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Properties of Rational & Irrational Numbers

Properties of Rational & Irrational Numbers
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  • 0:04 Rational & Irrational Numbers
  • 0:49 Identifying Rational Numbers
  • 3:50 Properties of Rational Numbers
  • 6:04 Adding/Multiplying…
  • 8:09 Lesson Summary
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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.

Rational & 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.

Identifying Rational Numbers

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 ≠ 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 √4. Roots of numbers may or may not be rational. In this case, √4 = 2 , and 2 / 1 is a rational number.

The next number, √5, is not a perfect square and cannot be written as a / b. Therefore, √5 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?

√2, √16, -31, π, and .7142857142857142. . .

Your decisions are:

  • √2 is not a perfect square, so it's irrational.
  • √16 is a perfect square equal to 4 / 1, so it's rational.
  • π can't be written as a / b (even though π ≅ 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
106x - x = 714285
x = 714285 / (106 - 1)
x = 714285 / 999999
If we factor and cancel, we'll end up with 5 / 7

factoring_714285_and_999999

Properties of Rational Numbers

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 ∈ Q.
    The symbol ∈ means ''is in'' or ''belongs to.''
  • Property 2: The product of two rational numbers is rational.
    Q x Q ∈ 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 √4 and √5:

Once again, we'll add:
√4 + √5 = 2 + √5, 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 √5, 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 ∈ I
  • Property 4:The product of a rational number with an irrational number is an irrational number.
    Q * I ∈ I

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