Any number of the form p/q where p and q are
integers and q > 0 is called a
rational number. Any real number which is not a rational number is an
irrational number. Amongst irra-tional numbers, of particular interest to us
are SURDS. Amongst surds, we will specifically be looking at ‘quadratic surds’
— surds of the type a + 
and a + + 
where the terms involve only square roots and not any higher
roots. We do not need to go very deep into the area of surds — what is required
is a basic understanding of some of the operations on surds.
If there is a surd of the form a + , then a surd of the form a is called the conjugate of the initial surd. The product of a
surd and its conjugate will always be a rational number.
Rationalisation of a surd
When there is a surd of the form 
it is difficult to perform arithmetic operations on it.
Hence, the denominator is converted into a
rational number thereby facilitating ease of handling the surd. This process of
converting the denominator into a rational number without changing the value of
the surd is called rationalisation. .
To convert the denominator of a surd into a
rational number, multiply the denominator and the numerator simultaneously with
the conjugate of the surd in the denominator so that the denominator gets
converted to a rational number without changing the value of the fraction. That
is, if there is a surd of the type a + in the denominator, then both the numerator and the denominator
have to multiplied with a surd of the form a - or a surd of the type -a + to convert the denominator into a rational number.
Square root of a surd
If there exists a square root of a surd of
the type a + , then it will be of the form x + 
. We can equate the square of x + to a + and thus solve for x and y. Here, one point should be noted: When
there is an equation with rational and irrational terms, the rational part on
the left hand side is equal to the rational part on the right hand side and,
the irrational part on the left hand side is equal to the irrational part on
the right hand side of the equation.
Comparison of surds
Sometimes we need to compare two or more
surds either to identify the largest one or to arrange the given surds in
ascending/descending order. The surds given in such cases will be such that
they will be close to each other and hence we will not be able to identify the
largest one by taking the approximate square root of each of the terms. In such
a case, the surds can both be squared and the common rational part be
subtracted. At this stage, normally one will be able to make out the order of
the surds. If even at this stage, it is not possible to identify the larger of
the two, then the numbers should be squared once more. .
