What is a Surd?
A surd is an irrational number.
We know √4 = 2, √9 = 3, √16 = 4,
But what is √2 =? and √5 =?
√2 and √5 are not rational numbers such as √4, √9 and √16 and others which are all rational numbers.
½ is a terminating decimal, 0.5 and
1/3 is a non-terminating decimal, 0.3333….
But √2 = 1.414215…….. can be expressed as neither a terminating nor a non-terminating decimal.
So, √2 is a surd.
Definition of a Surd:
The nth root of a number a is denoted as n√a (also as a1/n)
Let a be a rational number and n a positive integer.
Then n√a is called a Surd if it, i.e. n√ais an irrational number.
n√ais spoken as “nth root of a”
nthroot of a number a is a surd, if it is an irrational number.
Examples of Surds:
2√2 or just √2 is a surd.
3√9, 3√16, 4√25, 5√100 are all numbers whose given roots are not rational numbers. They are therefore Surds.
What is Not a Surd?
In the surd n√a, a is a rational number.
In n√a, if a is irrational, then n√a is not called a Surd.
From this definition:
Order of a Surd:
In the surd n√a, n is called order of the surd.
If the order of surd is 2, it is optionally dropped.
2√3 is same as √3.
Types of Surds:
The following are various types of Surds:
Surds such as √3, 3√9 which are entirely irrational numbers are called pure surds
Surds such as 2√3, 3√9 are called mixed surds as they containrational numbers such as 2, 3 and surds such as √3 and √9
Surds such as √2 + √3, √3 - √2 are called Compound Surds.
Compound surds are sum or difference of two other surds.
Surds that are different multiples of same surds are called similar surds.
√80 and √45, because
√80 = √ (5 × 16) = 4√5 and
√45 = √ (5 × 9) = 3 √5
Two surds of the form x + √y and x - √y are called conjugate surds.
2 + √3 and 2 - √3 are called conjugate surds.
The sum or difference of two conjugate surds is a rational number.
(2 + √3) + (2 - √3) = 4, a rational number.