{ 4 } = \pm 2 \]

#### Part (c)

\[ \sqrt{2} + 5 \sqrt{2} = (1 + 5) \sqrt{2} = 6 \sqrt{ 2 } \]

*All images were created with GeoGebra.*