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Math Pacing

Square Roots and Real Numbers. Math Pacing. 1. 2. Square Roots and Real Numbers. Statistics. Square Roots and Real Numbers. A square root is one of two equal factors of a number. For example, one square root of 64 is 8 because 8 • 8 = 64 .

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Math Pacing

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  1. Square Roots and Real Numbers Math Pacing 1. 2.

  2. Square Roots and Real Numbers Statistics

  3. Square Roots and Real Numbers A square root is one of two equal factors of a number. For example, one square root of 64 is 8 because8 • 8 = 64. Another square root of 64 is – 8 because(– 8) • (– 8) = 64. A number like 64, whose square root is a rational number is called a perfect square. Statistics

  4. Square Roots and Real Numbers indicates the principal square root of 64 indicates the negative square root of 64 indicates both square root of 64 The symbol , called a radical sign, is used to indicate a nonnegative or principal square root. Statistics

  5. Square Roots and Real Numbers Note that is NOT the same as . The notation represents the negative square root of 64. The notation represents the square root of – 64, which is NOT a real number because no real number multiplied by itself is negative. Statistics

  6. Find . Answer: represents the positive and negative square roots of Find Square Roots Example 7-1a

  7. Find . represents the positive square root of 0.0144. Answer: Find Square Roots Example 7-1c

  8. Find each square root. a. b. Answer: Find Square Roots Example 7-1d Answer: 0.6

  9. Square Roots and Real Numbers Recall that rational numbers are numbers that can be expressed as terminating or repeating decimals or in the form , where a and b are integers and b≠ 0. As you have seen, the square roots of perfect squares are rational numbers. However, numbers such as and are the square roots of numbers that are not perfect squares. Statistics

  10. Square Roots and Real Numbers Numbers like these cannot be expressed as a terminating or repeating decimal. Numbers that are not rational numbers are called irrational numbers. Irrational numbers and rational numbers together form the set of real numbers. Statistics

  11. Square Roots and Real Numbers Statistics natural numbers (N) whole numbers (W) integers (Z) rational numbers (Q) real numbers (R) irrational numbers (R – Q) Keep this handout in your notes.

  12. Name the set or sets of numbers to which belongs. Answer: Because , which is neither a repeating nor terminating decimal, this number is irrational (R – Q). Classify Real Numbers Example 7-2a

  13. Name the set or sets of numbers to which belongs. Answer: Because 1 and 6 are integers and , whichis a repeating decimal, the number is a rationalnumber (Q). Classify Real Numbers Example 7-2b

  14. Name the set or sets of numbers to which belongs. Answer: Because this number is a natural number (N), a whole number (W), an integer (Z) and a rational number (Q). Classify Real Numbers Example 7-2c

  15. Classify Real Numbers Example 7-2d Name the set or sets of numbers to which –327 belongs. Answer: This number is an integer (Z) and a rational number (Q).

  16. Name the set or sets of numbers to which each real number belongs.a. b. c. d. Classify Real Numbers Example 7-2e Answer: rationals (Q) Answer: naturals (N), whole (W), integers (Z), rationals (Q) Answer: irrationals (R – Q) Answer: integers (Z), rationals (Q)

  17. Square Roots and Real Numbers In lesson 2-1 you graphed rational numbers on a number line. However, the rational numbers alone do not complete the number line. By including irrational numbers, the number line is complete. This is illustrated by the Completeness Property which states that each point on the number line corresponds to exactly one real number. Statistics

  18. Square Roots and Real Numbers Recall that inequalities like x < 7 are open sentences. To solve the inequality, determine what replacement values for x make the sentence true. This can be shown by the solutions set:{all real numbers less than 7}. Not only does this include integers like 5 and – 2, but it also includes rational numbers like and and irrational numbers like and . Statistics

  19. Graph . Graph Real Numbers Example 7-3a The heavy arrow indicates that all numbers to the left of 8 are included in the graph. The dot at 8 indicates that 8 is included in the graph.

  20. Graph . Graph Real Numbers Example 7-3b The heavy arrow indicates that all the points to the right of –5 are included in the graph. The circle at –5 indicates that –5 is not included in the graph.

  21. Graph each solution set. a. b. Answer: Answer: Graph Real Numbers Example 7-3c

  22. Square Roots and Real Numbers To express irrational numbers as decimals, you need to use rational approximation. A rational approximation of an irrational number is a rational number that is close to, but not equal to, the value of the irrational number. For example, a rational approximation of is 1.41 when rounded to the nearest hundredth. Statistics

  23. Since the numbers are equal. Answer: Compare Real Numbers Example 7-4a Replace the  with <, >, or = to make the sentence true.

  24. Answer: Compare Real Numbers Example 7-4b Replace the  with <, >, or = to make the sentence true.

  25. Replace each  with <, >, or = to make each sentence true. a. b. Do these in your notes, PLEASE! Example 7-4c Compare Real Numbers Answer: < Answer: <

  26. Square Roots and Real Numbers You can write a set of real numbers in order from greatest to least or from least to greatest. To do so, find a decimal approximation for each number in the set and compare. Statistics

  27. Write in order from least to greatest. Write each number as a decimal. or about 2.4495 Order Real Numbers Example 7-5a

  28. Answer:The numbers arranged in order from least to greatest are or about 2.4444 Order Real Numbers Example 7-5b

  29. Write in order from least to greatest. Answer: Do this in your notes, PLEASE! Example 7-5c Order Real Numbers

  30. Square Roots and Real Numbers You can use rational approximations to test the validity of some algebraic statements involving real numbers. Statistics

  31. Multiple-Choice Test ItemFor what value of x is true? A –5 B 0 C D 5 Read the Test Item The expression is an open sentence, and the set of choices is thereplacement set. Rational Approximation Example 7-6a

  32. Solve the Test Item Replace x in with each given value. A False; and are not real numbers. Rational Approximation Example 7-6b

  33. B C False; is not a real number. Rational Approximation Example 7-6c Use a calculator. 0.447214 < 1 < 2.236068 True

  34. D The inequality is true for Rational Approximation Example 7-6e Use a calculator. 2.236068 < 1 < 0.447214 False Answer: The correct answer is C.

  35. Multiple-Choice Test ItemFor what value of x is true? A 3 B –3 C 0 D Rational Approximation Example 7-6f Answer: A

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