Surds

The word surd comes from the Latin word surdus, meaning deaf or mute. In the early days of mathematics, Arabians used to consider irrational numbers as mute. This means there was no value in these irrational numbers. Only rational numbers were considered audible. Over time, surds became a vital part of today’s mathematics.

Surds represent those numbers that do not have roots. For example, 4, 25, 36 have proper roots as 2, 5, and 6, whereas numbers 7, 12, 20 do not. These improper numbers, which do not have proper roots, are termed as surds.

These surds can be represented with exponential powers with fractional powers as well, like √6=6¹/², √2=2¹/².

Types of surds

Pure surds – Surds having only one irrational number as √7, 4√11, √x3 are called pure surds.
Mixed surds – Surds having both rational and irrational numbers as x√y, 4√3, 8√5 are called mixed surds.
Compound surds – Two or more surds in one mathematical expression are called compound surds. E.g. 4+√3, √5+√2,, √a+b√c

Rules of surds

Surds cannot be added. √a+√b ≠ √(a+b)
We can add such surds - m√a+n√a = (m+n)√a
3√5+2√5 = 5√5
Surds cannot be subtracted. √a−√b ≠ √(a−b)
Surds can be multiplied. √a×√b = √(a×b)
Surds can be divided. √a / √b = √(a/b)
Surds can be represented in exponential form. √a=a¹/², n√a=a¹/ⁿ

Simplification of surds

We can represent surds in the simplified way by taking out the roots of the number. A pure surd will then become a mixed surd.

e.g. √50 = √(5×5×2) = 5√2

Other example of solving surds

Solve 16³/² + 16-³/² = ?

Solution – We know from the law of exponents,

aᵐ x aⁿ = aᵐ⁺ⁿ

a-ᵐ = 1/aᵐ

⇒16³/² + 1/16³/²

⇒(16¹/²)³ + 1/(16¹/²)³

⇒(4²*¹/²)³ + 1/(4²*¹/²)³

⇒4³ + 1/4³

⇒ 64 + 1/64

⇒ (64 x 64+ 1)/64

= (4096+1)/64

= 4097/64

Rationalising surds

Rationalise 1/[(8√11 )- (7√5)]

Solution – We know that the conjugate of (8√11 ) - (7√5) is (8√11 )+(7√5). Change the addition sign in the expression and vice versa to find the conjugate of an expression.

To rationalize the denominator of the given fraction, we need to multiply the conjugate of the denominator on both numerator and denominator.

= [1/[(8√11 ) - (7√5)]] × [[(8√11 ) + (7√5)]/[(8√11 ) + (7√5)]]

= [(8√11 ) + (7√5)]/[(8√11 )² - (7√5)²]

= [(8√11 ) + (7√5)]/[704- 245]

= [(8√11 ) + (7√5)]/459

Applications of surds

Surds are essential in computer programming.
They play a vital role in designing buildings and architecture.
They are helpful in Pythagoras theorem and trigonometric calculations.
Surds are used to maintain algebraic fluency.
They are used in differential equations, Fourier series, differential and computational geometry, probability, and statistics.

Fun fact: Surds was invented by a European mathematician, Gherardo of Cremona, in 1150 BC. He used the Pythagoras theorem to find the diagonal of a square and the value of the first surd. He termed this value ‘voiceless’ because the root value had no meaning at that time.