Rationalization

Rationalization is a process of converting an irrational number into a rational number, which is one which can be expressed as the ratio of two integers. The numbers 1.003, -1¹⁄₃, and 22/7 are all rational numbers. Irrational numbers are those that cannot be so expressed. The ratio pi, the square root of 5, and the cube root of 4 are all irrational numbers.

Rationalization is a process applied most often to the denominators of fractions, such as 5/(1 + √+2). There are two reasons for this. If someone wanted to compute a rational approximation for such an expression, doing so would entail dividing by a many-place decimal, in this case 2.41421... With a calculator it would be easy to do, but if it must be done without a calculator, the process is long, tedious, and subject to errors. If the denominator were rationalized, however, the calculations would be far shorter.

The second and mathematically more important reason for rationalizing a denominator has to do with "fields," which are sets of numbers which are closed with respect to addition, subtraction, multiplication, and division. If one is working with the field of rational numbers and if one introduces a single irrational square root into the field, forming all possible sums, differences, products, and quotients, what happens? Are the resulting numbers made more complex in an unlimited sort of way, or does the complexity reach a particular level and stop?

The answer with respect to sums, differences, and products is simple. If the irrational square root which is introduced happens to be √2, then any possible sum, difference, or product can be put into the form p + q √2, where p and q are rational. The cube of 1 + √2, for example, can be reduced to 7 + 5 √2.

To check quotients, one can first put the numerator and denominator in the form p + q √2 (thinking of a quotient as a fraction). Then one rationalizes the denominator. This will result in a fraction whose numerator is in the form p + q √2, and whose denominator is a simple rational number. This can in turn be used with the distributive law to put the entire quotient into the form p + q √2 .

How does one rationalize a denominator? The procedure relies on the algebraic identity (x + y)(x - y) = x² - y², which converts two linear expressions into an expression having no linear terms. If x or y happens to be a square root, the radical will disappear.

Using this identity can be illustrated with the example given earlier:

The procedure is not limited to expressions involving √2.

If any irrational square root, √7, √80, or √n is introduced into the field of rational numbers, expressions involving it can be put into the form p + q √n . Then quotients involving such a form as a divisor can be computed by multiplying numerator and denominator by p - q √n, which will turn the denominator into p² - nq², a rational number. From there, ordinary arithmetic will finish the job.

Fields can be extended by introducing more than one irrational square root, or by introducing roots other than square roots, but everything becomes more complicated.

One analogous extension that is of great mathematical and practical importance is the extension of the field of real numbers to include √−1 or i. A process similar to the one used to rationalize denominators is used to convert a denominator from a complex number involving i into a real number.