How Can I Prove This Rational Number Is Greater Than Or Less Than $\frac{1}{13}$?

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Introduction

In mathematics, comparing the magnitude of two rational numbers can be a challenging task, especially when dealing with complex expressions. In this article, we will explore a method to determine whether a given rational number, represented by the expression $\varphi$, is greater than or less than $\frac{1}{13}$. The expression $\varphi$ is defined as:

$$\varphi := \frac{3 \cdot 7 \cdot 11 \cdot 15 \cdots 599}{5 \cdot 9 \cdot 13 \cdot 17 \cdots 601}$$

Understanding the Problem

To begin, let's analyze the given expression $\varphi$. It is a product of two sequences of numbers, one in the numerator and the other in the denominator. The numerator consists of odd numbers starting from 3 and ending at 599, while the denominator consists of odd numbers starting from 5 and ending at 601. The presence of these sequences makes the expression $\varphi$ a rational number.

Comparing Rational Numbers

When comparing two rational numbers, we can use various methods such as finding a common denominator, comparing the fractions, or using inequalities. In this case, we are interested in determining whether $\varphi$ is greater than or less than $\frac{1}{13}$. To do this, we need to analyze the properties of the expression $\varphi$ and its relationship with $\frac{1}{13}$.

Analyzing the Expression $\varphi$

Let's examine the expression $\varphi$ more closely. We can rewrite it as:

$$\varphi = \frac{3 \cdot 7 \cdot 11 \cdots 599}{5 \cdot 9 \cdot 13 \cdots 601}$$

Notice that the numerator and denominator have a common factor of 3. We can cancel out this factor to simplify the expression:

$$\varphi = \frac{7 \cdot 11 \cdots 599}{5 \cdot 9 \cdot 13 \cdots 601}$$

Using Inequalities

One way to compare $\varphi$ with $\frac{1}{13}$ is to use inequalities. We can start by analyzing the numerator and denominator separately. Let's consider the numerator:

$$7 \cdot 11 \cdots 599$$

This is a product of consecutive odd numbers. We can use the fact that the product of consecutive odd numbers is always greater than or equal to the product of the first and last numbers. In this case, the first number is 7 and the last number is 599. Therefore, we can write:

$$7 \cdot 11 \cdots 599 \geq 7 \cdot 599$$

Similarly, we can analyze the denominator:

$$5 \cdot 9 \cdot 13 \cdots 601$$

This is a product of consecutive odd numbers. We can use the same fact as before to write:

$$5 \cdot 9 \cdot 13 \cdots 601 \leq 5 \cdot 601$$

Comparing the Numerator and Denominator

Now that we have simplified the expression $\varphi$, we can compare the numerator and denominator. We can see that the numerator is greater than or equal to $7 \cdot 599$, while the denominator is less than or equal to $5 \cdot 601$. Therefore, we can write:

$$\varphi = \frac{7 \cdot 11 \cdots 599}{5 \cdot 9 \cdot 13 \cdots 601} \geq \frac{7 \cdot 599}{5 \cdot 601}$$

Using the Inequality

We can now use the inequality to compare $\varphi$ with $\frac{1}{13}$. We can rewrite the inequality as:

$$\varphi \geq \frac{7 \cdot 599}{5 \cdot 601}$$

We can simplify the right-hand side of the inequality by canceling out the common factor of 7:

$$\varphi \geq \frac{599}{5 \cdot 601}$$

Evaluating the Inequality

Now that we have simplified the inequality, we can evaluate it. We can see that the numerator is 599, while the denominator is $5 \cdot 601$. Therefore, we can write:

$$\varphi \geq \frac{599}{5 \cdot 601}$$

We can simplify the right-hand side of the inequality by canceling out the common factor of 5:

$$\varphi \geq \frac{599}{601}$$

Conclusion

In conclusion, we have shown that $\varphi \geq \frac{1}{13}$. This is because the numerator of $\varphi$ is greater than or equal to $7 \cdot 599$, while the denominator of $\varphi$ is less than or equal to $5 \cdot 601$. Therefore, we can conclude that $\varphi$ is greater than or equal to $\frac{1}{13}$.

Final Answer

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Q: What is the expression $\varphi$ and how is it defined?

A: The expression $\varphi$ is defined as:

$$\varphi := \frac{3 \cdot 7 \cdot 11 \cdot 15 \cdots 599}{5 \cdot 9 \cdot 13 \cdot 17 \cdots 601}$$

This is a product of two sequences of numbers, one in the numerator and the other in the denominator.

Q: How can I compare $\varphi$ with $\frac{1}{13}$?

A: To compare $\varphi$ with $\frac{1}{13}$, we can use inequalities. We can start by analyzing the numerator and denominator separately.

Q: What is the numerator of $\varphi$ and how can I simplify it?

A: The numerator of $\varphi$ is:

$$7 \cdot 11 \cdots 599$$

We can simplify this expression by using the fact that the product of consecutive odd numbers is always greater than or equal to the product of the first and last numbers. In this case, the first number is 7 and the last number is 599. Therefore, we can write:

$$7 \cdot 11 \cdots 599 \geq 7 \cdot 599$$

Q: What is the denominator of $\varphi$ and how can I simplify it?

A: The denominator of $\varphi$ is:

$$5 \cdot 9 \cdot 13 \cdots 601$$

We can simplify this expression by using the same fact as before. We can write:

$$5 \cdot 9 \cdot 13 \cdots 601 \leq 5 \cdot 601$$

Q: How can I compare the numerator and denominator of $\varphi$?

A: We can compare the numerator and denominator of $\varphi$ by using the inequalities we derived earlier. We can see that the numerator is greater than or equal to $7 \cdot 599$, while the denominator is less than or equal to $5 \cdot 601$. Therefore, we can write:

$$\varphi = \frac{7 \cdot 11 \cdots 599}{5 \cdot 9 \cdot 13 \cdots 601} \geq \frac{7 \cdot 599}{5 \cdot 601}$$

Q: How can I use the inequality to compare $\varphi$ with $\frac{1}{13}$?

A: We can use the inequality to compare $\varphi$ with $\frac{1}{13}$ by rewriting the inequality as:

$$\varphi \geq \frac{7 \cdot 599}{5 \cdot 601}$$

We can simplify the right-hand side of the inequality by canceling out the common factor of 7:

$$\varphi \geq \frac{599}{5 \cdot 601}$$

Q: What is the final answer?

A: The final answer is that $\varphi \geq \frac{1}{13}$.

Q: What is the significance of this result?

A: This result shows that $\varphi$ is greater than or equal to $\frac{1}{13}$. This is a useful result because it provides a bound on the value of $\varphi$.

Q: How can I apply this result in real-world problems?

A: This result can be applied in real-world problems where we need to compare the magnitude of two rational numbers. For example, in finance, we may need to compare the value of two investments to determine which one is more profitable.

Q: What are some common mistakes to avoid when comparing rational numbers?

A: Some common mistakes to avoid when comparing rational numbers include:

• Not simplifying the expressions before comparing them
• Not using inequalities to compare the expressions
• Not canceling out common factors in the expressions

By avoiding these mistakes, we can ensure that our comparisons are accurate and reliable.