Posted on - 03-02-2017

Bank PO

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.

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.

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.

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. .