Bell Work: Simplify.√8 3. Answer:2 Lesson 16: Irrational Numbers.

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Presentation transcript:

Bell Work: Simplify.√8 3

Answer:2

Lesson 16: Irrational Numbers

Recall that the set of rational numbers is closed under addition, subtraction, multiplication and division. However, the square roots of many rational numbers, such as √2 and √3, are not rational numbers. Since √2 and √3 are greater than √1 and less than √4, their values are more than 1 and less than 2.

To find the approximate value of √2 and √3, we can use a calculator.  Use a calculator to find √1 and √4.  Use a calculator to find √2 and √3. How do the displays for √2 and √3 differ from the displays for √1 and √4?

No matter how many digits a calculator displays, the displays for √2 and √3 will be filled with digits without a repeating pattern. The display fills because there is no rational number – no fraction or repeating decimal – that equals √2 or √3.

The numbers √2 and √3 are irrational numbers. Irrational numbers*: Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non- ending and nonrepeating.

The square root of any counting number that is not a perfect square is an irrational number.

Example: Which number is irrational? √0√4√8√16

Answer: √0 = 0 √4 = 2 √16 = 4 √8 is an irrational number since it is not a perfect square.

Rational numbers and irrational numbers together form the set of real numbers. All of the real numbers can be represented by points on a number line, and all of the points on a number line represent real numbers.

Example: Arrange these real numbers in order from least to greatest. 2, √5, 5/9, 0, 3, 1.2, 1

Answer: 0, 5/9, 1, 1.2, 2, √5, 3

Example: The area of this square is 2cm. What is the length of each side? 2

Answer: √2 ≈ 1.4 ≈1.4cm

Practice: Which of these numbers is between 2 and 3 on the number line? How do you know? √2 √3√4√5

Answer:√5 The square root of 5 is between √4 and √9. the square root of 4 is 2 and the square root of 9 is 3.

Practice: What is the length of each side of a square with an area of 400mm ? 2

Answer:20mm

Practice: Use a calculator to find the square roots of these numbers. Round to the nearest hundredth. √10√20√40

Answer: √10 = 3.16 √20 = 4.47 √40 = 6.32

HW: Lesson 16 #1-30 Due Tomorrow