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1.3 Exploring Real Numbers Real Numbers: ◦ The set of numbers containing both rational and irrational numbers ◦ All the numbers we deal with are REAL *All numbers are either Rational or Irrational* Rational Numbers: Irrational Numbers: ◦ Any number you can write as a fraction ◦ Any decimal that either ends or repeats with a pattern ◦ ◦ ◦ ◦ Any number that cannot be written as a fraction Non-terminating, non-repeating wacky decimals Examples? If a number is irrational, it cannot belong to any other set Vocabulary If a number is rational, it could also be one of the following: Natural Numbers: Whole Numbers: Integers: ◦ 1,2,3,4…. ◦ Also called Counting Numbers - These are the numbers we naturally count with ◦ 0,1,2,3…. ◦ Only positive numbers and 0 (the word “whole” has an 0 in the middle!) ◦ ◦ ◦ ◦ Negatives and positive numbers and 0 No fractions or decimals All the numbers you see on my number line -2, -1, 0, 1, 2… More Vocabulary… a) -13 ◦ Rational numbers ◦ Integers b) 3.28 ◦ Rational numbers c) Rational numbers d) 42 ◦ ◦ ◦ ◦ Rational numbers Natural numbers Whole Numbers Integers 1. Identifying Sets of Numbers a) outdoor temperatures ◦ Integers or Rational Numbers b) the number of beans in a bag ◦ Whole Numbers ◦ Why not Natural Numbers? 2. Identifying Sets of Numbers A counterexample is an example that proves a statement false. True or False: All negative numbers are integers. ◦ False ◦ Counterexample? 3. Counterexamples Inequality: compares the value of two quantities, using <, >, ≤, or ≥. Compare the following fractions: 4 4 18 24/7 13 15 4. Inequalities 7 Write - , - , and to greatest. in order from least ◦ Find the decimal approximation of each. ◦- ,- ,- 5. Order fractions Opposites are on opposites sides of zero and are the same distance away from zero. Examples? ◦ -2 and 2, -6 and 6, etc. Absolute Value is the distance a number is from zero. a) l-2.5l ◦ 2.5 b) l7l ◦7 4. Opposites and Absolute Value Keep your assignment out. We are going to grade. Please bring a pen to class to use. Remember, no stars, just a mark if the answer is wrong. Wednesday, August 24 1. 2. 3. 4. 5. 6. 7. 8. Integers, rational Rational Rational Natural, whole, integers, rational Rational Integers, rational Whole, integers, rational Rational Rational 10. Irrational 11. Answers vary 12. Answers vary 13. Answers vary 14. Rational 15. Whole 16. Integers 17. Whole 18. Rational 9. Answers for 1-18 20. False 22. True 24. > 26. > 28. 2.001, 2.01, 2.1 30. -5/6, -1/2, 2/3 32. 0.63, 0.636, 7/11 34. 4 36. 9/14 38. 3/5 40. 1295 Answers for 20-40 even