Irrational Numbers.pdf
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Name: _________________________________________________________ Period: ______ Date: ___________________________
11th Grade Mathematics PSSA Preparation Program o Mastered On: _____________________
Rational and Irrational Numbers Anchors Addressed M11.A.1.3.1 – Locate/identify irrational numbers at the approximate location on a number line. M11.A.1.3.2 – Compare and/or order any real numbers (rational and irrational may be mixed). Concepts Real numbers are further classified as rational or irrational numbers. Irrational numbers cannot be written as a fraction using only integers. Rational numbers in decimal form either terminate or repeat, but irrational numbers continue on forever without repeating. Examples of irrational numbers include: 3, 2𝜋, and 0.1221122211112222 … Since irrational numbers never terminate or repeat, the decimal form of an irrational number is always an approximation. Square roots of numbers, except perfect squares, and all solutions involving 𝜋 are approximate values. For example: 3 = 1.73205 … and 2𝜋 = 6.28318 … When solving problems involving 𝜋, the approximation 3.14 is acceptable on the PSSA and SAT. Classifying Numbers Example 1: Classify the following numbers as rational or irrational. A. 2.371732 … B. 0.625 C. 12.56637… Solution: To classify each of the values, determine if they can be written as a fraction. A. 2.371732 … cannot be written as a fraction because the decimal does not repeat or terminate, therefore the number is irrational. B. 0.625 can be written as a fraction and the value does terminate, therefore the number is rational. !"# ! = ! !""" C. 12.56637… cannot be written as a fraction because the decimal does not repeat or terminate, therefore the number is irrational. This number is 4𝜋.
Calculator Tip: You can use the calculator to turn decimals into fractions. On the TI-‐30x, type the decimal value and then press % j to change the decimal to a fraction. On a TI-‐83 calculator, from the Math menu, select Frac to change a decimal to a fraction.
Estimating Irrational Numbers Since the decimal form of an irrational number is an approximate value, we can approximate where the values appear on the number line. Example 1: Place the values 12, 𝜋, and 3.7671921 … on the number line. Solution: First, convert each value to a decimal. Therefore, 12 = 3.464101 …, 𝜋 = 3.14 …, and 3.7671921 is already a decimal. Once in decimal form, estimate the location on the number line.
Exercises A. Find the decimal form of each value to the nearest ten thousandth (3 decimal places) and determine if the following values are rational or irrational. 1. 16 !
2. !
3. 8
5. 324
!
4. !
!!
B. Determine if an exact solution can be found for the following measures.
6. !!
7. y n The area of a rectangle.
11. y n The surface area of a cylinder.
8. y n The area of a circle.
12. y n The area of a rectangle.
9. y n The perimeter of a triangle.
13. y n The volume of a sphere.
10. y n The volume of a cube.
14. y n The hypotenuse of any triangle
C. Answer the following questions about irrational numbers. 15. Can the area of a rectangle ever be irrational? If it is possible, provide an example.
D. Use the number line below to determine where each of the following values would be located.
16. 10 is between _____ and _____.
21. 3 3 is between _____ and _____.
17. 4 + 𝜋 is between _____ and _____.
22. 5 + 7 is between _____ and _____.
18.
! !" !"
is between _____ and _____.
23.
!" !
is between _____ and _____.
19. 2𝜋 is between _____ and _____.
24. 3 5 − 2 2 is between _____ and _____.
20. 45 is between _____ and _____.
25. 6 5 + 2 3 − 4 2 is between _____ and _____.
E. Order the following sets of numbers from least to greatest. 26. 14, 18, 4𝜋
31. 3 15, 2 21, 4 12
27. 22.2321, 625, 5! − 1
32. 125, 34 + 20, 10𝜋
28. (−1)! ,
!"# ! !"
, !
29. 3! , 95, 2𝜋
30. 132,
!
8 , 5 25
33.
! !! !
, 196, 200
34. 25 + 100, 50 + 75, 35 + 90
35. 2! , 272,
!! !
F. Without using a calculator, plot the following values on the number line. 36. 2.2! , 26,
!! !
37. 2𝜋, 38, 52
38. 2.75! , 18,
! !" !
!
39. 1.5! , − 2, !
40. (−1.4)! , − 3, 0.1!
11th Grade Mathematics PSSA Preparation Program o Mastered On: _____________________
Rational and Irrational Numbers Anchors Addressed M11.A.1.3.1 – Locate/identify irrational numbers at the approximate location on a number line. M11.A.1.3.2 – Compare and/or order any real numbers (rational and irrational may be mixed). Concepts Real numbers are further classified as rational or irrational numbers. Irrational numbers cannot be written as a fraction using only integers. Rational numbers in decimal form either terminate or repeat, but irrational numbers continue on forever without repeating. Examples of irrational numbers include: 3, 2𝜋, and 0.1221122211112222 … Since irrational numbers never terminate or repeat, the decimal form of an irrational number is always an approximation. Square roots of numbers, except perfect squares, and all solutions involving 𝜋 are approximate values. For example: 3 = 1.73205 … and 2𝜋 = 6.28318 … When solving problems involving 𝜋, the approximation 3.14 is acceptable on the PSSA and SAT. Classifying Numbers Example 1: Classify the following numbers as rational or irrational. A. 2.371732 … B. 0.625 C. 12.56637… Solution: To classify each of the values, determine if they can be written as a fraction. A. 2.371732 … cannot be written as a fraction because the decimal does not repeat or terminate, therefore the number is irrational. B. 0.625 can be written as a fraction and the value does terminate, therefore the number is rational. !"# ! = ! !""" C. 12.56637… cannot be written as a fraction because the decimal does not repeat or terminate, therefore the number is irrational. This number is 4𝜋.
Calculator Tip: You can use the calculator to turn decimals into fractions. On the TI-‐30x, type the decimal value and then press % j to change the decimal to a fraction. On a TI-‐83 calculator, from the Math menu, select Frac to change a decimal to a fraction.
Estimating Irrational Numbers Since the decimal form of an irrational number is an approximate value, we can approximate where the values appear on the number line. Example 1: Place the values 12, 𝜋, and 3.7671921 … on the number line. Solution: First, convert each value to a decimal. Therefore, 12 = 3.464101 …, 𝜋 = 3.14 …, and 3.7671921 is already a decimal. Once in decimal form, estimate the location on the number line.
Exercises A. Find the decimal form of each value to the nearest ten thousandth (3 decimal places) and determine if the following values are rational or irrational. 1. 16 !
2. !
3. 8
5. 324
!
4. !
!!
B. Determine if an exact solution can be found for the following measures.
6. !!
7. y n The area of a rectangle.
11. y n The surface area of a cylinder.
8. y n The area of a circle.
12. y n The area of a rectangle.
9. y n The perimeter of a triangle.
13. y n The volume of a sphere.
10. y n The volume of a cube.
14. y n The hypotenuse of any triangle
C. Answer the following questions about irrational numbers. 15. Can the area of a rectangle ever be irrational? If it is possible, provide an example.
D. Use the number line below to determine where each of the following values would be located.
16. 10 is between _____ and _____.
21. 3 3 is between _____ and _____.
17. 4 + 𝜋 is between _____ and _____.
22. 5 + 7 is between _____ and _____.
18.
! !" !"
is between _____ and _____.
23.
!" !
is between _____ and _____.
19. 2𝜋 is between _____ and _____.
24. 3 5 − 2 2 is between _____ and _____.
20. 45 is between _____ and _____.
25. 6 5 + 2 3 − 4 2 is between _____ and _____.
E. Order the following sets of numbers from least to greatest. 26. 14, 18, 4𝜋
31. 3 15, 2 21, 4 12
27. 22.2321, 625, 5! − 1
32. 125, 34 + 20, 10𝜋
28. (−1)! ,
!"# ! !"
, !
29. 3! , 95, 2𝜋
30. 132,
!
8 , 5 25
33.
! !! !
, 196, 200
34. 25 + 100, 50 + 75, 35 + 90
35. 2! , 272,
!! !
F. Without using a calculator, plot the following values on the number line. 36. 2.2! , 26,
!! !
37. 2𝜋, 38, 52
38. 2.75! , 18,
! !" !
!
39. 1.5! , − 2, !
40. (−1.4)! , − 3, 0.1!
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