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NY Regents Exam - Algebra I: Test Prep & Practice14 chapters | 93 lessons | 6 flashcard sets

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Lesson Transcript

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this video, we'll explore rational and irrational numbers by looking at what happens when they're added and multiplied. We'll use examples to explain these properties and learn more about rational and irrational numbers.

Having found seasonal work in town, Fred is delighted to start his new job at the Real Number Emporium. His first assignment is to sort the real numbers coming down the conveyor belt into two boxes, labeled Q and I. The letter ''Q'' is for the set of rational numbers, where ''Q'' stands for ''quotient.'' The letter ''I'' is for the set of irrational numbers. You see, real numbers are either **rational** or **irrational**. Fred knows the word ''irrational'' means ''not rational,'' but he's not clear on how to spot rational numbers. If only he'd stayed awake in algebra class. Let's see if we can refresh Fred's skills and help him keep his job.

The word ''rational'' includes the word ''ratio.'' A **ratio** is the quotient of two numbers. A ratio *a* / *b*, where *a* and *b* are integers like {. . ., -3, -2, -1, 0, 1, 2, 3,. . .} is the key to identifying rational numbers. The only restriction on *b* is *b* â‰ 0, because dividing by 0 is undefined. If a number can be written as *a* / *b*, it's a rational number, and goes into the Q box.

Imagine the first number Fred sees is 7. Okay, 7 = 7 / 1, so 7 is a rational number.

The next number is 2.56. Okay, 2.56 = 256 / 100, so again it goes into the Q box.

The third number is 0.3. Hmmm, 0.33. . . is a repeating decimal which equals 1/3, so it too is a rational number.

Will Fred ever see an irrational number?

The next number is âˆš4. Roots of numbers may or may not be rational. In this case, âˆš4 = 2 , and 2 / 1 is a rational number.

The next number, âˆš5, is not a perfect square and cannot be written as *a* / *b*. Therefore, âˆš5 is an irrational number, and it goes in the I box.

Fred has been on the job 15 minutes, and already he needs a break. He asks you to take over. How would you sort the following real numbers?

âˆš2, âˆš16, -31, Ï€, and .7142857142857142. . .

Your decisions are:

- âˆš2 is not a perfect square, so it's irrational.
- âˆš16 is a perfect square equal to 4 / 1, so it's rational.
- Ï€ can't be written as
*a*/*b*(even though Ï€ â‰… 22 / 7), which makes it irrational. - .7142857142857142. . . is a repeating decimal. It can be written as 5 / 7, which makes it rational.

There is a method for converting this repeating decimal into a ratio. Let's take a look.

*x* = .714285

106*x* - *x* = 714285*x* = 714285 / (106 - 1)*x* = 714285 / 999999

If we factor and cancel, we'll end up with 5 / 7

Fred is back on the job and finishes his first day. The boss, Mrs. Real, is impressed with your work and offers you a job in quality control. That's right, you're going to take samples out of the rational box and test them.

You decide to use some properties:

**Property 1:**The sum of two rational numbers is rational.

In short-hand form: Q + Q âˆˆ Q.

The symbol âˆˆ means ''is in'' or ''belongs to.''**Property 2:**The product of two rational numbers is rational.

Q x Q âˆˆ Q

If we add or multiply two rational numbers, the result is still a rational number.

The quality control starts with the numbers 7 and 2.56.

Let's add:

7 + 2.56 = 9.56 = 956 / 100, which is rational.

Now let's multiply:

7 x 2.56 = 17.92 = 1,792 / 100, which is rational.

The next two numbers to check are âˆš4 and âˆš5:

Once again, we'll add:

âˆš4 + âˆš5 = 2 + âˆš5, which can't be written as the ratio of *a* / *b*. So, one of the numbers was mistakenly put in the Q box. It's the âˆš5, which is an irrational number.

This leads to two more properties:

**Property 3:**The sum of a rational number with an irrational number is an irrational number.

Q + I âˆˆ I**Property 4:**The product of a rational number with an irrational number is an irrational number.

Q * I âˆˆ I

There is an exception to keep in mind! 0 is a rational number. Multiplying the rational number 0 times any irrational number gives 0. So, Property 4 is good, provided the rational number is not 0.

Mrs. Real is ready to give you a promotion, but she has some interview questions for you. What if you add or multiply two irrational numbers? Can you say anything about the result?

So far, we know adding or multiplying rational numbers gives a rational number:

Q + Q âˆˆ Q and Q * Q âˆˆ Q

We also know adding or multiplying a rational number with an irrational number gives an irrational number:

Q + I âˆˆ I and Q * I âˆˆ I

The first question posed by Mrs. Real: ''What can you say about I + I?''

For example, âˆš7 is irrational. So is (6 - âˆš7); but added together, the result is:

âˆš7 + (6 - âˆš7) = 6, which is rational.

On the other hand, âˆš 7 + âˆš7 = 2âˆš7, which is irrational.

So the answer is:

**Property 5:**The sum of two irrational numbers is sometimes rational and sometimes irrational.

Mrs. Real likes your answer and would now like you to ponder what happens when two irrational numbers are multiplied.

For example, âˆš3 is irrational. So is âˆš5. The product is:

âˆš3 x âˆš5 = âˆš(15), which is not a perfect square; thus, the result is irrational.

How about âˆš3 and âˆš3? Two irrational numbers multiplied together give:

âˆš3 x âˆš3 = âˆš9 = 3, a rational number.

How about the two irrational numbers âˆš8 and âˆš2? Multiplied together:

âˆš8 x âˆš2 = âˆš(16) = 4, a rational number.

Your answer to Mrs. Real is:

**Property 6:**The product of two irrational numbers is sometimes rational and sometimes irrational.

Mrs. Real is overjoyed! You start tomorrow. Fred, on the other hand, is already looking for other work that will let him sleep later in the morning and take off on 3-day weekends.

Real numbers are **rational** if they can be written as the ratio *a* / *b* where *a* and *b* are integers and where *b* â‰ 0. If a real number is not rational, it is **irrational**. The numerical interactions of rational and irrational numbers can be stated in the following six properties:

**Property 1:**The sum of two rational numbers is rational.**Property 2:**The product of two rational numbers is rational.**Property 3:**The sum of a rational number with an irrational number is an irrational number.**Property 4:**The product of a rational number with an irrational number is an irrational number.**Property 5:**The sum of two irrational numbers is sometimes rational and sometimes irrational.**Property 6:**The product of two irrational numbers is sometimes rational and sometimes irrational.

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NY Regents Exam - Algebra I: Test Prep & Practice14 chapters | 93 lessons | 6 flashcard sets

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