Why Do We Rationalise the Denominator? Standard Notation While Using Surds/ Radicals

The same surd fraction in two different forms

Rationalising the denominator​

When learning about surds/radicals at school, we are told that it is always best to rationalise the denominator. Nowadays, when I'm working with surds on a scientific calculator, it automatically gives me answers with rationalised denominators. But what does it mean to rationalise the denominator, and why do we do it?
Firstly, we should look at what rationalising the denominator means.

Surds, radicals and rationalisation​

When we have a number featuring roots, whether that's square roots, cube roots or any other form of root, we call it a surd in the UK or radical in the US. This could be a number such as 1+√2 or a fraction such as √3 / 2 or 1 / (1+√5).
Surds are part of a group of numbers called irrational numbers. When you square root any number other than a square number, you get an answer that cannot be written as a fraction of whole numbers and forms a never-ending decimal with no pattern or repetition to the numbers. e.g. √2 = 1.414213562...
Any number that can be written as a fraction of whole numbers is called a rational number and its decimal form will either end like 0.5 or 0.3647, or repeat itself (recur) like 0.3333... or 0.272727....
When we talk about rationalising the denominator, we mean converting a surd fraction into such a form that the denominator (the bottom of the fraction) is a rational number.
An example of this is shown in the picture above. √2 / 2 and 1 / √2 are exactly the same number (both equal to 0.7071067812...) but √2 / 2 has a rationalised denominator and 1 / √2 does not.
It is generally preferable to write this number as √2 / 2 with the rational denominator. But why is this the case? Let's look at some reasons.

Standard notation

One important part of mathematics is to have standard notation and procedure. That is how mathematics is a universal language, spoken around the world and understood by billions of people whether their spoken language is English, French, Japanese etc.
One example of this standardisation is when using BODMAS/PEMDAS. We are taught to use brackets first in an expression and then indices, before following on to multiplication and division, and finally completing any addition or subtraction. Because of this rule, we can give different people the sum 2 + 4 × 3 and they should all come back with the answer 14, not 18.
It is the same with surd fractions. We need a standard way of writing them and so it was decided that we would write them with a rational denominator.

Adding fractions including surds

Ease of computation

Having standard notation is a good reason for deciding on a preferred format. but so far we haven't seen a reason why a rational denominator was chosen as this standard.
There are two good reasons for this. One is that it is far easier to calculate with rational denominators. Have a look at the example above. We have two fractions with irrational denominators and adding these together without the help of a calculator looks like a horrible task.
If we first rationalise the denominator however, we get the following sum.

The same sum with rationalised denominators

Completing the addition

Now we have a much simpler sum to complete. The denominators are both rational numbers and we can use our addition method of multiplying top and bottom of each fraction to make the denominators the same. In this example we would multiply the first fraction by 2 top and bottom and multiply the second fraction by 5 top and bottom to give both fractions denominators of 10 before adding together as shown below.

Adding our surd fractions together

Converting to decimal form

The second reason for writing surds / radicals with rational denominators is to make it easier to convert into decimal form.
Before the use of calculators, any division needed to be done by hand. Let's look at how this affects us when trying to convert 1 / √2 into decimal form.
We would start by using the appropriate algorithm to calculate that √2 = 1.414213562.... To then convert 1 / √2 into decimal we would need to divide 1 by 1.414213562.

Trying to divide by a surd

We can see when trying to use the bus stop method that 1 ÷ 1.414213562... is not going to be easy to do.
What about if we rationalised the denominator first?
1 / √2 = √2 / 2
Now we just need to do 1.414213562... ÷ 2.

Dividing a surd by a whole number

You can see that this is a much easier division to do and quickly gives us an answer of 0.707106781...

Summary

So there we have it. It is usual in mathematics to choose a standard format for presenting and calculating. In the case for fractions involving surds / radicals there are two reasons for choosing the version with the rationalised denominator to be this standard:

1. They are easier to calculate with, especially when doing addition and subtraction.
2. They are easier to manually convert into decimal format.

Comments

​Don't forget to leave your comments below.