Rationalizing the numerator of a fraction is necessary when you are working with an irrational number. This lesson will focus on identifying irrational numbers in your fraction and using that irrational number to manipulate your fraction.
Rationalizing the Numerator
When thinking about what you’ve learned about the different categories of numbers, you might remember the broad terms, rational and irrational numbers. It’s important to understand how these two groups differ from each other before diving into rationalizing either the numerator or denominator of a fraction.
The major distinction between a rational and irrational number is whether or not a number can be written as the ratio of two integers, or whole numbers, in the form of decimals, fractions, or whole numbers. A rational number, no matter how large, meets this requirement, while an irrational number does not.Irrational numbers can come in the form of a decimal with endless non-repeating digits or a non-perfect integral, such as an integral that does not spit out a whole number.
A common example of an irrational number is pi, or 3.1415926….In terms of integrals, the square root of four results in the whole number, two, making it a rational number. However, the square root of five results in the non-terminating and non-repeating decimal 2.
23606 etc, making it an irrational number. Because we sometimes see fractions with an irrational number in the numerator or denominator, it is important to have a set of procedures to rationalize, or turn that irrational number into a rational one. While it is more common to rationalize a denominator, there are still cases where rationalizing the numerator is necessary.
Say you are given the following fraction and you are asked to rationalize your numerator.
Step #1: Take a look at your numerator and decide if it is irrational or rational. In this case, we have an irrational number, the square root of 8.
Step #2: Once you have determined that your numerator is irrational, you are going to create a new fraction using your irrational number as both the numerator and denominator. Remember, when you have the same number as the numerator and denominator, your fraction is equivalent to the number one.
Step #3: Because your created fraction is equivalent to one, and the identity property tells us that we can multiply any number by one and get the same number, you can multiply your original fraction by your created fraction and your answer will be equal to your original number.
Step #4: One of the rules of integrals tells us that when we multiply an integral by itself, the integral sign goes away, and we are left with the number inside of the integral.
In the case of our numerator, we multiply the square root of 8 by the square root of 8 to eliminate our integral. Now we see why we need to multiply by our irrational number.
Step #5: This leaves us with a rational number in our numerator, but it is very important for us to NOT stop there. You must always simplify your answer! In the case of our solution, we can reduce the fraction by a factor of two to get our final answer with the numerator rationalized.
Since we multiplied our original fraction by a fraction equivalent to one, we can safely say that our final answer is the exact same as our original fraction, just in a different form so that our numerator is a rational number.
There are going to be cases where you are presented with an irrational fraction, but there are multiple terms attached to the irrational number. Let’s take the following fraction into consideration:
We must follow the same steps as above, but instead we are going to be using something called a conjugate. The conjugate is a fancy way of saying, change the sign. So in the case of our example, the conjugate would be: 9 + square root of 6.
All we did was change our minus sign to a plus sign.Step #1: We can see that our irrational number is the square root of 6. Here is where the procedure starts to deviate. Instead of identifying just the irrational number, we are going to look at the big picture: 9 – square root of 6.Step #2: Using our conjugate as both our numerator and denominator, we’ll create a new fraction. But instead of just using the irrational number, we’ll use the conjugate.
Step #3: As before, we must multiply our original fraction by the created fraction.
Step #4: When multiplying, we must use FOIL in our numerator, which means:
- Multiply the First
- Multiply the Outer
- Multiply the Inner
- Multiply the Last
After using our FOIL method, we can reduce the numerator by eliminating the + 9 square roots of 6 and – 9 square roots of 6. This leaves us with 81 – 6 = 75.
In this lesson you not only reviewed the characteristics of an irrational number, but you also learned the steps necessary for rationalizing the numerator of a fraction. The steps include creating a new fraction with the conjugate, multiplying through the numerator and denominator, and simplifying your answer.