Conjugates and Rationalization

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Introduction:

The term conjugate means a pair of things joined together. Let's look at these smileys: 

smiley  sad

These two smileys are exactly the same except for one pair of features that are actually opposite of each other. If you look at these smileys, you will notice that they are the same except that they have opposite facial expressions: one has a smile and the other has a frown.


What are the math Conjugates?

math conjugate is formed by changing the sign between two terms in a binomial. For instance, the conjugate of \(x + y\) is \(x - y\). We can also say that \(x + y\) is a conjugate of \(x - y\). In other words, the two binomials are conjugates of each other. Instead of a smile and a frown, math conjugates have a positive sign and a negative sign, respectively.

Let's consider a simple example: The conjugate of \(3 + 4x\) is \(3 - 4x\). 

Now, Consider the surd \(3 + \sqrt 2 \),

If we change the plus sign to minus, we get the conjugate of this surd: \(3 - \sqrt 2 \).

What is special about conjugate of surds?


Rationalization of surds:

The special thing about conjugate of surds is that if you multiply the two (the surd and it's conjugate), you get a rational number. For example,

\[\left( {3 + \sqrt 2 } \right)\left( {3 - \sqrt 2 } \right) = {3^2} - 2 = 7\]

We note that for every surd of the form \(a + b\sqrt c \), we can multiply it by its conjugate \(a - b\sqrt c \)  and obtain a rational number:

\[\left( {a + b\sqrt c } \right)\left( {a - b\sqrt c } \right) = {a^2} - {b^2}c\]

Let’s call this process of multiplying a surd by something to make it rational – the process of rationalization. In the example above, that something with which we multiplied the original surd was its conjugate surd.

How will we rationalize the surd \(\sqrt 2 + \sqrt 3 \)? The conjugate surd (in the sense we have defined) in this case will be \(\sqrt 2 - \sqrt 3 \), and we have,

\[\left( {\sqrt 2 + \sqrt 3 } \right)\left( {\sqrt 2 - \sqrt 3 } \right) = 2 - 3 = - 1\]

How about rationalizing \(2 - \sqrt[3]{7}\) ? The conjugate surd in this case will be  \(2 + \sqrt[3]{7}\), but if we multiply the two, we have

\[\left( {2 - \sqrt[3]{7}} \right)\left( {2 + \sqrt[3]{7}} \right) = 4 - \sqrt[3]{{{7^2}}} = 4 - \sqrt[3]{{49}}\]

which is not a rational number. Thus, the process of rationalization could not be accomplished in this case by multiplying with the conjugate. The rationalizing factor (the something with which we have to multiply to rationalize) in this case will be something else. But what?

✍Note: The process of rationalization of surds by multiplying the two (the surd and it's conjugate) to get a rational number will work only if the surds have square roots.


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