1. Rational Vs. Irrational Making sense of rational and Irrational numbers

2. Learning Goal Decompose the Real Number system into parts to see the differences between sets of numbers.

3. Success Criteria I can identify subsets of the real number system I can sort rational and irrational numbers I can locate rational and irrational numbers on the number line I can identify subsets of the real number system I can sort rational and irrational numbers I can locate rational and irrational numbers on the number line

4. Animal Reptile Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number. Lizard Gecko

5. The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers. Irrational numbersRational numbers Real Numbers Integers Whole numbers

6. Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 3 = 3.8 4545 = 0.6 2323 1.44 = 1.2

7. A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution! Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational.

9. Check It Out! Example 1 Write all classifications that apply to each number. 9 whole, integer, rational, real –35.9 is a terminating decimal. –35.9 rational, real 81 3 whole, integer, rational, real = = 3 9393 81 3 A. B. C. 9 = 3

10. Additional Example 1: Classifying Real Numbers Write all classifications that apply to each number. 5 is a whole number that is not a perfect square. 5 irrational, real –12.75 is a terminating decimal. –12.75 rational, real 16 2 whole, integer, rational, real = = 2 4242 16 2 A. B. C.

11. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

12. 23 is a whole number that is not a perfect square. 23 irrational 9090 undefined, so not a real number Check It Out! Example 2 A. B. State if each number is rational, irrational, or not a real number.

13. 64 81 rational 8989 = 8989 64 81 C. Check It Out! Example 2 State if each number is rational, irrational, or not a real number.

14. 21 irrational 0303 rational 0303 = 0 Additional Example 2: Determining the Classification of All Numbers A. B.

15. not a real number Additional Example 2: Determining the Classification of All Numbers 4040 C. State if each number is rational, irrational, or not a real number.

16. The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.

17. Additional Example 3: Applying the Density Property of Real Numbers 2525 3 + 3 ÷ 2 3535 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.5 = 6 ÷ 2 1212 = 7 ÷ 2 = 3 3 1212 333 1515 2525 433 3535 4545 Find a real number between 3 and 3. 3535 2525 A real number between 3 and 3 is 3. 3535 2525 1212 Check: Use a graph.

18. Check It Out! Example 3 3737 4 + 4 ÷ 2 4747 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.7 = 8 ÷ 2 1212 = 9 ÷ 2 = 4 4 1212 444444 2727 3737 4747 5757 1717 6767 Find a real number between 4 and 4. 4747 3737 A real number between 4 and 4 is 4. 4747 3737 1212 Check: Use a graph.

19. Lesson Quiz Write all classifications that apply to each number. 1. 2. – State if each number is rational, irrational, or not a real number. 3. 4. Find a real number between –2 and –2. 3838 3434 5. 2 4 9 16 2 25 0 not a real number rational real, irrational real, integer, rational Possible answer: –2 5858

20. Sorting Activity Sort the cards from the envelope into rational numbers and irrational numbers Can the rational numbers be broken into smaller categories? Sort the cards from the envelope into rational numbers and irrational numbers Can the rational numbers be broken into smaller categories?

21. Number Line Activity Draw a number line on the construction paper Use the colored stickers to label each number from the set Green-Rational only Pink-Rational, but could be a whole number or an integer also Yellow-Irrational Draw a number line on the construction paper Use the colored stickers to label each number from the set Green-Rational only Pink-Rational, but could be a whole number or an integer also Yellow-Irrational

22. How’d we do with our Success Criteria? I can identify subsets of the real number system I can sort rational and irrational numbers I can locate rational and irrational numbers on the number line I can identify subsets of the real number system I can sort rational and irrational numbers I can locate rational and irrational numbers on the number line

23. So, did we meet our Learning Goal? Decompose the Real Number system into parts to see the differences between sets of numbers.