No other number in the set is a positive integer

Oct 13, 2025 by recoveryshake.com 89 views

In mathematics, numbers can be classified into various sets based on their properties and characteristics. The set of natural numbers, denoted by $\mathbb{N}$ , consists of all positive integers starting from 1 and extending to infinity. In this article, we will explore which numbers from the given set belong to the set of natural numbers.

The Given Set of Numbers

The given set of numbers is:

{-5, -1.8, 0, 0.8888\ldots, 2, \pi, \frac{-2}{3}}

A) Natural Numbers ( $\mathbb{N}$ )

Natural numbers are positive integers starting from 1 and extending to infinity. To determine which numbers from the given set belong to the set of natural numbers, we need to identify the positive integers in the set.

2: This is a positive integer and belongs to the set of natural numbers.
No other number in the set is a positive integer, so the only number that belongs to the set of natural numbers is 2.

B) Which of these numbers belong to the set of integers ( $\mathbb{Z}$ )?

The set of integers, denoted by $\mathbb{Z}$ , consists of all whole numbers, including positive, negative, and zero. To determine which numbers from the given set belong to the set of integers, we need to identify the whole numbers in the set.

-5: This is a negative integer and belongs to the set of integers.
-1.8: This is a negative decimal number and does not belong to the set of integers.
0: This is a whole number and belongs to the set of integers.
0.8888\ldots: This is a non-terminating decimal number and does not belong to the set of integers.
2: This is a positive integer and belongs to the set of integers.
\pi: This is an irrational number and does not belong to the set of integers.
\frac{-2}{3}: This is a negative fraction and belongs to the set of integers.

C) Which of these numbers belong to the set of rational numbers ( $\mathbb{Q}$ )?

The set of rational numbers, denoted by $\mathbb{Q}$ , consists of all numbers that can be expressed as the ratio of two integers. To determine which numbers from the given set belong to the set of rational numbers, we need to identify the numbers that can be expressed as the ratio of two integers.

-5: This can be expressed as the ratio of two integers, $\frac{-5}{1}$ , and belongs to the set of rational numbers.
-1.8: This can be expressed as the ratio of two integers, $\frac{-18}{10}$ , and belongs to the set of rational numbers.
0: This can be expressed as the ratio of two integers, $\frac{0}{1}$ , and belongs to the set of rational numbers.
0.8888\ldots: This can be expressed as the ratio of two integers, $\frac{8}{9}$ , and belongs to the set of rational numbers.
2: This can be expressed as the ratio of two integers, $\frac{2}{1}$ , and belongs to the set of rational numbers.
\pi: This is an irrational number and does not belong to the set of rational numbers.
**\frac-2}{3}** This can be expressed as the ratio of two integers, $\frac{-2{3}$, and belongs to the set of rational numbers.

D) Which of these numbers belong to the set of irrational numbers ( $\mathbb{Q}^c$ )?

The set of irrational numbers, denoted by $\mathbb{Q}^c$ , consists of all numbers that cannot be expressed as the ratio of two integers. To determine which numbers from the given set belong to the set of irrational numbers, we need to identify the numbers that cannot be expressed as the ratio of two integers.

-5: This can be expressed as the ratio of two integers, $\frac{-5}{1}$ , and does not belong to the set of irrational numbers.
-1.8: This can be expressed as the ratio of two integers, $\frac{-18}{10}$ , and does not belong to the set of irrational numbers.
0: This can be expressed as the ratio of two integers, $\frac{0}{1}$ , and does not belong to the set of irrational numbers.
0.8888\ldots: This can be expressed as the ratio of two integers, $\frac{8}{9}$ , and does not belong to the set of irrational numbers.
2: This can be expressed as the ratio of two integers, $\frac{2}{1}$ , and does not belong to the set of irrational numbers.
\pi: This is an irrational number and belongs to the set of irrational numbers.
**\frac-2}{3}** This can be expressed as the ratio of two integers, $\frac{-2{3}$, and does not belong to the set of irrational numbers.

Conclusion

In this article, we will address some of the most frequently asked questions related to the sets of numbers.

Q: What is the difference between the set of natural numbers and the set of integers?

A: The set of natural numbers consists of all positive integers starting from 1 and extending to infinity. The set of integers, on the other hand, consists of all whole numbers, including positive, negative, and zero.

Q: Which set does the number 0 belong to?

A: The number 0 belongs to the set of integers.

Q: Can a number be both rational and irrational?

A: No, a number cannot be both rational and irrational. A number is either rational or irrational, but not both.

Q: What is an example of an irrational number?

A: An example of an irrational number is the number \pi. This number cannot be expressed as the ratio of two integers and has an infinite number of decimal places.

Q: Can a number be a member of multiple sets?

A: Yes, a number can be a member of multiple sets. For example, the number 2 is a member of the set of natural numbers, the set of integers, and the set of rational numbers.

Q: How do I determine which set a number belongs to?

A: To determine which set a number belongs to, you need to understand the properties of each set. For example, if a number is a positive integer, it belongs to the set of natural numbers. If a number is a whole number, it belongs to the set of integers. If a number can be expressed as the ratio of two integers, it belongs to the set of rational numbers.

Q: Can a number be a member of the set of rational numbers if it is not a fraction?

A: Yes, a number can be a member of the set of rational numbers even if it is not a fraction. For example, the number 2 can be expressed as the ratio of two integers, 2/1, and therefore belongs to the set of rational numbers.

Q: What is the difference between the set of rational numbers and the set of real numbers?

A: The set of rational numbers consists of all numbers that can be expressed as the ratio of two integers. The set of real numbers, on the other hand, consists of all numbers, including rational and irrational numbers.

Q: Can a number be a member of the set of real numbers if it is not a rational number?

A: Yes, a number can be a member of the set of real numbers even if it is not a rational number. For example, the number \pi is an irrational number and therefore belongs to the set of real numbers.

Q: How do I determine which set a number belongs to if it is a decimal number?

A: To determine which set a decimal number belongs to, you need to understand the properties of each set. For example, if a decimal number can be expressed as the ratio of two integers, it belongs to the set of rational numbers. If a decimal number is a non-terminating decimal, it may be an irrational number and therefore belongs to the set of real numbers.

Q: Can a number be a member of the set of real numbers if it is not a decimal number?

A: Yes, a number can be a member of the set of real numbers even if it is not a decimal number. For example, the number 2 is a rational number and therefore belongs to the set of real numbers.

Conclusion

In conclusion, understanding the sets of numbers is an essential part of mathematics. By knowing which set a number belongs to, you can better understand the properties and characteristics of that number. We hope that this article has helped you to better understand the sets of numbers and how to determine which set a number belongs to.