Identifying Rational Numbers but Not Integers

Oct 13, 2025 by wbrum.org 135 views

Introduction

In mathematics, rational numbers are those that can be expressed as the ratio of two integers, i.e., in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is non-zero. On the other hand, integers are whole numbers, either positive, negative, or zero, without a fractional part. In this article, we will explore the given list of numbers and identify which ones are rational numbers but not integers.

What are Rational Numbers?

Rational numbers are a subset of real numbers and include all integers, fractions, and decimals that can be expressed as a ratio of two integers. For example, $\frac{1}{2}$ , $2$ , and $-3$ are all rational numbers. Rational numbers can be expressed in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is non-zero.

What are Integers?

Integers are whole numbers, either positive, negative, or zero, without a fractional part. Examples of integers include $1$ , $-2$ , $0$ , and $3$ . Integers are a subset of rational numbers, but not all rational numbers are integers.

Identifying Rational Numbers but Not Integers

Now, let's examine the given list of numbers and identify which ones are rational numbers but not integers.

$\frac{2}{9}$

The number $\frac{2}{9}$ is a rational number because it can be expressed as a ratio of two integers, i.e., $\frac{2}{9}$ . However, it is not an integer because it has a fractional part.

$4$

The number $4$ is an integer because it is a whole number without a fractional part. Therefore, it is not a rational number but not an integer.

$\sqrt{5}$

The number $\sqrt{5}$ is an irrational number because it cannot be expressed as a ratio of two integers. Therefore, it is not a rational number but not an integer.

$-2.01$

The number $-2.01$ is a rational number because it can be expressed as a decimal, which can be converted to a fraction. However, it is not an integer because it has a fractional part.

$\sqrt[4]{22}$

The number $\sqrt[4]{22}$ is an irrational number because it cannot be expressed as a ratio of two integers. Therefore, it is not a rational number but not an integer.

$16.\overline{5}$

The number $16.\overline{5}$ is a rational number because it can be expressed as a repeating decimal, which can be converted to a fraction. However, it is not an integer because it has a fractional part.

$-13$

The number $-13$ is an integer because it is a whole number without a fractional part. Therefore, it is not a rational number but not an integer.

$\sqrt[3]{27}$

The number $\sqrt[3]{27}$ is a rational number because it can be expressed as a ratio of two integers, i.e., $\sqrt[3]{27} = 3$ . However, it is not an integer because it has a fractional part.

$2 \frac{1}{6}$

The number $2 \frac{1}{6}$ is a rational number because it can be expressed as a mixed number, which can be converted to an improper fraction. However, it is not an integer because it has a fractional part.

$8.151551555\ldots$

The number $8.151551555\ldots$ is a rational number because it can be expressed as a repeating decimal, which can be converted to a fraction. However, it is not an integer because it has a fractional part.

$-\sqrt{35}$

The number $-\sqrt{35}$ is an irrational number because it cannot be expressed as a ratio of two integers. Therefore, it is not a rational number but not an integer.

$\sqrt[7]{5}$

The number $\sqrt[7]{5}$ is an irrational number because it cannot be expressed as a ratio of two integers. Therefore, it is not a rational number but not an integer.

$-\frac{8}{7}$

The number $-\frac{8}{7}$ is a rational number because it can be expressed as a ratio of two integers. However, it is not an integer because it has a fractional part.

$0$

The number $0$ is an integer because it is a whole number without a fractional part. Therefore, it is not a rational number but not an integer.

$\sqrt{25}$

The number $\sqrt{25}$ is a rational number because it can be expressed as a ratio of two integers, i.e., $\sqrt{25} = 5$ . However, it is not an integer because it has a fractional part.

Conclusion

In conclusion, the rational numbers but not integers from the given list are:

$\frac{2}{9}$
$-2.01$
$16.\overline{5}$
$\sqrt[3]{27}$
$2 \frac{1}{6}$
$8.151551555\ldots$
$-\frac{8}{7}$
$\sqrt{25}$

Introduction

In our previous article, we explored the concept of rational numbers and identified which numbers from a given list are rational numbers but not integers. In this article, we will provide a Q&A guide to help you better understand the concept of rational numbers and integers.

Q: What is the difference between rational numbers and integers?

A: Rational numbers are those that can be expressed as the ratio of two integers, i.e., in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is non-zero. Integers, on the other hand, are whole numbers, either positive, negative, or zero, without a fractional part.

Q: Can all rational numbers be expressed as a decimal?

A: Yes, all rational numbers can be expressed as a decimal. However, not all decimals are rational numbers. For example, the decimal $0.12345678910111213\ldots$ is not a rational number because it cannot be expressed as a ratio of two integers.

Q: Can all rational numbers be expressed as a fraction?

A: Yes, all rational numbers can be expressed as a fraction. For example, the decimal $0.5$ can be expressed as the fraction $\frac{1}{2}$ .

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a ratio of two integers, i.e., in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is non-zero. An irrational number, on the other hand, is a number that cannot be expressed as a ratio of two integers.

Q: Can all irrational numbers be expressed as a decimal?

A: Yes, all irrational numbers can be expressed as a decimal. However, not all decimals are irrational numbers. For example, the decimal $0.5$ is a rational number because it can be expressed as the fraction $\frac{1}{2}$ .

Q: Can all rational numbers be expressed as a mixed number?

A: Yes, all rational numbers can be expressed as a mixed number. For example, the fraction $\frac{3}{4}$ can be expressed as the mixed number $0.75$ .

Q: Can all rational numbers be expressed as a repeating decimal?

A: Yes, all rational numbers can be expressed as a repeating decimal. For example, the fraction $\frac{1}{3}$ can be expressed as the repeating decimal $0.\overline{3}$ .

Q: Can all rational numbers be expressed as a non-repeating decimal?

A: Yes, all rational numbers can be expressed as a non-repeating decimal. For example, the fraction $\frac{1}{2}$ can be expressed as the non-repeating decimal $0.5$ .

Q: Can all rational numbers be expressed as a negative number?

A: Yes, all rational numbers can be expressed as a negative number. For example, the fraction $-\frac{1}{2}$ is a rational number because it can be expressed as a ratio of two integers.

Q: Can all rational numbers be expressed as a positive number?

A: Yes, all rational numbers can be expressed as a positive number. For example, the fraction $\frac{1}{2}$ is a rational number because it can be expressed as a ratio of two integers.

Conclusion

In conclusion, rational numbers and integers are two distinct concepts in mathematics. Rational numbers are those that can be expressed as a ratio of two integers, while integers are whole numbers, either positive, negative, or zero, without a fractional part. We hope that this Q&A guide has helped you better understand the concept of rational numbers and integers.

Frequently Asked Questions

Q: What is the difference between a rational number and an irrational number?
A: A rational number is a number that can be expressed as a ratio of two integers, while an irrational number is a number that cannot be expressed as a ratio of two integers.
Q: Can all rational numbers be expressed as a decimal?
A: Yes, all rational numbers can be expressed as a decimal.
Q: Can all rational numbers be expressed as a fraction?
A: Yes, all rational numbers can be expressed as a fraction.
Q: Can all irrational numbers be expressed as a decimal?
A: Yes, all irrational numbers can be expressed as a decimal.
Q: Can all rational numbers be expressed as a mixed number?
A: Yes, all rational numbers can be expressed as a mixed number.
Q: Can all rational numbers be expressed as a repeating decimal?
A: Yes, all rational numbers can be expressed as a repeating decimal.
Q: Can all rational numbers be expressed as a non-repeating decimal?
A: Yes, all rational numbers can be expressed as a non-repeating decimal.
Q: Can all rational numbers be expressed as a negative number?
A: Yes, all rational numbers can be expressed as a negative number.
Q: Can all rational numbers be expressed as a positive number?
A: Yes, all rational numbers can be expressed as a positive number.