Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. In grade school we learn to rationalize denominators of fractions when possible. An answer on this site says that "there is a bias against roots in the denominator of a fraction".
So why does this bias against roots in the denominator exist and what is its justification? The only reason I can think of is that the bias is a relic of a time before the reals were understood well enough for mathematicians to be comfortable dividing by irrationals, but I have been unable to find a source to corroborate or contradict this guess. This was very important before computers in problems where you had to do something else after computing an answer. One simple example is the following: When you calculate the angle between two vectors, often you get a fraction containing roots.
The simplest way to define a standard form is by making the denominator or numerator integer. If you wonder why the denominator is the choice, it is the natural choice: As I said often you need to make computations with fractions. Note that bringing fractions to the same denominator is usually easier if the denominator is an integer. But at the end of the day, it is just a convention. The one which looks simpler is often relative The historical reason for rationalizing the denominator is that before calculators were invented, square roots had to be approximated by hand.
I may have missed it, but there is an important reason that I think has been omitted from the other answers. Ahaan Rungta mentioned it, but did not explain in detail. And so on. The difficulty of the calculations depends only on the complexity of the divisor , which is To extract a result with any required degree of precision one needs only continue the calculation until the required number of digits have been emitted.
But the operations themselves are determined by the divisor. Using an exact value for the divisor is impossible because of the way the algorithm works, so you must truncate the divisor. It's not clear how much error will be introduced by this truncation.
If you need more digits later you can easily produce them when you need them. The main reason I'd guess our math teacher culture tells us to require rationalizing the denominator is so that there is one set universal nomenclature among students about what a standard form means. Historical thing: before calculators, you had to do things by hand duh.
But I developed that facility after rationalizing a bunch of denominators, which makes me think that it's perfectly useful as a pedagogical bias. This is quite related but not identical to making the denominator real for complex valued fractions such as. Of course that can be intermingled with non-rational numbers, e. Like so many things it is nothing to get particularly obsessed about, but knowing how to rationalise denominators is quite a useful tool to have at one's disposition.
In fact more so in the general context of manipulating expressions than just for simplifying numbers. It is based on a small trick that is easy to understand, but which most people would probably not have thought of if it were not taught to them. Also I think that a similar method though maybe better called realising than rationalising is used in the most striaghtforward proof of the fact that the complex numbers are a field.
Rationalizing the denominator RTD a special case of the method of simpler multiples is useful because it often serves to simplify problems, e. This can lead to all sorts of simplifications, e. Here's another example from number theory showing how RTD serves to reduce divisibility of algebraic integers to rational integers.
Harold Edwards: Divisor Theory. The process of rationalizing the denominator with its conjugate is as follows. Another way to rationalize the denominator is to use algebraic identities. Let us understand this with an example. Let's rationalize the denominator in the following way:.
The same procedure that we followed to rationalize the denominator with 2 terms, we can follow those steps but with a little variation. Here is a list of a few points that should be remembered while studying about rationalize the denominator. To rationalize a denominator with a square root, we multiply both the numerator and the denominator with the same square root.
Rationalizing the denominator means the process of moving a root, for instance, a cube root or a square root from the bottom of a fraction to the top of the fraction. Learn Practice Download. Rationalize the Denominator We rationalize the denominator to ensure that it becomes easier to perform any calculation on the rational number. What is Rationalizing? Rationalize the Denominator Using Conjugates 3.
So when we rationalize either the denominator or numerator we want to rid it of radicals. Note that the phrase "perfect square" means that you can take the square root of it.
Just as "perfect cube" means we can take the cube root of the number, and so forth. Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent. Step 2: Make sure all radicals are simplified. Some radicals will already be in a simplified form, but make sure you simplify the ones that are not.
If you need a review on this, go to Tutorial Simplifying Radical Expressions. Step 3: Simplify the fraction if needed. Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical. Example 1 : Rationalize the denominator. Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.
Since we have a square root in the denominator, then we need to multiply by the square root of an expression that will give us a perfect square under the radical in the denominator. Square roots are nice to work with in this type of problem because if the radicand is not a perfect square to begin with, we just have to multiply it by itself and then we have a perfect square.
It is real tempting to cancel the 3 which is on the outside of the radical with the 6 which is inside the radical on the last fraction.
You cannot do that unless they are both inside the same radical or both outside the radical like the 4 in the numerator and the 6 in the denominator were in the second to the last fraction. Example 2 : Rationalize the denominator. Since we have a cube root in the denominator, we need to multiply by the cube root of an expression that will give us a perfect cube under the radical in the denominator. Also, we cannot take the cube root of anything under the radical. So, the answer we have is as simplified as we can get it.
Rationalizing the Numerator with one term As mentioned above, when a radical cannot be evaluated, for example, the square root of 3 or cube root of 5, it is called an irrational number. So, in order to rationalize the numerator, we need to get rid of all radicals that are in the numerator. Note that these are the same basic steps for rationalizing a denominator, we are just applying to the numerator now. Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the numerator.
If the radical in the numerator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the numerator.
If the radical in the numerator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the numerator and so forth Example 3 : Rationalize the numerator.
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