Identify The Two Rational Numbers.A. 3 \sqrt{3} 3 ​ B. − 7 3 -\frac{7}{3} − 3 7 ​ C. 2.7182818459 D. 2.777777...

Introduction

Rational numbers are a fundamental concept in mathematics, and understanding them is crucial for various mathematical operations and applications. In this article, we will delve into the world of rational numbers and identify two such numbers from a given set of options.

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 a fraction. They can be either positive or negative and include all integers and fractions. Rational numbers are denoted by the letter 'Q' and are a subset of real numbers.

Properties of Rational Numbers

Rational numbers have several properties that make them unique and useful in mathematics. Some of the key properties include:

• Closure: Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero).
• Commutativity: Rational numbers are commutative under addition and multiplication.
• Associativity: Rational numbers are associative under addition and multiplication.
• Distributivity: Rational numbers are distributive under multiplication over addition.

Examples of Rational Numbers

Some examples of rational numbers include:

• Integers: 3, -5, 0
• Fractions: 3/4, -2/3, 1/2
• Decimals: 0.5, 0.25, 0.75

Identifying Rational Numbers

Now that we have a basic understanding of rational numbers, let's identify two rational numbers from the given set of options.

Option A: $\sqrt{3}$

$\sqrt{3}$ is an irrational number, not a rational number. It cannot be expressed as a ratio of two integers and is a non-terminating, non-repeating decimal.

Option B: $-\frac{7}{3}$

$-\frac{7}{3}$ is a rational number. It can be expressed as a ratio of two integers, -7 and 3, and is a terminating decimal.

Option C: 2.7182818459

2.7182818459 is an irrational number, not a rational number. It is a non-terminating, non-repeating decimal and cannot be expressed as a ratio of two integers.

Option D: 2.777777...

2.777777... is a rational number. It can be expressed as a ratio of two integers, 7/3, and is a repeating decimal.

Conclusion

In conclusion, rational numbers are a fundamental concept in mathematics, and understanding them is crucial for various mathematical operations and applications. We identified two rational numbers from the given set of options, $-\frac{7}{3}$ and 2.777777..., and discussed their properties and examples.

Key Takeaways

• Rational numbers are a set of numbers that can be expressed as the ratio of two integers.
• Rational numbers have several properties, including closure, commutativity, associativity, and distributivity.
• Examples of rational numbers include integers, fractions, and decimals.
• Rational numbers can be identified by checking if they can be expressed as a ratio of two integers.

Further Reading

For further reading on rational numbers, we recommend the following resources:

• Khan Academy: Rational Numbers
• Math Is Fun: Rational Numbers
• Wikipedia: Rational Number

References

• [1] Khan Academy. (n.d.). Rational Numbers. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-rational-numbers
• [2] Math Is Fun. (n.d.). Rational Numbers. Retrieved from https://www.mathisfun.com/rational-numbers.html
• [3] Wikipedia. (n.d.). Rational Number. Retrieved from https://en.wikipedia.org/wiki/Rational_number
  Rational Numbers: Frequently Asked Questions =====================================================

Introduction

Rational numbers are a fundamental concept in mathematics, and understanding them is crucial for various mathematical operations and applications. In this article, we will address some frequently asked questions about rational numbers.

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

A: Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., in the form of a fraction. Irrational numbers, on the other hand, are numbers that cannot be expressed as a ratio of two integers and are non-terminating, non-repeating decimals.

Q: Can all rational numbers be expressed as decimals?

A: Yes, all rational numbers can be expressed as decimals. However, not all decimals are rational numbers. For example, the decimal 0.123456789101112... is an irrational number because it is a non-terminating, non-repeating decimal.

Q: Can all rational numbers be expressed as fractions?

A: Yes, all rational numbers can be expressed as fractions. For example, the decimal 0.5 can be expressed as the fraction 1/2.

Q: What is the difference between terminating and non-terminating decimals?

A: Terminating decimals are decimals that have a finite number of digits after the decimal point. For example, 0.5 is a terminating decimal. Non-terminating decimals, on the other hand, are decimals that have an infinite number of digits after the decimal point. For example, 0.123456789101112... is a non-terminating decimal.

Q: Can all non-terminating decimals be expressed as rational numbers?

A: No, not all non-terminating decimals can be expressed as rational numbers. For example, the decimal 0.123456789101112... is an irrational number because it is a non-terminating, non-repeating decimal.

Q: Can all rational numbers be expressed as integers?

A: No, not all rational numbers can be expressed as integers. For example, the fraction 1/2 is a rational number, but it is not an integer.

Q: Can all integers be expressed as rational numbers?

A: Yes, all integers can be expressed as rational numbers. For example, the integer 5 can be expressed as the fraction 5/1.

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

A: Rational numbers are a subset of real numbers. Real numbers include all rational numbers, as well as irrational numbers.

Q: Can all real numbers be expressed as rational numbers?

A: No, not all real numbers can be expressed as rational numbers. For example, the number pi is a real number, but it is an irrational number and cannot be expressed as a ratio of two integers.

Conclusion

In conclusion, rational numbers are a fundamental concept in mathematics, and understanding them is crucial for various mathematical operations and applications. We addressed some frequently asked questions about rational numbers and provided examples to illustrate the concepts.

Key Takeaways

• Rational numbers are numbers that can be expressed as the ratio of two integers.
• Rational numbers can be expressed as decimals, but not all decimals are rational numbers.
• Rational numbers can be expressed as fractions, but not all fractions are rational numbers.
• Terminating decimals are decimals that have a finite number of digits after the decimal point.
• Non-terminating decimals are decimals that have an infinite number of digits after the decimal point.

Further Reading

For further reading on rational numbers, we recommend the following resources:

• Khan Academy: Rational Numbers
• Math Is Fun: Rational Numbers
• Wikipedia: Rational Number

References

• [1] Khan Academy. (n.d.). Rational Numbers. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-rational-numbers
• [2] Math Is Fun. (n.d.). Rational Numbers. Retrieved from https://www.mathisfun.com/rational-numbers.html
• [3] Wikipedia. (n.d.). Rational Number. Retrieved from https://en.wikipedia.org/wiki/Rational_number