Arranging the Numbers in a List

Oct 13, 2025 by recoveryshake.com 44 views

Introduction

In mathematics, comparing and ordering numbers is a fundamental concept that helps us understand various mathematical operations and relationships. In this article, we will explore how to arrange a list of given numbers in a specific order. The numbers we will be working with are: $-\sqrt{7}$ , $-0.\overline{7}$ , $1.5$ , and $\sqrt{25}$ . We will delve into the world of mathematics to understand the properties of these numbers and arrange them in a list.

Understanding the Numbers

Before we can arrange the numbers in a list, we need to understand their properties. Let's start by analyzing each number individually.

$-\sqrt{7}$

The number $-\sqrt{7}$ is a negative square root of 7. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of 7 is approximately 2.645751311. However, since we are dealing with a negative square root, the value is negative.

$-0.\overline{7}$

The number $-0.\overline{7}$ is a negative repeating decimal. A repeating decimal is a decimal that goes on forever in a repeating pattern. In this case, the decimal 0.777... repeats indefinitely. To convert this decimal to a fraction, we can use the formula for infinite geometric series:

-0.\overline{7} = -\frac{7}{9}

$1.5$

The number $1.5$ is a simple decimal. It can be written as a fraction:

1.5 = \frac{3}{2}

$\sqrt{25}$

The number $\sqrt{25}$ is the square root of 25. Since 25 is a perfect square (5^2 = 25), the square root of 25 is simply 5.

Arranging the Numbers in a List

Now that we have a good understanding of each number, let's arrange them in a list from smallest to largest.

Number	Value
$-\sqrt{7}$	-2.645751311
$-0.\overline{7}$	-0.777...
$1.5$	1.5
$\sqrt{25}$	5

As we can see, the numbers are already in order from smallest to largest. However, let's take a closer look at the values to confirm.

Comparing the Numbers

To confirm that the numbers are in the correct order, let's compare them using mathematical operations.

Comparing $-\sqrt{7}$ and $-0.\overline{7}$

We can compare these two numbers by subtracting one from the other:

-\sqrt{7} - (-0.\overline{7}) = -\sqrt{7} + 0.\overline{7}

Using a calculator, we can approximate the value of $-\sqrt{7}$ as -2.645751311 and the value of $0.\overline{7}$ as -0.777... . Since $-\sqrt{7}$ is greater than $-0.\overline{7}$ , we can conclude that $-\sqrt{7}$ is larger than $-0.\overline{7}$ .

Comparing $1.5$ and $\sqrt{25}$

We can compare these two numbers by subtracting one from the other:

\sqrt{25} - 1.5 = 5 - 1.5 = 3.5

Since $\sqrt{25}$ is greater than $1.5$ , we can conclude that $\sqrt{25}$ is larger than $1.5$ .

Conclusion

In conclusion, the numbers $-\sqrt{7}$ , $-0.\overline{7}$ , $1.5$ , and $\sqrt{25}$ can be arranged in a list from smallest to largest as follows:

Number	Value
$-0.\overline{7}$	-0.777...
$-\sqrt{7}$	-2.645751311
$1.5$	1.5
$\sqrt{25}$	5

Introduction

In our previous article, we explored how to arrange a list of numbers in a specific order. We analyzed the properties of each number and compared them using mathematical operations. In this article, we will answer some frequently asked questions related to arranging numbers in a list.

Q&A

Q: What is the order of the numbers $-\sqrt{7}$ , $-0.\overline{7}$ , $1.5$ , and $\sqrt{25}$ ?

A: The order of the numbers is $-0.\overline{7}$ , $-\sqrt{7}$ , $1.5$ , and $\sqrt{25}$ .

Q: Why is $-0.\overline{7}$ smaller than $-\sqrt{7}$ ?

A: $-0.\overline{7}$ is smaller than $-\sqrt{7}$ because $-0.\overline{7}$ is equal to $-\frac{7}{9}$ , which is approximately -0.777..., while $-\sqrt{7}$ is approximately -2.645751311.

Q: How do you compare $1.5$ and $\sqrt{25}$ ?

A: To compare $1.5$ and $\sqrt{25}$ , we can subtract one from the other. Since $\sqrt{25}$ is greater than $1.5$ , we can conclude that $\sqrt{25}$ is larger than $1.5$ .

Q: What is the value of $-\sqrt{7}$ ?

A: The value of $-\sqrt{7}$ is approximately -2.645751311.

Q: What is the value of $-0.\overline{7}$ ?

A: The value of $-0.\overline{7}$ is approximately -0.777...

Q: What is the value of $1.5$ ?

A: The value of $1.5$ is equal to $\frac{3}{2}$ .

Q: What is the value of $\sqrt{25}$ ?

A: The value of $\sqrt{25}$ is equal to 5.

Common Mistakes

When arranging numbers in a list, it's easy to make mistakes. Here are some common mistakes to avoid:

Not understanding the properties of each number: Make sure you understand the properties of each number before comparing them.
Not comparing numbers using mathematical operations: Comparing numbers using mathematical operations is essential to ensure that the numbers are in the correct order.
Not considering the order of operations: When comparing numbers, make sure to consider the order of operations (e.g., multiplication before addition).

Conclusion

In conclusion, arranging numbers in a list requires a good understanding of the properties of each number and the ability to compare them using mathematical operations. By avoiding common mistakes and following the steps outlined in this article, you can confidently arrange numbers in a list.