Which Of The Following Are True Statements? Select All That Apply.- All Integers Are Rational Numbers.- Repeating Decimals Are Not Rational Numbers.- 715 − 14 \frac{715}{-14} − 14 715 ​ Is Not A Rational Number.- Terminating Decimals Are Rational Numbers.-

Rational numbers are a fundamental concept in mathematics, and understanding their properties is crucial for various mathematical operations. In this article, we will delve into the world of rational numbers and examine the given statements to determine which ones are true.

What are Rational Numbers?

Rational numbers are a set of numbers 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. This definition encompasses all integers, fractions, and decimals that can be expressed as a ratio of integers.

Statement 1: All integers are rational numbers.

This statement is TRUE. Integers, such as 5, -3, and 0, can be expressed as a ratio of integers, e.g., $\frac{5}{1}$, $\frac{-3}{1}$, and $\frac{0}{1}$, respectively. Therefore, all integers are indeed rational numbers.

Statement 2: Repeating decimals are not rational numbers.

This statement is FALSE. Repeating decimals, such as 0.333..., 0.142857..., and 0.123456..., can be expressed as a ratio of integers. For example, 0.333... can be written as $\frac{1}{3}$, and 0.142857... can be written as $\frac{1}{7}$. Therefore, repeating decimals are indeed rational numbers.

Statement 3: $\frac{715}{-14}$ is not a rational number.

This statement is FALSE. The expression $\frac{715}{-14}$ is a rational number because it can be expressed as a ratio of integers. The negative sign in the denominator does not affect its rationality.

Statement 4: Terminating decimals are rational numbers.

This statement is TRUE. Terminating decimals, such as 0.5, 0.25, and 0.75, can be expressed as a ratio of integers. For example, 0.5 can be written as $\frac{1}{2}$, 0.25 can be written as $\frac{1}{4}$, and 0.75 can be written as $\frac{3}{4}$. Therefore, terminating decimals are indeed rational numbers.

Conclusion

In conclusion, the true statements are:

• All integers are rational numbers.
• Terminating decimals are rational numbers.

The false statements are:

• Repeating decimals are not rational numbers.
• $\frac{715}{-14}$ is not a rational number.

Understanding the properties of rational numbers is essential for various mathematical operations, and recognizing the true and false statements will help you navigate the world of mathematics with confidence.

Additional Examples

To further illustrate the concept of rational numbers, let's consider a few more examples:

• The number $\frac{22}{7}$ is a rational number because it can be expressed as a ratio of integers.
• The number $0.666...$ is a rational number because it can be expressed as a ratio of integers, i.e., $\frac{2}{3}$.
• The number $-0.5$ is a rational number because it can be expressed as a ratio of integers, i.e., $\frac{-1}{2}$.

Real-World Applications

Rational numbers have numerous real-world applications, including:

• Finance: Rational numbers are used to represent interest rates, exchange rates, and other financial ratios.
• Science: Rational numbers are used to represent physical quantities, such as speed, distance, and time.
• Engineering: Rational numbers are used to represent geometric shapes, such as lengths, widths, and heights.

Conclusion

In our previous article, we explored the concept of rational numbers and examined the given statements to determine which ones are true. In this article, we will delve into a Q&A session to further clarify the properties of rational numbers.

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

A: Rational numbers are a set of numbers 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. Irrational numbers, on the other hand, are a set of numbers that cannot be expressed as a ratio of integers.

Q: Can all rational numbers be expressed as decimals?

A: Yes, all rational numbers can be expressed as decimals. In fact, rational numbers can be expressed as either terminating or repeating decimals.

Q: What is the difference between terminating and repeating decimals?

A: Terminating decimals are decimals that have a finite number of digits after the decimal point, such as 0.5, 0.25, and 0.75. Repeating decimals, on the other hand, are decimals that have a repeating pattern of digits after the decimal point, such as 0.333..., 0.142857..., and 0.123456....

Q: Can all repeating decimals be expressed as rational numbers?

A: Yes, all repeating decimals can be expressed as rational numbers. For example, 0.333... can be written as $\frac{1}{3}$, and 0.142857... can be written as $\frac{1}{7}$.

Q: Can all rational numbers be expressed as fractions?

A: Yes, all rational numbers can be expressed as fractions. In fact, rational numbers are often represented as fractions, such as $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is non-zero.

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

A: Rational numbers include all integers, as well as fractions and decimals that can be expressed as a ratio of integers. In other words, all integers are rational numbers, but not all rational numbers are integers.

Q: Can rational numbers be negative?

A: Yes, rational numbers can be negative. For example, the rational number $\frac{-3}{4}$ is a negative rational number.

Q: Can rational numbers be expressed as percentages?

A: Yes, rational numbers can be expressed as percentages. In fact, percentages are a way of expressing rational numbers as a fraction of 100.

Q: What is the importance of rational numbers in real-world applications?

A: Rational numbers have numerous real-world applications, including finance, science, and engineering. They are used to represent interest rates, exchange rates, physical quantities, and geometric shapes.

Q: Can rational numbers be used to solve equations?

A: Yes, rational numbers can be used to solve equations. In fact, rational numbers are often used to solve linear equations, quadratic equations, and other types of equations.

Conclusion

In conclusion, rational numbers are a fundamental concept in mathematics, and understanding their properties is crucial for various mathematical operations. By answering these frequently asked questions, we hope to have provided a better understanding of rational numbers and their applications.

Additional Resources

For further learning, we recommend the following resources:

• Khan Academy: Rational Numbers
• Mathway: Rational Numbers
• Wolfram MathWorld: Rational Numbers

Conclusion

In conclusion, rational numbers are a fundamental concept in mathematics, and understanding their properties is crucial for various mathematical operations. By recognizing the true and false statements and answering these frequently asked questions, you will be better equipped to navigate the world of mathematics with confidence.