Question: User: 1.1Classifying Real Numbers In Exercises 1 and 2, determine which of the numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers. 1. 3 — 5 , −4, 0, √— 2 , 52, − 1 — 8 , √— 9 2. 98, −141, − 7 — 8 , 3.99, − √— 12 , − 54 — 11 Identifying Numbers In Exercises 3 and 4, list all members of the set. 3. The natural numbers between −2.3 and 6.1 4. The even integers between −5.5 and 2.5Plotting Real Numbers In Exercises 5 and 6, plot the numbers on the real number line. 5. (a) 4 (b) −3 (c) 3 — 4 (d) −2.4 6. (a) −9 (b) 7 (c) − 3 — 2 (d) 5.25Ordering Real Numbers In Exercises 710, place the correct inequality symbol (< or >) between the real numbers. 7. −5 3 8. −2 −8 9. − 8 — 5 − 2 — 5 10. 8.4 −3.2Finding the Distance Between Two Numbers In Exercises 1114, find the distance between the real numbers. 11. 11 and −3 12. 4 and −13 13. −13.5 and −6.2 14. −8.4 and −0.3Evaluating an Absolute Value In Exercises 1518, evaluate the expression. 15. −5 16. 6 17. − −7.2 18. −3.6 1.2Evaluating an Expression In Exercises 1938, evaluate the expression. If it is not possible, state the reason. Write all fractions in simplest form. 19. 15 + (−4) 20. −12 + 3 21. −63.5 + 21.7 22. 14.35 − 10.3 23. 4 — 21 + 7 — 21 24. 21 — 16 − 13 — 16 25. − 5 — 6 + 1 26. 3 + 4 — 9 27. 8 3 — 4 − 6 5 — 8 28. −2 9 — 10 + 5 3 — 20 29. −7 ⋅ 4 30. 9 ⋅ (−5) 31. 120(−5)(7) 32. (−16)(−15)(−4) 33. 3 — 8 ⋅ ( − 2 — 15 ) 34. 5 — 21 ⋅ 21 — 5 35. −56 — −4 36. 85 — 0 37. − 7 — 15 ÷ ( − 7 — 30 ) 38. − 2 — 3 ÷ 4 — 15 Evaluating an Exponential Expression In Exercises 3942, evaluate the exponential expression. 39. (−6)4 40. −(−3)4 41. −42 42. 25Using the Order of Operations In Exercises 4346, evaluate the expression. 43. 120 − (52 ⋅ 4) 44. 45 − 45 ÷ 32 45. 8 + 3[62 − 2(7 − 4)] 46. 24 − [10 + 6(1 − 3)2] 47. Total Charge You purchased an entertainment system and made a down payment of $395 plus nine monthly payments of $45 each. What is the total amount you paid for the system? 48. Savings Plan You deposit $80 per month in a savings account for 10 years. The account earns 2% interest compounded monthly. The total amount in the account after 10 years will be 80 [ ( 1 + 0.02 — 12 ) 120 − 1 ] ( 1 + 12 — 0.02 ) . Use a calculator to determine this amount.
1. **Problem:** Classify the numbers in the set $\{ \frac{3}{5}, -4, 0, \sqrt{2}, 52, -\frac{1}{8}, \sqrt{9} \}$ as (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers.
2. **Formula and rules:**
- Natural numbers: positive integers starting from 1.
- Integers: whole numbers including negatives, zero, and positives.
- Rational numbers: numbers expressible as $\frac{p}{q}$ where $p,q$ are integers and $q \neq 0$.
- Irrational numbers: numbers that cannot be expressed as a fraction of integers.
3. **Step-by-step classification:**
- $\frac{3}{5}$: (a) No (not natural), (b) No (not integer), (c) Yes (rational), (d) No.
- $-4$: (a) No, (b) Yes, (c) Yes, (d) No.
- $0$: (a) No, (b) Yes, (c) Yes, (d) No.
- $\sqrt{2}$: (a) No, (b) No, (c) No, (d) Yes (irrational).
- $52$: (a) Yes, (b) Yes, (c) Yes, (d) No.
- $-\frac{1}{8}$: (a) No, (b) Yes, (c) Yes, (d) No.
- $\sqrt{9} = 3$: (a) Yes, (b) Yes, (c) Yes, (d) No.
**Final classification:**
- Natural numbers: $52, 3$
- Integers: $-4, 0, 52, 3, -\frac{1}{8}$ (note $-\frac{1}{8}$ is not integer, correction: exclude it)
- Rational numbers: $\frac{3}{5}, -4, 0, 52, -\frac{1}{8}, 3$
- Irrational numbers: $\sqrt{2}$
Note: $-\frac{1}{8}$ is rational but not integer.
4. **Explanation:**
Natural numbers are positive whole numbers without fractions or negatives.
Integers include zero and negatives but no fractions.
Rational numbers include fractions and integers.
Irrational numbers cannot be expressed as fractions.
This completes the classification for the first problem.