Number System: Natural, Whole, Rational & Irrational Numbers

Natural Numbers

1. What is the smallest natural number?
   • The smallest natural number is 1.
2. Which of the following is a natural number: 0, -1, 1, 2.5?
   • 1 is a natural number.
3. How many natural numbers are there between 5 and 15?
   • The natural numbers between 5 and 15 are: 6, 7, 8, 9, 10, 11, 12, 13, 14. So, there are 9 natural numbers.
4. Is 0 a natural number? Justify your answer.
   • No, 0 is not considered a natural number, as natural numbers start from 1.
5. If you add two natural numbers, will the result always be a natural number? Explain with an example.
   • Yes, the sum of two natural numbers will always be a natural number. For example, $3 + 5 = 8$3+5=8, which is a natural number.

Whole Numbers

6. Which of the following is not a whole number: -5, 0, 7, 2.5?
   • -5 and 2.5 are not whole numbers. Whole numbers are non-negative integers.
7. What is the difference between a whole number and a natural number?
   • Whole numbers include 0, whereas natural numbers start from 1. All natural numbers are whole numbers, but 0 is not a natural number.
8. If a whole number is divided by 5, can the result be a whole number? Provide an example.
   • Yes, if the whole number is divisible by 5. For example, $10 \div 5 = 2$10÷5=2, which is a whole number.
9. What is the largest whole number less than 10?
   • The largest whole number less than 10 is 9.
10. How many whole numbers are there between 0 and 10?

• The whole numbers between 0 and 10 are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So, there are 10 whole numbers.

Integers

11. Which of the following is an integer: -2.5, 4, 0, 5.5?

• 4 and 0 are integers. -2.5 and 5.5 are not integers.

12. Is the sum of two negative integers always negative? Give an example.

• Yes, the sum of two negative integers is always negative. For example, $-3 + (-4) = -7$−3+(−4)=−7.

13. What is the absolute value of -7?

• The absolute value of -7 is 7.

14. Arrange the following integers in ascending order: -5, 2, -8, 0.

• -8, -5, 0, 2.

15. Can you divide an integer by 0? Explain why or why not.

• No, division by 0 is undefined because it does not give a meaningful result.

Rational Numbers

16. Which of the following is a rational number: 34\frac{3}{4}43​, 2\sqrt{2}2​, π\piπ, −1.5-1.5−1.5?

• 34\frac{3}{4}43​ and -1.5 are rational numbers. 2\sqrt{2}2​ and π\piπ are irrational.

17. What is the decimal expansion of 78\frac{7}{8}87​?

• The decimal expansion of $\frac{7}{8}$87​ is 0.875.

18. Express 0.75 as a rational number.

• 0.75 can be expressed as 34\frac{3}{4}43​.

19. Can a rational number be negative? Provide an example.

• Yes, a rational number can be negative. For example, – 53\frac{5}{3}35​ is a negative rational number.

20. Which of the following is not a rational number: 50\frac{5}{0}05​, 1.5, -49\frac{4}{9}94​, 0.25?

• 50\frac{5}{0}05​ is not a rational number because division by 0 is undefined.

Irrational Numbers

21. Which of the following numbers is irrational: 3\sqrt{3}3​, 5, 79\frac{7}{9}97​, 16\sqrt{16}16​?

• 3\sqrt{3}3​ is irrational. The others are rational.

22. Prove that π\piπ is an irrational number.

• π\piπ cannot be expressed as a fraction of two integers, and its decimal expansion goes on forever without repeating, making it irrational.

23. If the square root of a number is irrational, what can we say about that number?

• If the square root of a number is irrational, the number is not a perfect square. For example, $\sqrt{2}$2​ is irrational, and 2 is not a perfect square.

24. Is 5\sqrt{5}5​ rational or irrational? Explain your answer.

• 5\sqrt{5}5​ is irrational because it cannot be expressed as a fraction of two integers and its decimal expansion is non-terminating and non-repeating.

25. What is the decimal expansion of 2\sqrt{2}2​? Does it terminate or repeat?

The decimal expansion of $\sqrt{2}$2​ is approximately 1.414213562…, and it does not terminate or repeat. Therefore, it is an irrational number.