1.2 Write Down Two Rational Numbers Between 2 \sqrt{2} 2 ​ And 10 \sqrt{10} 10 ​ . Show Your Working.1.3 Write Down 0.11111111 … 0.11111111 \ldots 0.11111111 … As A Fraction.

Rational Numbers and Decimal Expansions: A Mathematical Exploration

In mathematics, rational numbers and decimal expansions are two fundamental concepts that are often used in various mathematical operations. Rational numbers are those that can be expressed as the ratio of two integers, while decimal expansions represent numbers in a base-10 system. In this article, we will explore two problems related to rational numbers and decimal expansions, and provide step-by-step solutions to each problem.

Problem 1: Rational Numbers between $\sqrt{2}$ and $\sqrt{10}$

Problem Statement

Write down two rational numbers between $\sqrt{2}$ and $\sqrt{10}$.

Solution

To find two rational numbers between $\sqrt{2}$ and $\sqrt{10}$, we can start by finding the square roots of these numbers.

$\sqrt{2} \approx 1.4142$

$\sqrt{10} \approx 3.1623$

Now, we need to find two rational numbers between these two values. We can start by finding the midpoint of these two values.

Midpoint = $\frac{\sqrt{2} + \sqrt{10}}{2} \approx \frac{1.4142 + 3.1623}{2} \approx 2.28825$

The two rational numbers between $\sqrt{2}$ and $\sqrt{10}$ are $2.25$ and $2.5$.

Explanation

We can see that $2.25$ and $2.5$ are both rational numbers, as they can be expressed as the ratio of two integers. For example, $2.25 = \frac{9}{4}$ and $2.5 = \frac{5}{2}$.

Problem 2: Decimal Expansion of $0.11111111 \ldots$

Problem Statement

Write down $0.11111111 \ldots$ as a fraction.

Solution

To write down $0.11111111 \ldots$ as a fraction, we can start by letting $x = 0.11111111 \ldots$. We can then multiply both sides of the equation by $10$ to get:

$10x = 1.11111111 \ldots$

Now, we can subtract the original equation from this new equation to get:

$10x - x = 1.11111111 \ldots - 0.11111111 \ldots$

Simplifying this equation, we get:

$9x = 1$

Now, we can solve for $x$ by dividing both sides of the equation by $9$:

$x = \frac{1}{9}$

Therefore, $0.11111111 \ldots = \frac{1}{9}$.

Explanation

We can see that $0.11111111 \ldots$ is a repeating decimal, which means that it can be expressed as a fraction. In this case, the fraction is $\frac{1}{9}$.

In this article, we have explored two problems related to rational numbers and decimal expansions. We have shown that rational numbers can be used to represent numbers between two given values, and that decimal expansions can be used to represent repeating decimals as fractions. By following the step-by-step solutions provided in this article, readers should be able to understand and apply these concepts in their own mathematical work.

Key Takeaways

• Rational numbers can be used to represent numbers between two given values.
• Decimal expansions can be used to represent repeating decimals as fractions.
• The midpoint of two values can be used to find rational numbers between those values.
• Repeating decimals can be expressed as fractions by using algebraic manipulation.

Further Reading

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

• "Rational Numbers" by Math Open Reference
• "Decimal Expansions" by Khan Academy
• "Algebraic Manipulation" by Wolfram Alpha

By following the resources provided in this article, readers should be able to gain a deeper understanding of rational numbers and decimal expansions, and apply these concepts in their own mathematical work.
Rational Numbers and Decimal Expansions: A Q&A Guide

In our previous article, we explored two problems related to rational numbers and decimal expansions. In this article, we will provide a Q&A guide to help readers understand and apply these concepts in their own mathematical work.

Q: What is a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. For example, 3/4, 22/7, and 1/2 are all rational numbers.

Q: What is a decimal expansion?

A: A decimal expansion is a way of representing a number in a base-10 system. It is a series of digits that follow a decimal point, such as 0.5, 0.25, or 0.11111111...

Q: How do I find rational numbers between two given values?

A: To find rational numbers between two given values, you can start by finding the midpoint of those values. The midpoint is the average of the two values, and it can be used to find rational numbers between them.

Q: How do I express a repeating decimal as a fraction?

A: To express a repeating decimal as a fraction, you can use algebraic manipulation. Let x be the repeating decimal, and multiply both sides of the equation by 10 to get 10x. Then, subtract the original equation from this new equation to get 9x = 1. Finally, solve for x by dividing both sides of the equation by 9.

Q: What is the difference between a rational number and a decimal expansion?

A: A rational number is a number that can be expressed as a fraction, while a decimal expansion is a way of representing a number in a base-10 system. While all rational numbers can be expressed as decimal expansions, not all decimal expansions are rational numbers.

Q: Can you provide an example of a rational number that is not a decimal expansion?

A: Yes, the number 1/2 is a rational number that is not a decimal expansion. While it can be expressed as the decimal 0.5, it is not a repeating decimal and therefore cannot be expressed as a decimal expansion.

Q: Can you provide an example of a decimal expansion that is not a rational number?

A: Yes, the number 0.123456789101112... is a decimal expansion that is not a rational number. This number is an irrational number, which means that it cannot be expressed as a fraction.

Q: How do I determine whether a decimal expansion is rational or irrational?

A: To determine whether a decimal expansion is rational or irrational, you can use the following test: if the decimal expansion repeats after a certain number of digits, it is likely to be a rational number. If the decimal expansion does not repeat, it is likely to be an irrational number.

Q: Can you provide an example of a decimal expansion that is a rational number?

A: Yes, the number 0.333333... is a decimal expansion that is a rational number. This number can be expressed as the fraction 1/3.

In this article, we have provided a Q&A guide to help readers understand and apply the concepts of rational numbers and decimal expansions. We have covered topics such as the definition of rational numbers, decimal expansions, and how to express repeating decimals as fractions. By following the answers provided in this article, readers should be able to gain a deeper understanding of these concepts and apply them in their own mathematical work.

Key Takeaways

• Rational numbers can be expressed as fractions.
• Decimal expansions can be used to represent numbers in a base-10 system.
• Repeating decimals can be expressed as fractions using algebraic manipulation.
• Not all decimal expansions are rational numbers.
• The midpoint of two values can be used to find rational numbers between those values.

Further Reading

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

• "Rational Numbers" by Math Open Reference
• "Decimal Expansions" by Khan Academy
• "Algebraic Manipulation" by Wolfram Alpha

By following the resources provided in this article, readers should be able to gain a deeper understanding of rational numbers and decimal expansions, and apply these concepts in their own mathematical work.