Surds

Surd is a mathematical term used to describe irrational numbers that can be expressed as the root of an integer. When a root cannot be simplified further, we call that a surd. For example, √4 is not a surd because it can be simplified to 2. When we simplify √4, we get 2 because the square root of 4 is 2. Surds help in keeping calculations exact rather than using decimal approximations.

When calculating surds, there are a few key properties that you must follow:
 

1. The sum or difference of a rational number and a quadratic surd cannot result in a quadratic surd.
    
2. If p ± √q = x ± √y, then p = x and q = y.
    
3. If \(\sqrt {(a+\sqrt b)} = \sqrt c + \sqrt d,\space \), then \( \sqrt {(a-\sqrt b)} = \sqrt c - \sqrt d,\) the same goes for the inverse.
    
4. Only like surds can be added or subtracted.
    
5. Surds can be multiplied if standard root rules are followed. 
    
6. Surds can be divided. For example, \( {\sqrt a \over \sqrt b} = { \sqrt{a\over b} }\)

We can classify surds into six different types:

1. Simple surds: A simple surd has only one term. For example, √7.
    
2. Pure surds: When surds are completely irrational, we call them pure surds. For example, √3.
    
3. Similar surds: They are surds with the same radicand. For example, √3, 2√3, 5√3.
    
4. Mixed surds: Mixed surd is a product of rational and irrational numbers. For example, 6√3 is a mixed surd because 6 is a rational number, while √3 is irrational.
    
5. Compound surds: Compound surds are the addition or subtraction of two or more surds. For example, √5 + √3 is the sum of two different surds.
    
6. Binomial surds: It takes two separate surds to form one binomial surd. For example, √3 + √7 is a binomial surd because it has two surds added together.

There are 6 rules that we use for the calculation of surds: 

Rule 1: \(\sqrt{a\times b} = \sqrt a \times \sqrt b\) 

Example: \(\sqrt{36} = \sqrt{9×4} = \sqrt9 × \sqrt4 = 3 × 2 = 6\)

Rule 2: \({\sqrt a \over \sqrt b} = \sqrt{a \over b}\)

Example: \({\sqrt{18} \over \sqrt2} = {\sqrt{18\over2}} = \sqrt9 = 3\)

Rule 3: \({b\over \sqrt a} = {{b\over \sqrt a} \times {\sqrt a\over \sqrt a}} = {b \sqrt a \over \ a}\)

Example: \({3\over \sqrt 5} = {{3\over \sqrt 5} \times {\sqrt 5\over \sqrt 5}} = {3 \sqrt 5 \over \ 5}\)

Rule 4: \(a \sqrt c \space \pm \space b\sqrt c = (a \space \pm \space b)\times \sqrt c\)

Example: \(5 \sqrt 5 \space \pm \space 3\sqrt 5 = (5 \space \pm \space 3)\times \sqrt 5\)

Rule 5: \({c \over {a\space+\space b\sqrt n}} = {{c \over {a\space+\space b\sqrt n}}} \times {{a\space-\space b\sqrt n}\over {a\space-\space b\sqrt n}} = {c \times ({a\space-\space b\sqrt n)}\over {a^2\space-\space b^2 n}}\)

Example: \({5\over {4\space+\space 2\sqrt 3}} = {{5 \over {4\space+\space 2\sqrt 3}}} \times {{4\space-\space 2\sqrt 3}\over {4\space-\space 2\sqrt 3}} = {5 \times ({4\space-\space 3\sqrt 2)}\over {4^2\space-\space (2^2 \times 3)}} = {20-15 \sqrt 2\over 4}\)

Rule 6: \({c \over {a\space-\space b\sqrt n}} = {{c \over {a\space-\space b\sqrt n}}} \times {{a\space+\space b\sqrt n}\over {a\space+\space b\sqrt n}} = {c \times ({a\space+\space b\sqrt n)}\over {a^2\space-\space b^2 n}}\)

Example: \({5\over {4\space-\space 2\sqrt 3}} = {{5 \over {4\space-\space 2\sqrt 3}}} \times {{4\space+\space 2\sqrt 3}\over {4\space+\space 2\sqrt 3}} = {5 \times ({4\space+\space 3\sqrt 2)}\over {4^2\space-\space (2^2 \times 3)}} ={20+15 \sqrt 2\over 4} \)

When solving for surds, there are a few steps we need to look out during each operation:
 

• Simplify: We factor out the perfect squares. Example: \(\sqrt{72} = 6\sqrt 2\).

• Addition or subtraction: Only like surds can be combined. Example: 3√5 + 7√5 = 10√5.

• Multiplication: Multiply the radicands inside the root. Example: \(\sqrt 3 \times \sqrt {12} = \sqrt{36} = 6\)

• Division: Divide the radicands before simplifying the root. Example: \(\sqrt{18} \space/ \sqrt2 = \sqrt 9 = 3\)

These are some of the few ways we can solve for surds when using the basic arithmetic operations. 
 

Performing operations of surds can be tough and confusing. Here are a few tips and tricks to simplify calculations:

1. Memorize the numbers that are perfect squares like 4, 16, 9, 25, etc.
    
2. For finding the square root of a large number, find all its factors and group them into pairs. Pick one from each pair and multiply it outside the root. If any factor is not in pair, then keep that number inside square root.
   Example: \(\sqrt {180} = \sqrt {(2 \times 2)\times (3\times 3) \times5} \)  \(= (2 \times 3) {\sqrt {5}} \space = \space 6 \sqrt 5\)
    
3. To rationalize the denominator, multiply both the numerator and denominator by the denominator.
   Example: \({\sqrt 4 \over \sqrt 5} = {{\sqrt 5 \times \sqrt 4} \over {\sqrt 5 \times \sqrt 5}} = {{\sqrt {20}} \over {5}}\)
    
4. Remember square and cube formulas:
   (a + b)² = a² + b² + 2ab
   (a – b)² = a² + b² – 2ab
   a³ + b³ = (a + b) (a² + b² – ab)
   a³ – b³ = (a – b) (a² + b² + ab)
    
5. For adding or subtracting like surds, add or subtract the numbers which are outside the square root. For example, \(a\sqrt n \pm b \sqrt n = {(a \pm b) \sqrt n} \)

Surds are used in fields where precise calculations involving irrational numbers are required:
 

• Engineering and construction: We use surds to calculate problems involving diagonal distances, slopes, and structural designs.
   
• Finance and banking: Compound interest often includes surds when working with non-repeating decimal growth rates.
   
• Navigation and GPS systems: Surds are used in distance formulas when calculating precise locations on Earth’s curved surface.
   
• Computer Graphics: Surds are used to model 3D shapes in a two-dimensional space to program precise movements of objects in animations and graphics.
   
• Physics: Surds are widely applied in the field of physics such as optics and waves, to design lenses and studying sound waves.