Compare − 2. 3 ‾ -2 . \overline{3} − 2. 3 And − 8 3 -\frac{8}{3} − 3 8 ​ Using Symbols $\ \textless \ $, $\ \textgreater \ $, Or = = = .A. − 2. 3 ‾ \textless − 8 3 -2 . \overline{3} \ \textless \ -\frac{8}{3} − 2. 3 \textless − 3 8 ​ B. $-2 . \overline{3} \

Introduction

In mathematics, decimals and fractions are two fundamental ways to represent numbers. While decimals are often used in everyday applications, fractions provide a more precise and elegant way to express ratios. In this article, we will compare two numbers: $-2 . \overline{3}$ and $-\frac{8}{3}$. We will use the symbols $\ \textless \ $, $\ \textgreater \ $, or $=$ to determine their relative values.

Understanding Repeating Decimals

A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. In the case of $-2 . \overline{3}$, the block of digits "3" repeats indefinitely. To understand the value of this number, we can use the following formula:

$$x = -2 + \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots$$

This is an infinite geometric series with a first term of $\frac{3}{10}$ and a common ratio of $\frac{1}{10}$. Using the formula for the sum of an infinite geometric series, we can calculate the value of $x$:

$$x = -2 + \frac{\frac{3}{10}}{1 - \frac{1}{10}} = -2 + \frac{3}{9} = -2 + \frac{1}{3} = -\frac{5}{3}$$

Therefore, $-2 . \overline{3} = -\frac{5}{3}$.

Understanding Fractions

A fraction is a way to represent a ratio of two numbers. In the case of $-\frac{8}{3}$, the numerator is $-8$ and the denominator is $3$. To understand the value of this number, we can simply divide the numerator by the denominator:

$$-\frac{8}{3} = -2\frac{2}{3}$$

Comparing the Numbers

Now that we have understood the values of both numbers, we can compare them. We will use the symbols $\ \textless \ $, $\ \textgreater \ $, or $=$ to determine their relative values.

A. $-2 . \overline{3} \ \textless \ -\frac{8}{3}$

To determine if $-2 . \overline{3}$ is less than $-\frac{8}{3}$, we can compare their values:

$$-2 . \overline{3} = -\frac{5}{3}$$

$$-\frac{8}{3} = -2\frac{2}{3}$$

Since $-\frac{5}{3}$ is greater than $-2\frac{2}{3}$, we can conclude that $-2 . \overline{3} \ \textless \ -\frac{8}{3}$ is false.

B. $-2 . \overline{3} \ \textgreater \ -\frac{8}{3}$

To determine if $-2 . \overline{3}$ is greater than $-\frac{8}{3}$, we can compare their values:

$$-2 . \overline{3} = -\frac{5}{3}$$

$$-\frac{8}{3} = -2\frac{2}{3}$$

Since $-\frac{5}{3}$ is less than $-2\frac{2}{3}$, we can conclude that $-2 . \overline{3} \ \textgreater \ -\frac{8}{3}$ is false.

C. $-2 . \overline{3} = -\frac{8}{3}$

To determine if $-2 . \overline{3}$ is equal to $-\frac{8}{3}$, we can compare their values:

$$-2 . \overline{3} = -\frac{5}{3}$$

$$-\frac{8}{3} = -2\frac{2}{3}$$

Since $-\frac{5}{3}$ is not equal to $-2\frac{2}{3}$, we can conclude that $-2 . \overline{3} = -\frac{8}{3}$ is false.

Conclusion

In conclusion, we have compared two numbers: $-2 . \overline{3}$ and $-\frac{8}{3}$. We have used the symbols $\ \textless \ $, $\ \textgreater \ $, or $=$ to determine their relative values. Based on our analysis, we can conclude that:

• $-2 . \overline{3} \ \textless \ -\frac{8}{3}$ is false
• $-2 . \overline{3} \ \textgreater \ -\frac{8}{3}$ is false
• $-2 . \overline{3} = -\frac{8}{3}$ is false

Therefore, $-2 . \overline{3}$ and $-\frac{8}{3}$ are not equal, and $-2 . \overline{3}$ is not less than or greater than $-\frac{8}{3}$.

References

• [1] "Repeating Decimals" by Math Open Reference. Retrieved from https://www.mathopenref.com/repeatingdecimals.html
• [2] "Fractions" by Math Is Fun. Retrieved from https://www.mathisfun.com/fractions.html
  Frequently Asked Questions: Comparing Repeating Decimals and Fractions ====================================================================

Q: What is a repeating decimal?

A: A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, $-2 . \overline{3}$ is a repeating decimal where the block of digits "3" repeats indefinitely.

Q: How do I convert a repeating decimal to a fraction?

A: To convert a repeating decimal to a fraction, you can use the following formula:

$$x = a + \frac{b}{10} + \frac{c}{100} + \frac{d}{1000} + \cdots$$

where $a$ is the non-repeating part of the decimal, and $b$, $c$, $d$, etc. are the repeating digits. You can then use the formula for the sum of an infinite geometric series to calculate the value of $x$.

Q: What is the difference between a repeating decimal and a non-repeating decimal?

A: A non-repeating decimal is a decimal number that does not have a block of digits that repeats indefinitely. For example, $-2.5$ is a non-repeating decimal. A repeating decimal, on the other hand, is a decimal number that has a block of digits that repeats indefinitely.

Q: Can I compare two repeating decimals?

A: Yes, you can compare two repeating decimals by converting them to fractions and then comparing the fractions. For example, to compare $-2 . \overline{3}$ and $-\frac{8}{3}$, you can convert the repeating decimal to a fraction and then compare the fractions.

Q: How do I determine if a repeating decimal is less than, greater than, or equal to a fraction?

A: To determine if a repeating decimal is less than, greater than, or equal to a fraction, you can convert the repeating decimal to a fraction and then compare the fractions. If the fraction is less than the other fraction, then the repeating decimal is less than the other fraction. If the fraction is greater than the other fraction, then the repeating decimal is greater than the other fraction. If the fractions are equal, then the repeating decimals are equal.

Q: Can I use a calculator to compare two repeating decimals?

A: Yes, you can use a calculator to compare two repeating decimals. However, you should be aware that some calculators may not be able to handle repeating decimals accurately. It's always a good idea to double-check your calculations by converting the repeating decimals to fractions and then comparing the fractions.

Q: What are some common mistakes to avoid when comparing repeating decimals and fractions?

A: Some common mistakes to avoid when comparing repeating decimals and fractions include:

• Not converting the repeating decimal to a fraction before comparing it to the fraction
• Not using the correct formula to convert the repeating decimal to a fraction
• Not comparing the fractions accurately
• Not double-checking your calculations

Q: How can I practice comparing repeating decimals and fractions?

A: You can practice comparing repeating decimals and fractions by working through examples and exercises. You can also use online resources and calculators to help you practice. Additionally, you can try creating your own examples and exercises to challenge yourself.

Q: What are some real-world applications of comparing repeating decimals and fractions?

A: Comparing repeating decimals and fractions has many real-world applications, including:

• Calculating interest rates and investments
• Determining the cost of goods and services
• Calculating the area and perimeter of shapes
• Determining the volume and surface area of objects
• Calculating the speed and distance of objects

By understanding how to compare repeating decimals and fractions, you can make more accurate calculations and make better decisions in a variety of real-world situations.