1. Between Which Two Consecutive Integers Does $\sqrt{138}$ Lie?2. Rewrite 0.26 As A Proper Fraction (in The Form Of $\frac{a}{b}$), Showing All Steps.

Mathematical Conundrums: Unraveling the Mysteries of Numbers

Mathematics is a vast and intricate subject that has been a cornerstone of human understanding for centuries. From the simplest arithmetic operations to the most complex mathematical theories, numbers play a crucial role in shaping our understanding of the world around us. In this article, we will delve into two intriguing mathematical problems that will test our skills and challenge our perceptions.

Problem 1: Locating $\sqrt{138}$ between Two Consecutive Integers

The first problem we will tackle is to find the two consecutive integers between which $\sqrt{138}$ lies. To begin, we need to understand that $\sqrt{138}$ is the square root of 138, which is a positive number. We can start by finding the square root of 138 using a calculator or by approximating it using the square root of nearby perfect squares.

Finding the Square Root of 138

To find the square root of 138, we can start by finding the square root of nearby perfect squares. We know that $11^2 = 121$ and $12^2 = 144$. Since 138 lies between these two perfect squares, we can conclude that $\sqrt{138}$ lies between 11 and 12.

Why Does $\sqrt{138}$ Lie Between 11 and 12?

To understand why $\sqrt{138}$ lies between 11 and 12, let's consider the following:

• $11^2 = 121$
• $12^2 = 144$
• $138$ lies between $121$ and $144$

Since $138$ lies between $121$ and $144$, we can conclude that $\sqrt{138}$ lies between $11$ and $12$. This is because the square root of a number is always greater than or equal to the square root of the lower perfect square and less than or equal to the square root of the higher perfect square.

Problem 2: Rewriting 0.26 as a Proper Fraction

The second problem we will tackle is to rewrite 0.26 as a proper fraction in the form of $\frac{a}{b}$. To begin, we need to understand that 0.26 is a decimal number that can be rewritten as a fraction.

Rewriting 0.26 as a Fraction

To rewrite 0.26 as a fraction, we can start by considering it as a sum of two decimal numbers: 0.2 and 0.06.

• $0.2 = \frac{2}{10} = \frac{1}{5}$
• $0.06 = \frac{6}{100} = \frac{3}{50}$

Since $0.26 = 0.2 + 0.06$, we can rewrite it as:

$0.26 = \frac{1}{5} + \frac{3}{50}$

To add these two fractions, we need to find a common denominator, which is 50.

• $\frac{1}{5} = \frac{10}{50}$
• $\frac{3}{50}$

Now we can add the two fractions:

$0.26 = \frac{10}{50} + \frac{3}{50} = \frac{13}{50}$

Why Does 0.26 Equal $\frac{13}{50}$?

To understand why 0.26 equals $\frac{13}{50}$, let's consider the following:

• $0.26 = 0.2 + 0.06$
• $0.2 = \frac{1}{5} = \frac{10}{50}$
• $0.06 = \frac{3}{50}$

Since $0.26 = 0.2 + 0.06$, we can rewrite it as:

$0.26 = \frac{10}{50} + \frac{3}{50} = \frac{13}{50}$

This is because the sum of two decimal numbers is equal to the sum of their corresponding fractions.

Conclusion

In conclusion, we have tackled two intriguing mathematical problems that have tested our skills and challenged our perceptions. The first problem involved finding the two consecutive integers between which $\sqrt{138}$ lies, while the second problem involved rewriting 0.26 as a proper fraction in the form of $\frac{a}{b}$. Through careful analysis and reasoning, we have been able to solve these problems and gain a deeper understanding of the underlying mathematical concepts.

Key Takeaways

• $\sqrt{138}$ lies between 11 and 12.
• 0.26 can be rewritten as a proper fraction in the form of $\frac{a}{b}$.
• The sum of two decimal numbers is equal to the sum of their corresponding fractions.

Final Thoughts

Mathematics is a vast and intricate subject that has been a cornerstone of human understanding for centuries. From the simplest arithmetic operations to the most complex mathematical theories, numbers play a crucial role in shaping our understanding of the world around us. Through careful analysis and reasoning, we can tackle even the most challenging mathematical problems and gain a deeper understanding of the underlying mathematical concepts.
Mathematical Conundrums: Unraveling the Mysteries of Numbers

Q&A: Mathematical Conundrums

In this article, we will continue to explore the world of mathematics and tackle some of the most intriguing questions and problems. From the basics of arithmetic to the complexities of calculus, we will delve into the world of numbers and uncover the secrets that lie within.

Q: What is the square root of 138?

A: The square root of 138 is a positive number that lies between 11 and 12. This is because 138 lies between the perfect squares of 11 and 12, and the square root of a number is always greater than or equal to the square root of the lower perfect square and less than or equal to the square root of the higher perfect square.

Q: How do I rewrite 0.26 as a proper fraction?

A: To rewrite 0.26 as a proper fraction, we can start by considering it as a sum of two decimal numbers: 0.2 and 0.06. We can then rewrite these decimal numbers as fractions and add them together to get the final answer.

Q: Why does 0.26 equal $\frac{13}{50}$?

A: 0.26 equals $\frac{13}{50}$ because we can rewrite 0.26 as a sum of two decimal numbers: 0.2 and 0.06. We can then rewrite these decimal numbers as fractions and add them together to get the final answer.

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

A: A decimal number is a number that is written in the form of a decimal point, such as 0.25 or 0.75. A fraction, on the other hand, is a number that is written in the form of a ratio of two integers, such as $\frac{1}{2}$ or $\frac{3}{4}$.

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

A: To convert a decimal number to a fraction, we can start by considering it as a sum of two decimal numbers. We can then rewrite these decimal numbers as fractions and add them together to get the final answer.

Q: What is the importance of fractions in mathematics?

A: Fractions are an essential part of mathematics, as they allow us to represent and manipulate ratios and proportions. They are used in a wide range of mathematical operations, from addition and subtraction to multiplication and division.

Q: How do I add and subtract fractions?

A: To add and subtract fractions, we need to have a common denominator. We can then add or subtract the numerators while keeping the denominator the same.

Q: What is the difference between a proper fraction and an improper fraction?

A: A proper fraction is a fraction where the numerator is less than the denominator, such as $\frac{1}{2}$ or $\frac{3}{4}$. An improper fraction, on the other hand, is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{5}{2}$ or $\frac{7}{4}$.

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, we can divide the numerator by the denominator and write the remainder as the new numerator.

Q: What is the importance of mixed numbers in mathematics?

A: Mixed numbers are an essential part of mathematics, as they allow us to represent and manipulate ratios and proportions in a more intuitive and visual way. They are used in a wide range of mathematical operations, from addition and subtraction to multiplication and division.

Conclusion

In conclusion, we have tackled some of the most intriguing questions and problems in mathematics, from the basics of arithmetic to the complexities of calculus. Through careful analysis and reasoning, we have been able to uncover the secrets that lie within the world of numbers and gain a deeper understanding of the underlying mathematical concepts.

Key Takeaways

• The square root of 138 lies between 11 and 12.
• 0.26 can be rewritten as a proper fraction in the form of $\frac{a}{b}$.
• Fractions are an essential part of mathematics, as they allow us to represent and manipulate ratios and proportions.
• Mixed numbers are an essential part of mathematics, as they allow us to represent and manipulate ratios and proportions in a more intuitive and visual way.

Final Thoughts

Mathematics is a vast and intricate subject that has been a cornerstone of human understanding for centuries. From the simplest arithmetic operations to the most complex mathematical theories, numbers play a crucial role in shaping our understanding of the world around us. Through careful analysis and reasoning, we can tackle even the most challenging mathematical problems and gain a deeper understanding of the underlying mathematical concepts.