Rational Numbers in Mathematics. Lecture 2.1: Fractions INTRODUCTION 2.0

Transcription

1 Rational Numbers in Mathematics INTRODUCTION 2.0 In this lesson, we will focus on the exploration of rational numbers in all their forms and operation on rational numbers. We will explore fractions and decimals and go into detail on how to convert from one form to another. Lecture 2.1: Fractions A fraction represents part of a whole. For example, let's say that you order a large pizza with 10 slices to share with 2 friends. Once everyone's taken a slice, what fraction of the pizza is left? There were 10 slices to begin with. Each person takes one slice each, which amounts to 3 slices total. This leaves 7 slices remaining in the pizza box. In other words, there's 7 out of 10 slices left, or 7/10 of the pizza:

2 There are three key parts to a fraction: the denominator, the numerator, and the fraction line: The denominator is the number in the bottom of the fraction and it represents the number of parts in the whole. For our pizza example, there are a total of 10 slices in 1 pizza, so the denominator is 10. The numerator is the number in the top of the fraction. It tells us how many parts of the whole we have. For instance, we have 7 slices of pizza left. The fraction line separates the numerator and denominator. In words, it means "out of" or "divided by", such as 7 out of 10 slices. Lecture 2.2: Multiplication of Fractions We encounter multiplication of fractions when we want to find part of a subset or as operations to other calculations. For an example of the former, let's say that 1/2 of the students at Grand Canyon University are female and that 2/3 of the female students are under 21. What fraction of the student population is female and under 21? To calculate this, we take the following product: How do we multiply these fractions? We simply multiply the numbers in the numerator and multiply the numbers in the denominator: Before continuing with the multiplication, we can cancel like values, such as the 2. This will reduce the problem to the following: Thus, 1/3 of the students that attend Grand Canyon University are female students under the age of 21.

3 Lecture 2.3: Addition and Subtraction of Fractions In order to add or subtract fractions, they must have the same number in the denominator, called a common denominator. For example, the following fractions have a common denominator: To add or subtract fractions with common denominators, we sum or take the difference of the number in the numerator and keep the denominator the same: So what do we do when we want to add or subtract fractions with different denominators? The first step is to find a common denominator. We can find the common denominator by looking at the prime factors of each denominator. Recall that a prime number is a number that is divisible by 1 and itself, and no other. For example, 3 is a prime number. Now, let's see how prime numbers can aid in the discovery of a common denominator. Suppose that we are asked to determine the sum of the following fractions: If we look at the denominators for each tern in the equation above, we see that the denominators are 4, 18, and 72. Let's take a look at the prime factors of each.

4 We can see that the prime factors for 4 are 2 and 2. In other words, if we multiply 2 times 2 we get 4; 2 times 3 times 3 gives 18. Let's now look at only the prime factors for each: We now want to compare the three above to one another. Let's compare the prime factors of 18 and 72 first: Notice the highlighted portion for the prime factors of both denominators. They are exactly the same, except that 72 has two additional prime factors. Our goal is to get them to be identical; therefore, we need to include the prime factors of 72 that are not included for the prime factors of 18: Now both have the same prime factors. We also need to do the same for the prime factors of 4: Now that all the prime factors are equal to one another, we can multiply all the factors together to determine the least common devisor, which in this case is 72, one of our original denominators. Keep in mind, this will not always be the case. Now that we have determined the least common devisor, let's use this information in performing the necessary operations. We were asked to determine the following: We need to convert each denominator into the least common devisor 72. Therefore, we need to determine what we need to multiply ever denominator by to change it from what it is currently to 72: Recall though, we have done the majority of this work already. For example, recall the work we did with the prime factors earlier, specifically, the prime factors of 4: 2 x 2

5 In the process of including all prime factors across all the denominators, we had to include other prime factors: 2 x 2 x 2 x 3 x 3 To get 72 using the prime factors of 4, we had to multiply by, or 18. Thus, we need to multiply the denominator by 18: But, what we do to the bottom, we must also do to the top: We use the same prime factor reasoning to change 18 to 72: 2 x 3 x 3 2 x 2 x 2 x 3 x 3 Thus, we need to multiply the denominator by 4: But, what we do to the bottom, we must also do to the top: Finally, what do we need to multiply 72 by to get 72? We just need to multiply by 1: And what we do to the bottom, we must also do to the top:

6 Notice that we did nothing to the original equation: You may be telling yourself, "These two equation are clearly different!", but let's take a closer look at this. First, if I were to ask, "What is 18 divided by 18?" the answer is 1. And 4/4? This is a mathematician's trick, in that if we multiply by something we also divide it out. So, now let's complete the multiplication, and simplify: Now that we have had a chance to go through a step by step strategy for adding and subtracting rational numbers, let's practice using the following interactive guide. MEDIA HERE:

7 Lecture 2.4: Division of Fractions Let's say that you need to perform the following division problem: How do you complete such a division problem? In order to divide these fractions, your strategy should be to turn this into a multiplication problem. Then, you can use the rules of multiplication to solve the problem. In order to change a division problem into a multiplication problem, there are two steps: 1. Flip the sign, then 2. Flip the 2nd fraction. "Flip the sign" changes division to multiplication. "Flip the 2nd fraction" swaps the numerator and denominator of the second fraction. Our example can be converted to the following multiplication problem: Mathematically, what's taking place is the following. We were given the original: Which is equivalent to: A complex fraction is a fraction that has for its numerator and denominator fractions itself. This is an complex fraction, so we need to remove the fraction from the denominator. In order to do this though, we need to multiply the

8 denominator by the reciprocal. The reciprocal of 3/5 is 5/3. Thus, by multiplying the numerator and the denominator by the reciprocal of 5/3, we remove the fraction from the denominator: Using the properties of fraction multiplication discussed earlier, we get the following: Lecture 2.5: Convert Fractions to Decimals To convert a number from a fraction to a decimal involves division. For example, the fraction 3/8 represents "3 out of 8," which has the same meaning as "3 divided by 8." The first step in changing the fraction to a decimal is to rewrite it as a long division problem: Note that the numerator is under the division sign and the denominator is outside of the division sign. Nest, we add our decimal points. We add the first decimal point directly after the number under the division sign. Then, we add a second decimal point (the one that will go in our answer) directly above the first one:

9 Continuing to apply methods used from Lecture 1, we can determine the quotient: Thus, the fraction 3/8 can be written as the decimal Conclusion: none References: none